Archive for year: 2023

Kinematics Carolina Distance Learning

Investigation Manual

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Table of Contents

Overview …………………………………………………………………………………………… 3

Objectives …………………………………………………………………………………………. 3

Time Requirements ……………………………………………………………………………. 3

Background ………………………………………………………………………………………. 4

Materials ……………………………………………………………………………………………. 8

Safety ………………………………………………………………………………………………… 9

Alternate Methods for Collecting Data using Digital Devices. ……….. 10

Preparation ……………………………………………………………………………………… 11

Activity 1: Graph and interpret motion data of a moving object ….. 11

Activity 2: Calculate the velocity of a moving object ……………………. 12

Activity 3: Graph the motion of an object traveling under constant

acceleration ……………………………………………………………………………………. 16

Activity 4: Predict the time for a steel sphere to roll down an incline 23

Activity 5: Demonstrate that a sphere rolling down the incline is

moving under constant acceleration …………………………………………….. 26

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Overview

Kinematics is the branch of physics that deals with the analysis of the motion of objects

wihout concern for the forces causing the motion. Scientists have developed

equations that describe the movement of objects within certain parameters, such as

objects moving with a constant velocity or a constant acceleration. Using these

equations, the future position and velocity of an object can be predicted. This

investigation will focus on objects moving with a constant velocity or a constant

acceleration. Data will be collected on these objects, and the motion of the objects

will be analyzed through graphing these data.

Objectives

 Explain linear motion for objects traveling with a constant velocity or constant

acceleration

 Utilize vector quantities such as displacement and acceleration, and scalar

quantities such as distance and speed.

 Analyze graphs that depict the motion of objects moving at a constant velocity

or constant acceleration.

 Use equations of motion to analyze and predict the motion of objects moving at

a constant velocity or constant acceleration.

Time Requirements

Preparation …………………………………………………………………………………5 minutes

Activity 1 …………………………………………………………………………………….15 minutes

Activity 2 …………………………………………………………………………………….20 minutes

Activity 3 …………………………………………………………………………………….20 minutes

Activity 4 …………………………………………………………………………………….10 minutes

Activity 5 …………………………………………………………………………………….20 minutes

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Background

Mechanics is the branch of physics that that studies the motion of objects and the

forces and energies that affect those motions. Classical Mechanics refers to the motion

of objects that are large compared to subatomic particles and slow compared to the

speed of light. The effects of quantum mechanics and relativity are negligible in

classical mechanics. Most objects and forces encountered in daily life can be

described by classical mechanics, such as the motion of a baseball, a train, or even a

bullet or the planets. Engineers and other scientists apply the principles of physics in

many scenarios. Physicists and engineers often collect data about an object and use

graphs of the data to describe the motion of objects.

Kinematics is a specific branch of mechanics that describes the motion of objects

without reference to the forces causing the motion. Examples of kinematics include

describing the motion of a race car moving on a track or an apple falling from a tree,

but only in terms of the object’s position, velocity, acceleration, and time without

describing the force from the engine of the car, the friction between the tires and the

track, or the gravity pulling the apple. For example, it is possible to predict the time it

would take for an object dropped from the roof of a building to fall to the ground using

the following kinematics equation:

𝒔 = 1

2 𝒂 𝑡2

Where s is the displacement from the starting position at a given time, a is the

acceleration of the object, and t is the time after the object is dropped. The equation

does not include any variables for the forces acting on the object or the mass or energy

of the object. As long as the some initial conditions are known, such an object’s

position, acceleration, and velocity at a given time, the motion or position of the object

at any future or previous time can be calculated by applying kinematics. This method

has many useful applications. One could calculate the path of a projectile such as a

golf ball or artillery shell, the time or distance for a decelerating object to come to rest,

or the speed an object would be traveling after falling a given distance.

Early scientists such as Galileo Galilee (1564-1642), Isaac Newton (1642-1746) and

Johannes Kepler (1571-1630) studied the motion of objects and developed

mathematical laws to describe and predict their motion. Until the late sixteenth

century, the idea that heavier objects fell faster than lighter objects was widely

accepted. This idea had been proposed by the Greek philosopher Aristotle, who lived

around the third century B.C. Because the idea seemed to be supported by

experience, it was generally accepted. A person watching a feather and a hammer

dropped simultaneously from the same height would certainly observe the hammer

falling faster than the feather. According to legend, Galileo Galilee, an Italian physicist

and mathematician, disproved this idea in a dramatic demonstration by dropping

objects of different mass from the tower of Pisa to demonstrate that they fell at the

same rate. In later experiments, Galileo rolled spheres down inclined planes to slow

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down the motion and get more accurate data. By analyzing the ordinary motion of

objects and graphing the results, it is possible to derive some simple equations that

predict their motion.

To study the motion of objects, a few definitions should be established. A vector refers

to a number with a direction and magnitude (or size). Numbers that have a magnitude

but not a direction are referred to as a scalar. In kinematics, vectors are important,

because the goal is to calculate the location and direction of movement of the object

at any time in the future or past. For example if an object is described as being 100

miles from a given position traveling at a speed of 50 miles per hour, that could mean

the object will reach the position in 2 hours. It could also mean the object could be

located up to 100 miles farther away in 1 hour, or somewhere between 100 and 200

miles away depending on the direction. The quantity speed, which refers to the rate of

change in position of an object, is a scalar quantity because no direction of travel is

defined. The quantity velocity, which refers to both the speed and direction of an

object, is a vector quantity.

Distance, or the amount of space between two objects, is a scalar quantity.

Displacement, which is distance in a given direction, is a vector quantity. If a bus

travels from Washington D.C. to New York City, the distance the bus traveled is

approximately 230 miles. The displacement of the bus is (roughly) 230 miles North-East.

If the bus travels from D.C to New York and back, the distance traveled is roughly 460

miles, but the displacement is zero because the bus begins and ends at the same point.

It is important to define the units of scalar and vector quantities when studying

mechanics. A person giving directions from Washington D.C. to New York might

describe the distance as being approximately 4 hours. This may be close to the actual

travel time, but this does not indicate actual distance.

To illustrate the difference between distance and displacement, consider the following

diagrams in Figures 1-3.

Consider the number line in Figure 1. The displacement from zero represented by the

arrowhead on the number line is -3, indicating both direction and magnitude. The

distance from zero indicated by the point on the number line equals three, which is the

magnitude of the displacement. For motion in one dimension, the + or‒ sign is sufficient

to represent the direction of the vector.

Figure 1.

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Figure 2 Figure 3

The arrows in Figures 2 and 3 represent displacement vectors for an object. The long

lines represent a displacement with a magnitude of five. This displacement vector can

be resolved into two component vectors along the x and y axes. In all four diagrams

the object is moved some distance in either the positive or negative x direction, and

then some distance in the positive y direction; however, the final position of the object

is different in each diagram. The total distance between the object’s initial and final

position in each instance is 5 meters, however to describe the displacement, s, from the

initial position more information is needed.

In Figure 2, the displacement vector can be given by 5 meters (m) at 53.1°. This vector

is found by vector addition of the two component vectors, 3 m at 0° and 4 m at 90°, using conventional polar coordinates that assign 0° to the positive x direction and

progress counterclockwise towards 360°. The displacement in Figure 3 is 5 m at 143.1°.

In each case the magnitude of the vector is length of the arrow, that is, the distance

that the object travels. Most texts will indicate that a variable represents a vector

quantity by placing an arrow over the variable or placing the variable in bold.

To indicate the magnitude of a vector, absolute value bars are used. For example the

magnitude of the displacement vector in each diagram is 5 m. In Figure 2 the

displacement is given by:

s = 5 m at 51.3°

The magnitude of this vector may be written as:

| 𝒔 | = d = 5 m

The displacement vector in Figure 2, s = 5 m at 53.1°, can be resolved into the

component vectors 3 m at 0° and 4 m at 90°.

Two more terms that are critical for the study of kinematics are velocity and

acceleration. Both terms are vector quantities.

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Velocity (v) is defined as the rate of change of the position of an object. For an object

moving in the x direction, the magnitude of the velocity (speed) may be described as:

𝒗 = 𝑥2 − 𝑥1

∆𝑡

Where x2 is the position at time t2 and x1 is the position of the object at time t1. The

variable ∆t represents the time interval t2 -t1. The symbol, ∆, is the Greek symbol delta,

and refers to a change or difference. ∆t is read, “delta t”. Time in the following

examples is provided in seconds (s). Please be sure that you do not confuse the “s” unit

for seconds, and the “s “ unit for displacement in these formulas.

For example if an object is located at a position designated x1 = 2 m and moves to

position x2 = 8 m over a time interval ∆t = 2 s, then the average speed could be

calculated: 8𝑚 − 2𝑚

2s = 3𝑚/s

The velocity could for this object could be indicated as:

𝒗 = 3 𝑚/s

Because velocity is a vector quantity, the positive sign indicates that the object was

traveling in the positive x direction, at a speed of 3 m/s.

Acceleration is defined as the rate of change of velocity. The magnitude of

acceleration may be described as:

𝒂 = 𝒗𝟐 − 𝒗1

∆𝑡

For example, an object with an initial velocity v1 = 10 m/s slows to a final velocity of v2 =

1 m/s over an interval of 3 s.

1 𝑚

s⁄ − 10 𝑚 𝑠⁄

3s = −3 𝑚

s s⁄⁄

The object has an average acceleration of ‒3 meters per second per second, which

can also be written as ‒3 meters per second squared, or ‒3 𝑚 s2⁄ .

Because only the initial and final positions or velocities over a given time interval are

used in these equations, the calculated values indicate the average velocity or

acceleration. Calculating the instantaneous velocity or acceleration of an object

requires the application of calculus. Only average velocity and acceleration are

considered in this investigation.

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Materials

Included in the Central Materials kit:

Tape Measure

Rubber Bands

Protractor

Included in the Mechanics Module materials kit

Constant Velocity Vehicle

Steel Sphere

Acrylic Sphere

Angle Bar

Foam Board

Block of Clay

Needed, but not supplied:

Scientific or Graphing Calculator

or Computer with Spreadsheet Software

Permanent Marker

Masking Tape

Stopwatch, or smartphone able to record

video

Reorder Information: Replacement supplies for the Kinematics investigation can be

ordered from Carolina Biological Supply Company, kit 580404 Mechanics Module.

Call 1-800-334-5551 to order.

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Safety

Safety goggles should be worn while conducting this investigation.

Read all the instructions for this laboratory activity before beginning. Follow the

instructions closely and observe established laboratory safety practices.

Do not eat, drink, or chew gum while performing this activity. Wash your hands with

soap and water before and after performing the activity. Clean up the work area

with soap and water after completing the investigation. Keep pets and children

away from lab materials and equipment.

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Alternate Methods for Collecting Data using Digital Devices.

Much of the uncertainty in these experiments arises from human error in measuring the

times of events. Some of the time intervals are very short, which increases the effect of

human error due to reaction time.

Observing the experiment from a good vantage point that removes parallax errors and

recording measurements for multiple trials helps to minimize error, but using a digital

device as an alternate method of data collection may further minimize error.

Many digital devices, smart phones, tablets, etc. have cameras and software that

allow the user to pause or slow down the video.

If you film the experiment against a scale, such as a tape measure, you can use your

video playback program to record position and time data for the carts. This can

provide more accurate data and may eliminate the need for multiple trials.

If the time on your device’s playback program is not sufficiently accurate, some

additional apps may be available for download.

Another option is to upload the video to your computer. Different video playback

programs may come with your operating system or software suite or may be available

for download.

Some apps for mobile devices and computer programs available for download are

listed below, with notes about their features.

Hudl Technique: http://get.hudl.com/products/technique/

 iPhone/iPad and Android

 FREE

 Measures times to the hundredth-second with slow motion features

QuickTime http://www.apple.com/quicktime/download/

 Free

 Install on computer

 30 frames per second

 Has auto scrubbing capability

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Preparation

1. Collect materials needed for this investigation.

2. Locate and clear an area of level floor space in order to conduct the constant

velocity experiment. The space should be free of obstruction and three to four

meters long with a surface which will allow the vehicle to maintain traction but not

impede the vehicle.

Activity 1: Graph and interpret motion data of a moving object

One way to analyze the motion of an object is to graph the position and time data.

The graph of an object’s motion can be interpreted and used to predict the object’s

position at a future time or calculate an object’s position at a previous time.

Table 1 represents the position of a train on a track. The train can only move in one

dimension, either forward (the positive x direction) or in reverse (the negative x

direction).

Table 1

Time (x-axis), seconds Position (y-axis), meters

0 0

5 20

10 40

15 50

20 55

30 60

35 70

40 70

45 70

50 55

1. Plot the data from Table 1 on a graph using the y-axis to represent the displacement

from the starting position (y = 0) and the time coordinate on the x-axis.

2. Connect all the coordinates on the graph with straight lines.

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Activity 2: Calculate the velocity of a moving object

In this activity you will graph the motion of an object moving with a constant velocity.

The speed of the object can be calculated by allowing the Constant Velocity Vehicle

to travel a given distance and measuring the time that it took to move this distance. As

seen in Activity 1, this measurement will only provide the average speed. In this activity,

you will collect time data at several travel distances, plot these data, and analyze the

graph

1. Find and clear a straight path approximately two meters long.

2. Install the batteries and test the vehicle.

3. Use your tape measure or ruler to measure a track two meters long. The track should

be level and smooth with no obstructions. Make sure the surface of the track

provides enough traction for the wheels to turn without slipping.

Place masking tape across the track at 25 cm intervals.

4. Set the car on the floor approximately 5 cm behind the start point of the track.

5. Set the stopwatch to the timing mode and reset the time to zero.

6. Start the car and allow the car to move along the track.

7. Start the stopwatch when the front edge of the car crosses the start point.

8. Stop the stopwatch when the front edge of the car crosses the first 25 cm point.

9. Recover the car, and switch the power off. Record the time and vehicle position on

the data table.

10. Repeat steps #5‒9 for each 25 cm interval marked. Each trial will have a distance

that is 25 cm longer than the previous trial, and the stopwatch will record the time

for the car to travel the individual trial distance.

11. Record the data in Data Table 1.

Note: The vehicle should be able to travel two meters in a generally straight path. If

the vehicle veers significantly to one side, you may need to allow the vehicle to

travel next to a wall. The friction will affect the vehicle’s speed, but the effect will be

uniform for each trial.

Note: Starting the car a short distance before the start point allows the vehicle to

reach its top speed before the time starts and prevents the short period of

acceleration from affecting the data.

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Data Table 1

Time (s) Displacement (m)

0.00

0.25

0.50

0.75

1.00

1.25

1.50

1.75

2.00

12. Graph the time and displacement data points on graph paper.

13. Draw a line of best fit through the data points.

14. Calculate the slope of the line.

15. Make a second data table, indicating the velocity of the car at any time.

Data Table 2

Time (s) Velocity (m/s)

1

2

3

4

5

6

7

8

Note: The points should generally fall in a straight line. If you have access to a

graphing calculator or a computer with spreadsheet software, the calculator or

spreadsheet can be programmed to draw the line of best fit, or trend line.

Note: Based on the equation of a line that cross the y-axis at y = 0, the slope of the

line, m, will be the velocity of the object. 𝑦 = 𝑚𝑥 𝑑 = 𝑣∆𝑡

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16. Graph the data points from the Data Table 2 on a second sheet of graph paper.

Label the y-axis Velocity and the x-axis Time.

17. Draw a vertical line from the x-axis at the point time = 2 seconds so that it intersects

the line representing the velocity of the car.

18. Draw a second vertical line from the x-axis at the point time = 4 seconds so that it

intersects the line representing the velocity of the car.

19. Calculate the area represented by the rectangle enclosed by the two vertical lines

you just drew, the line for the velocity of the car, and the x-axis. An example is shown

as the blue shaded area in Figure 4.

Figure 4

Note: Because the object in this example, the battery-powered car, moves with a

constant speed, all the values for the velocity of the car in the second table should

be the same. The value of the velocity for the car should be the slope of the line in

the previous graph.

Note: When the data points from this table are plotted on the second graph, the

motion of the car should generate a horizontal line. On a velocity vs. time graph, an

object moving with a constant speed is represented by a horizontal line.

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Note: In order to calculate the area of this rectangle, you must multiply the value

for the time interval between time t=2 s and time t=4 s, by the velocity of the car.

This area represents the distance traveled by the object during this time interval.

This technique is often referred to as calculating the “area under the curve”. The

graph of velocity vs. time for an object that is traveling with a constant

acceleration will not be a horizontal line, but using the same method of graphing

the velocity vs. time and finding the “area under the curve” in a given time

interval can allow the distance traveled by the object to be calculated.

Distance = velocity × time

In this equation, the time units (s) cancel out when velocity and time are

multiplied, leaving the distance unit in meters.

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Activity 3: Graph the motion of an object traveling under constant

acceleration

Collecting data on freefalling objects requires accurate timing instruments or access to

a building with heights of several meters where objects can safely be dropped over

heights large enough to allow accurate measurement with a stopwatch. To collect

usable data, in this activity you will record the time objects to roll down an incline. This

reduces acceleration to make it easier to record accurate data on the distance that

an object moves.

1. Collect the following materials:

Steel Sphere

Acrylic Sphere

Angle Bar

Clay

Tape Measure

Timing Device

Protractor

2. Use the permanent marker and the tape measure to mark the inside of the angle

bar at 1-cm increments.

3. Use the piece of clay and the protractor to set up the angle bar at an incline

between 5° to 10°. Use the clay to set the higher end of the anglebar and to

stabilize the system. (Figure 5)

Figure 5

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Set up the angle bar so that the lower end terminates against a book or a wall, to stop

the motion of the sphere (Figure 6.)

Figure 6

4. Place the steel sphere 10 cm from the lower end of the track.

5. Release the steel sphere and record the time it takes for the sphere to reach the

end of the track.

6. Repeat steps #4‒5 two more times for a total of three measurements at a starting

point of 10 cm.

7. Repeat steps #4‒6, increasing the distance between the starting point and the end

of the track by 10 cm each time.

8. Record your data in Data Table 3.

Note: You are recording the time it takes for the sphere to accelerate over an

increasing distance. Take three measurements for each distance, and average the

time for that distance. Record the time for each attempt and the average time in

Table 4.

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Data Table 3

Time (s) Average time (s) Average Time 2 (s2) Distance (m)

Trial 1 =

0.1 Trial 2 =

Trial 3 =

Trial 1 = 0.2

Trial 2 =

Trial 3 =

Trial 1 =

0.3 Trial 2 =

Trial 3 =

Trial 1 =

0.4 Trial 2 =

Trial 3 =

Trial 1 =

0.5 Trial 2 =

Trial 3 =

Trial 1 =

0.6 Trial 2 =

Trial 3 =

Trial 1 =

0.7 Trial 2 =

Trial 3 =

Trial 1 =

0.8 Trial 2 =

Trial 3 =

9. Calculate the average time for each distance and record this value in Table 4.

10. Create a graph of distance vs. time using the data from Table 4.

11. Complete Table 4 by calculating the square of the average time for each distance.

12. Create a graph of displacement vs. time squared from the data in Table 4.

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When data points generate a parabola, it means the y value is proportional to the

square of the x value, or:

𝒚 ∝ 𝑥2

That means the equation for a line that fits all the data points looks like:

𝑦 = 𝐴𝑥2 + 𝐵𝑥 + 𝐶.

In our experiment, the y-axis is displacement and the x-axis is time-; therefore

displacement is proportional to the time squared:

𝒔 ∝ 𝑡2

So, we can exchange y in the equation with displacement (s), to give a formula that

looks like:

𝒔 = 𝐴𝑡2 + 𝐵𝑡 + 𝐶.

We would know the displacement s, at any time t. We just need to find the

constants, A, B, and C.

The equation that describes the displacement of an object moving

with a constant acceleration is one of the kinematics equations:

𝒔 = 1

2 𝒂∆𝑡2 + 𝒗𝟏∆𝑡

The following section describes how to find this equation using the same method of

finding the “area under the curve” covered in Activity 2.

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Finding an Equation for the Motion of an Object with Constant Acceleration

The general form of a line is: 𝑦 = 𝑚𝑥 + 𝑏

Whre m is the slope of the line, and b is the y-intercept, the point where the line

crosses the y-axis. Because the first data point represents time zero and

displacement zero, the y-intercept is zero and the equation for the line simplifies

to:

y = mx

The data collected in Activity 3 showed that:

𝒔 ∝ 𝑡2

This means that the displacement for the object that rolls down an inclined plane

is can be represented mathematically as:

𝒔 = 𝑘𝑡2 + c

Wher k is an unknown constant representing the slope of the line, and c is an

unknown constant representing the y-intercept.

The displacement of the sphere as it rolls down the incline can be calculated

using this equation, if the constants k and c can be found.

Further experimentation indicates that the constant k for an object in freefall is

one-half the acceleration. If the object is released from rest, the constant c will

be zero.

So for an object that is released from rest, falling under the constant

acceleration due to gravity, the displacement from the point of release is given

by:

𝒔 = 1

2 𝒂𝑡2

Where s is the displacement, t is the time of freefall, and 𝒂 is the acceleration.

For objects in freefall near Earth’s Surface the acceleration due to gravity has a

value of 9.8 𝑚 s2⁄ .

Another way to derive this equation, and find the values for k and c, is to

consider the velocity vs. time graph for an object moving with a constant

acceleration. Remember the velocity vs. time graph for the object moving with

constant velocity from Activity 2. If velocity is constant, the equation of that

graph would be: 𝒗 = 𝑘

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Where v represents the velocity, plotted on the y-axis, and k is the constant

value of the velocity. Plotted against time on the x-axis, this graph is a horizontal

line, as depicted in Figure 7.

Figure 7

By definition, the shaded area is the distance traveled by the object during the

time interval: Δ𝑡 = 𝑡2 − 𝑡1

𝒗 = 𝒅𝒊𝒔𝒑𝒍𝒂𝒄𝒆𝒎𝒆𝒏𝒕

𝑡𝑖𝑚𝑒 =

𝒔

∆𝑡

∴ 𝒔 = 𝒗∆𝑡

If an object has a constant acceleration, then by definition:

𝒂 = 𝒗𝟐 − 𝒗1

∆𝑡

Or : 𝒗2 = 𝒂∆𝑡 + 𝒗𝟏

This is equation is in the general form of a line y = mx + b, with velocity on the y-

axis and time on the x-axis. The graph of this equation would look like the graph

in Figure 8.

Figure 8

Similar to how the shaded area A1 in Figure 7 represents the distance traveled by

the object during the time interval Δt = t2 – t1, the shaded area A2 combined with

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A1 equals the distance traveled by the object undergoing constant

acceleration

The area A1 can be given by:

𝐴1 = 𝒗𝟏∆𝑡

The area A2 can be given by:

𝐴2 = 1

2 (𝒗2 − 𝒗𝟏)∆𝑡

Because this is the area of the triangle, where the length of the base is Δt and the

height of the triangle is (𝒗𝟐 − 𝒗𝟏),

Adding these two expressions and rearranging:

𝒔 = 1

2 (𝒗2 − 𝒗𝟏)∆𝑡

And substituting: 𝒗2 = 𝒂∆𝑡 + 𝒗𝟏

Gives this equation:

𝒔 = 1

2 (𝒂Δ𝑡 + 𝑣1 + 𝑣2Δ𝑡 + 𝑣1Δ𝑡)

Simplifying gives:

𝒔 = 1

2 𝒂∆𝑡2 + 𝒗𝟏∆𝑡

This equation gives the theoretical displacement for an object undergoing a

constant acceleration, 𝒂, at any time t, where s is the displacement during the

time interval, Δ𝑡, and v1 is the initial velocity.

If the object is released from rest, as in our experiment, v1 = 0 and the equation

simplifies to:

𝒔 = 1

2 𝒂∆𝑡2

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Activity 4: Predict the time for a steel sphere to roll down an incline

In this activity you will use the kinematics equation:

𝒔 = 1

2 𝒂∆𝑡2

This will allow you to predict how long the sphere will take to roll down the

inclined track.

First you must solve the previous equation for time:

𝑡 = √ 2𝒔

𝒂

If the object in our experiment was in freefall you would just need to substitute

the distance it was falling for s and substitute the acceleration due to Earth’s

gravity for 𝒂, which is

g = 9.8 m/s2

In this experiment, however the object is not undergoing freefall, it is rolling down

an incline.

The acceleration of an object sliding, without friction down an incline is given by:

𝒂 = gSINθ

Where θ is the angle between the horizontal plane (the surface of your table)

and the inclined plane (the track), and g is the acceleration due to Earth’s

gravity.

When a solid sphere is rolling down an incline the acceleration is given by:

𝒂 = 0.71 gSINθ

The SIN (trigonometric sine) of an angle can be found by measuring the angle

with a protractor and using the SIN function on your calculator or simply by

dividing the length of the side opposite the angle (the height from which the

sphere starts) by the length of the hypotenuse of the right triangle (the length of

the track). Figure 9 shows the formula for deriving sines from triangles

Note: Read the following section carefully.

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Figure 9

sin 𝜃 = 𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒

ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒

Activity 4: Procedure

1. Set up the angle bar as a track. Measure the length of the track and the angle of

elevation between the track and the table.

2. Rearrange the kinematics equation to solve for time (second equation), and

substitute the value 0.71 g SINθ for 𝒂 (third equation). Use a distance of 80 cm for s.

𝒔 = 1

2 𝒂∆𝑡2

𝑡 = √ 2𝒔

𝒂

𝑡 = √ 2𝒔

(0.71𝐠 SINθ)

3. Release the steel sphere from the start point at the elevated end of the track and

measure the time it takes for the sphere to roll from position s = 0 to a final position s

= 80 cm.

4. Compare the measured value with the value predicted in Step 2. Calculate the

percent difference between these two numbers.

25

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5. Repeat Activity 4 with the acrylic sphere. What effect does the mass of the sphere

have on the acceleration of the object due to gravity?

26

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Activity 5: Demonstrate that a sphere rolling down the incline is

moving under constant acceleration

1. Collect the piece of foam board. Use a ruler and a pencil to draw lines across the

short dimension (width) of the board at 5 cm increments.

2. Collect rubber bands from the central materials kit. Wrap the rubber bands around

the width of the foam board so that the rubber bands line up with the pencil marks

you made at the 5 cm intervals. See Figure 10, left panel.

3. Use a book to prop up the foam board as an inclined plane at an angle from 5° to

10° from the horizontal.

4. Place the steel sphere at the top of the ramp and allow the sphere to roll down the

foam board.

5. Remove the rubber bands from the foam board.

6. On the reverse side of the foam board, use a pencil to mark a line across the short

dimension of the board 2 cm from the end. Label this line zero. Mark lines at the

distances listed in Table 5. Each measurement should be made from the zero line.

(see Figure 10).

Note: The sound as the steel sphere crosses the rubber bands will increase in

frequency as the steel sphere rolls down the ramp, indicating that the sphere is

accelerating. As the sphere continues to roll down the incline, it takes less time to

travel the same distance.

If the steel sphere is moving under a constant acceleration, then the displacement

of the sphere from the initial position, if the sphere is released from rest, is given by:

𝒔 = 1

2 𝒂∆𝑡2

The displacement at each time t should be proportional to 𝑡2

27

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Table 2

Displacement (cm)

1

4

9

16

25

36

49

64

81

7. Place rubber bands on the foam board, covering the pencil lines you just made.

8. Set the foam board up at the same angle as the previous trial.

9. Roll the steel sphere down the foam board.

Note: The sounds made as the sphere crosses the rubber bands on the foam board

in the second trial should be at equal intervals. The sphere is traveling a greater

distance each time it crosses a rubber band, but the time interval remains constant

meaning the sphere is moving with a constant acceleration.

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©2015 Carolina Biological Supply Company

Figure 10

Note: For more information on the Trigonometry, Kinematics Equations, and

Rotational Motion exercises, visit the Carolina Biological Supply website at the

following links:

Basic Right Triangle Trigonometry

Derivation of the Kinematics Equations

The Ring and Disc Demonstration

29

©2015 Carolina Biological Supply Company

Physics 226 Fall 2013

Problem Set #1

NOTE: Show ALL work and ALL answers on a piece of separate loose leaf paper, not on this sheet.

Due on Thursday, August 29th

1) Skid and Mitch are pushing on a sofa in opposite

directions with forces of 530 N and 370 N respectively. The mass of the sofa is 48 kg. The sofa is initially at rest before it accelerates. There is no friction acting on the sofa. (a) Calculate the acceleration of the sofa. (b) What velocity does the sofa have after it moves 2.5 m? (c) How long does it take to travel 2.5 m?

2) You have three force

vectors acting on a mass at the origin. Use the component method we covered in lecture to find the magnitude and direction of the re- sultant force acting on the mass.

3) You have three force

vectors acting on a mass at the origin. Use the component method we covered in lecture to find the magnitude and direction of the re- sultant force acting on the mass.

4) A bowling ball rolls off of a table that is 1.5 m tall. The

ball lands 2.5 m from the base of the table. At what speed did the ball leave the table?

5) Skid throws his guitar up

into the air with a velocity of 45 m/s. Calculate the maximum height that the guitar reaches from the point at which Skid lets go of the guitar. Use energy methods.

6) A beam of mass 12 kg and length 2 m is attached to a

hinge on the left. A box of 80 N is hung from the beam 50 cm from the left end. You hold the beam horizontally with your obviously powerful index finger. With what force do you push up on the beam?

Mitch Sofa Skid

7) The tennis ball of mass 57 g which

you have hung in your garage that lets you know where to stop your car so you don’t crush your garbage cans is entertaining you by swinging in a vertical circle of radius 75 cm. At the bottom of its swing it has a speed of 4 m/s. What is the tension in the string at this point?

y

8) Derivatives:

a) Given: y = (4x + L)(2×2 – L), find dx dy

.

b) Given:   

  

 

 Lx2 Lx2lny , find

dx dy

.

9) Integrals:

a) Given:   

o

o

45

45 d

r cosk

, evaluate.

b) Given:    R

0 2322 dr

xr

kxr2 , evaluate.

ANSWERS:

1) a) 3.33 m/s2 b) 4.08 m/s c) 1.23 s 2) 48.0 N, 61.0º N of W 3) 27.4 N, 16.1º S of E 4) 4.52 m/s 5) 103.3 m 6) 78.8 N

7) 1.78N 8) a) 24×2 + 4xL – 4L

b) 22 x4L L4



9) a) r k2 

b)  





 





 

22 xR

x1k2

F2 = 90 N

F1 = 40 N 35

45 x

F3 = 60 N

y

F1 = 45 N 60

F2 = 65 N

50 x

70

F3 = 85 N

Guitar

Skid

Physics 226 Fall 2013

Problem Set #2

1) A plastic rod has a charge of –2.0 C. How many

electrons must be removed so that the charge on the rod becomes +3.0C?

–

+

+

+

2)

Three identical metal spheres, A, B, and C initially have net charges as shown. The “q” is just any arbitrary amount of charge. Spheres A and B are now touched together and then separated. Sphere C is then touched to sphere A and separated from it. Lastly, sphere C is touched to sphere B and then separated from it. (a) How much charge ends up on sphere C? What is the total charge on the three spheres (b) before they are allowed to touch each other and (c) after they have touched? (d) Explain the relevance of the answers to (b) and (c).

3)

Skid of 40 kg and Mitch of 60 kg are standing on ice on opposite sides of an infinite black pit. They are each carrying neutral massless spheres while standing 8 m apart. Suppose that 3.0 x 1015 electrons are removed from one sphere and placed on the other. (a) Calculate the magnitude of the electrostatic force on each sphere. Are the forces the same or different? Explain. (b) Calculate the magnitude of the accelerations for Skid and Mitch at the moment they are 8 m apart. Are they the same or different? Explain. (c) As Skid and Mitch move closer together do their accelerations increase, decrease, or remain the same? Explain.

4) An electron travels in a circular orbit around a stationary

proton (i.e. a hydrogen atom). In order to move in a circle there needs to be a centripetal force acting on the electron. This centripetal force is due to the electrostatic force between the electron and the proton. The electron has a kinetic energy of 2.18 x 10–18 J. (a) What is the speed of the electron? (b) What is the radius of orbit of the electron?

5)

Three charges are arranged as shown. From the left to the right the values of the charges are 6 μC, – 1.5 μC, and – 2 μC. Calculate the magnitude and direction of the net electrostatic force on the charge on the far left.

6) For the same charge distribution of Problem #5, calculate

the magnitude and direction of the net electrostatic force on the charge on the far right.

7)

Two charged spheres are connected to a spring as shown. The unstretched length of the spring is 14 cm. (a) With Qa = 6 μC and Qb = – 7 μC, the spring compresses to an equilibrium length of 10 cm. Calculate the spring constant. (b) Qb is now replaced with a different charge Qc. The spring now has an equilibrium length of 20 cm. What is the magnitude of the charge Qc? (c) What is the sign of Qc? How do you know this?

8) The two charges above are fixed and cannot move. Find

the location in between the charges that you could put a proton so that the proton would have a net force of zero.

9) Three charges are fixed to an xy coordinate system.

A charge of –12 C is on the y axis at y = +3.0 m. A charge of +18 C is at the origin. Lastly, a charge of + 45 C is on the x axis at x = +3.0 m. Calculate the magnitude and direction of the net electrostatic force on the charge x = +3.0 m.

10) Four charges are situated

at the corners of a square each side of length 18 cm. The charges have the same magnitude of q = 4 μC but different signs. See diagram. Find the magnitude and direction of the net force on lower right charge.

+5q – 1q Neutral

C B A

Skid Mitch

Infinite Black Pit

– –

3 cm 2 cm

+

– + Qa Qb

+

8 cm

+ 4 μC 12 μC

11) For the same charge distribution of problem #10, find the magnitude and direction of the net force on upper right charge.

20

12)

All the charges above are multiples of “q” where q = 1μC. The horizontal and vertical distances between the charges are 15 cm. Find the magnitude and direction of the net electric force on the center charge.

13) Use the same charge distribution as in problem #12 but change all even-multiple charges to the opposite sign. Find the magnitude and direction of the net electric force on center charge.

14) Two small metallic spheres, each

of mass 0.30 g, are suspended by light strings from a common point as shown. The spheres are given the same electric charge and it is found that the two come to equilibrium when the two strings have an angle of 20 between them. If each string is 20.0 cm long, what is the magnitude of the charge on each sphere?

– 4q +9q +4q

+3q +3q +8q

15)

+6q R2 – 4q 12 cm m A meter stick of 15 kg is suspended by a string at the

60 cm location. A mass, m, is hung at the 80 cm mark. A massless charged sphere of + 4 μC is attached to the meter stick at the left end. Below this charge is another charge that is fixed 12 cm from the other when the meter stick is horizontal. It has a charge of – 4 μC. Calculate the mass, m, so that the meter stick remains horizontal.

ANSWERS:

7) a) 945 N/m b) 4.2 x 10–5 C 8) 2.93 cm 9) 0.648 N, 17.2º 10) 4.06 N, 45º 11) 6.66 N, 64.5º 12) 19.69 N, 80.1º 13) 18.5 N, 23.4º 14) 1.67 x 10–8 C 15) 10.56 kg

1) 3.1 x 1013 e–

2) a) +1.5q b) +4q c) +4q 3) a) FE, Skid = 32.4 N b) aSkid = 0.81 m/s2 4) a) 2.19 x 106 m/s b) 5.27 x 10–11 m 5) FE = 133.2 N, → 6) FE = 24.3 N, →

Physics 226 Fall 2013

Problem Set #3 1) A charge of –1.5 C is placed on the x axis at

x = +0.55 m, while a charge of +3.5 C is placed at the origin. (a) Calculate the magnitude and direction of the net electric field on the x-axis at x = +0.8 m. (b) Determine the magnitude and direction of the force that would act on a charge of –7.0 C if it was placed on the x axis at x = +0.8 m.

2) For the same charge distribution of problem #1, do the

following. (a) Calculate the magnitude and direction of the net electric field on the x-axis at x = +0.4 m. (b) Determine the magnitude and direction of the force that would act on a charge of –7.0 C if it was placed on the x axis at x = +0.4 m.

3)

Charges are placed at the three corners of a rectangle as shown. The charge values are q1 = 6 nC, q2 = – 4 nC, and q3 = 2.5 nC. Calculate the magnitude and direction of the electric field at the fourth corner.

4) For the same charge distribution of problem #3, with the

exception that you change both q1 and q2 to the opposite sign, calculate the magnitude and direction of the electric field at the fourth corner.

5) A drop of oil has a mass of 7.5 x 10–8 kg and a charge of

– 4.8 nC. The drop is floating close the to Earth’s surface because it is in an electric field. (a) Calculate the magnitude and direction of the electric field. (b) If the sign of the charge is changed to positive, then what is the acceleration of the oil drop? (c) If the oil drop starts from rest, then calculate the speed of the oil drop after it has traveled 25 cm.

6) A proton accelerates from rest in a uniform electric field

of magnitude 700 N/C. At a later time, its speed is 1.8 x 106 m/s. (a) Calculate the acceleration of the proton. (b) How much time is needed for the proton to reach this speed? (c) How far has the proton traveled during this time? (d) What is the proton’s kinetic energy at this time?

7) All the charges above are multiples of “q” where

q = 1μC. The horizontal and vertical distances between the charges are 25 cm. Find the magnitude and direction of the net electric field at point P.

8) Use the same charge distribution as in problem #7 but

change all even-multiple charges to the opposite sign. Find the magnitude and direction of the net electric field at point P.

9) In the above two diagrams, M & S, an electron is given an

initial velocity, vo, of 7.3 x 106 m/s in an electric field of 50 N/C. Ignore gravitation effects. (a) In diagram M, how far does the electron travel before it stops? (b) In diagram S, how far does the electron move vertically after it has traveled 6 cm horizontally? (Hint: Think projectile motion)

– +

+ P

q3 q2

q1

35 cm

20 cm

– 8q

– 4q

+9q

+9q

– 5q

+6q +6q

+2q

P

– – vo vo

M S

10) A 2 g plastic sphere is suspended by a 25 cm long piece of string. Do not ignore gravity. The sphere is hanging in a uniform electric field of magnitude 1100 N/C. See diagram. If the sphere is in equilibrium when the string makes a 20 angle with the vertical, what is the magnitude and sign of the net charge on the sphere?

11) You have an electric dipole of

opposite charges q and distance 2a apart. (a) Find an equation in terms of q, a, and y for the magnitude of the total electric field for an electric dipole at any distance y away from it. (b) Find an equation in terms of q, a, and y for the magnitude of the total electric field for an electric dipole at a distance y away from it for when y >> a.

12)

A dipole has an electric dipole moment of magnitude 4 μC·m. Another charge, 2q, is located a distance, d, away from the center of the dipole. In the diagram all variables of q = 20 μC and d = 80 cm. Calculate the net force on the 2q charge.

13) An electric dipole of charge 30 μC and separation 60 mm is put in a uniform electric field of strength 4 x 106 N/C. What is the magnitude of the torque on the dipole in a uniform field when (a) the dipole is parallel to the field, (b) the dipole is perpendicular to the field, and (c) the dipole makes an angle of 30º to the field. 20º

14) An electron of charge, – e, and mass, m, and a positron of charge, e, and mass, m, are in orbit around each other. They are a distance, d, apart. The center of their orbit is halfway between them. (a) Name the force that is acting as the centripetal force making them move in a circle. (b) Calculate the speed, v, of each charge in terms of e, m, k (Coulomb’s Constant), and d.

15) A ball of mass, m, and positive charge, q, is dropped from

rest in a uniform electric field, E, that points downward. If the ball falls through a height, h, and has a velocity of

gh2v  , find its mass in terms of q, g, and E.

16) The two charges above are fixed and cannot move. Find a

point in space where the total electric field will equal zero.

ANSWERS:

1) a) 1.67 x 105 N/C, WEST

b) 1.17 N, EAST 2) a) 7.97 x 105 N/C, EAST b) 5.6 N, EAST 3) 516 N/C, 61.3º 4) 717 N/C, 69.8º 5) a) 153.1 N/C, SOUTH b) 19.6 m/s2

c) 3.13 m/s 6) a) 6.71 x 1010 m/s2

b) 2.68 x 10–5 s c) 24.1 m d) 2.71 x 10–15 J 7) 1.23 x 106 N/C, 80.5º 8) 3.06 x 105 N/C, 48.4º

9) a) 3.04 m b) 0.297 mm 10) 6.49 x 10–6 C

11) a)  222 ay kqay4



b) 3y kqa4

12) 5.81 N 13) a) 0 b) 7.2 N·m c) 3.6 N·m

14) md2 kev 

15) g

Eq m 

16) 8.2 cm

–

+ y q

a

a

–q +

6 cm

– – 4 μC 12 μC

d

– + – q q 2q

+

Physics 226 Fall 2013

Problem Set #4

NOTE: Any answers of zero must have some kind of justification. 1) You have a thin straight wire of

charge and a solid sphere of charge. The amount of charge on each object is 8 mC and it is uniformly spread over each object. The length of the wire and the diameter of the sphere are both 13 cm. (a) Find the amount of charge on 3.5 cm of the wire. (b) For the sphere, how much charge is located within a radius of 3.5 cm from its center?

2) A uniform line of charge with density, λ, and length, L

is positioned so that its center is at the origin. See diagram above. (a) Determine an equation (using integration) for the magnitude of the total electric field at point P a distance, d, away from the origin. (b) Calculate the magnitude and direction of the electric field at P if d = 2 m, L = 1 m, and λ = 5 μC/m. (c) Show that if d >> L then you get an equation for the E-field that is equivalent to what you would get for a point charge. (We did this kind of thing in lecture.)

3)

A uniform line of charge with charge, Q, and length, L, is positioned so that its center is at the left end of the line. See diagram above. (a) Determine an equation (using integration) for the magnitude of the x-component of the total electric field at point P a distance, d, above the left end of the line. (b) Calculate the magnitude and direction of the x-component of the total electric field at point P if d = 1.5 m, L = 2.5 m, and Q = – 8 μC. (c) What happens to your equation from part (a) if d >> L? Conceptually explain why this is true.

4)

13 cm

You have a semi-infinite line of charge with a uniform linear density 8 μC/m. (a) Calculate the magnitude of the total electric field a distance of 7 cm above the left end of line. (You can use modified results from lecture and this homework if you like … no integration necessary.) (b) At what angle will this total E-field act? (c) Explain why this angle doesn’t change as you move far away from the wire. Can you wrap your brain around why this would be so?

d

5)

A uniform line of charge with charge, Q, and length, D, is positioned so that its center is directly below point P which is a distance, d, above. See diagram above. (a) Determine the magnitude of the x-component of the total electric field at point P. You must explain your answer or show calculations. (b) Calculate the magnitude and direction of the y-component of the total electric field at P if d = 2 m, D = 4.5 m, and Q = –12 μC. HINT: You can use integration to do this OR you can use one of the results (equations) we got in lecture and adapt it to this problem.

6) You have an infinite line of charge of constant linear

density, λ. (a) Determine an equation for the magnitude of the total electric field at point P a distance, d, away from the origin. Use any method you wish (except Gauss’ Law) to determine the equation. There’s at least three different ways you could approach this. You can use the diagram in #5 where D →  if you want a visual. (b) Calculate the electric field at d = 4 cm with λ = 3 μC/m.

P + + + + + +

0 2 L

2 L

P

0

d

– – – – – – – L

P

0

7 cm

 + + + + +

P

d

– – – – – – – D

7)

You have three lines of charge each with a length of 50 cm. The uniform charge densities are shown. The horizontal distance between the left plate and right ones is 120 cm. Find the magnitude and direction of the TOTAL E-field at P which is in the middle of the left plate and the right ones.

8) For the same charge distribution of problem #7, with the

exception that you change the sign of the 4 μC plate and you change the distance between the plates to 160 cm, find the magnitude and direction of the TOTAL E-field at P which is in the middle of the left plate and the right ones.

9) You have 3 arcs of charge, two ¼ arcs and one ½ arc.

The arcs form of circle of radius 5 cm. The uniform linear densities are shown in the diagram. (a) Using an integral and showing your work, determine the equation for the electric field at point P due to the ½ arc. (b) Calculate the magnitude and direction of the total electric field at point P.

10) For this problem use the same charge distribution as

problem #9, with the exception of changing all even charges to the opposite sign. (a) Using an integral and showing your work, determine the equation for the electric field at point P due to the ½ arc. (b) Calculate the magnitude and direction of the total electric field at point P.

11) You have two thin discs both

of diameter 26 cm. They also have the same magnitude surface charge density of, 20 μC/m2, but opposite sign. The charge is uniformly distributed on the discs. The discs are parallel to each

other and are separated by a distance of 30 cm. (a) Calculate the magnitude and direction of the total electric field at a point halfway between the discs along their central axes. (b) Calculate the magnitude and direction of the total electric field at a point halfway between the discs along their central axes if the diameter of the discs goes to infinity. (c) Determine the total electric field at a point halfway between the discs along their central axes if discs have charge of the same sign.

– 5 μC/m

+ + + +

+ +

– – –

3 μC/m

4 μC/m

P

12) You have two concentric thin rings of

charge. The outer ring has a dia- meter of 50 cm with a uniformly spread charge of – 15 μC. The inner ring has a diameter of 22 cm with a uniform linear charge density of 15 μC/m. Calculate the magnitude and direction of the total E-field at point P which lies 40 cm away from the rings along their central axes.

P

13) A proton is released from rest 5 cm away from an infinite

disc with uniform surface charge density of 0.4 pC/m2. (a) What is the acceleration of the proton once it’s released? (b) Calculate the kinetic energy of the proton after 2.5 s. [See Conversion Sheet for metric prefixes.]

2 μC/m 14) In the above two diagrams, G & L, an electron is given

an initial velocity, vo, of 7.3 x 106 m/s above infinite discs with uniform surface charge density of –0.15 fC/m2. (a) In diagram G, how much time passes before the electron stops? (b) In diagram L, how far does the electron move horizontally after it has traveled 20 m vertically? (Hint: Think projectile motion)

15) Two thin infinite planes

of surface charge density 6 nC/cm2 intersect at 45º to each other. See the diagram in which the planes are coming out of the page (edge on view). Point P lies 15 cm from each plane. Calculate the magnitude and direction of the total electric field at P.

–

+ + +

– – +

– – 2 μC/m 5 μC/m

+

+

+

P – –

L G

vo vo

P

45º

P – +

ANSWERS:

1) a) 2.15 mC b) 1.25 mC

2) a) 22 Ld4 Lk4 

 b) 1.2 x 104 N/C,

EAST

3) a)  





 





 

22x Ld

d 1

dL Qk

E

b) 9322 N/C, EAST c) 0 4) a) 1.46 x 106 N/C b) 45º c) Because Ex = Ey 5) a) 0 b) 1.79 x 104 N/C, SOUTH 6) 1.35 x 106 N/C, NORTH

7) 5.93 x 104 N/C, 13.6º 8) 2.37 x 104 N/C, 59.8º

9) a) R k2Ey 



b) 4.85 x 105 N/C, 22.0º

10) a) R k2E y 



b) 2.05 x 106 N/C, 74.8º 11) a) 5.53 x 105 N/C, WEST b) 2.26 x 106 N/C, WEST c) 0 12) 1.01 x 105 N/C, WEST 13) a) 2.17 x 106 m/s2 b) 2.45 x 10–14 J 14) a) 4.9 s b) 3780 m 15) 2.6 x 106 N/C, 22.5º

Physics 226 Fall 2013

Problem Set #5

NOTE: Any answers of zero must have some kind of justification. 1)

A uniform electric field of strength 300 N/C at an angle of 30º with respect to the x-axis goes through a cube of sides 5 cm. (a) Calculate the flux through each cube face: Front, Back, Left, Right, Top, and Bottom. (b) Calculate the net flux through the entire surface. (c) An electron is placed centered 10 cm from the left surface. What is the net flux through the entire surface? Explain your answer.

2)

A right circular cone of height 25 cm and radius 10 cm is enclosing an electron, centered 12 cm up from the base. See Figure G. (a) Using integration and showing all work, find the net flux through the cone’s surface. The electron is now centered in the base of the cone. See Figure L. (b) Calculate the net flux through the surface of the cone.

3) Using the cube in #1, you place a 4μC charge directly in the center of the cube. What is the flux through the top face? (Hint: Consider that this problem would be MUCH more difficult if the charge was not centered in the cube.)

4) Using the cube in #1, you place a 4μC charge at the lower,

left, front corner. What is the net flux through the cube? (Hint: Think symmetry.)

5) You have a thin spherical shell

of radius 10 cm with a uni- form surface charge density of – 42 μC/m2. Centered inside the sphere is a point charge of 4 μC. Find the magnitude and direction of the total electric field at: (a) r = 6 cm and (b) r = 12 cm.

6) You have a solid sphere of radius 6 cm and uniform volume charge density of – 6 mC/m3. Enclosing this is a thin spherical shell of radius 10 cm with a total charge of 7 μC that is uniformly spread over the surface. (a) What is the discontinuity of the E-field at the surface of the shell. (b) What is the discontinuity of the E-field at the surface of the solid sphere? Also, find the magnitude and direction of the total electric field at: (c) r = 4 cm, (d) r = 8 cm, and (e) r = 13 cm.

x 30º

y

7) Use the same set-up in #6 with the following exceptions:

The solid sphere has a total charge of 5 μC and the shell has uniform surface charge density of 60 μC/m2. Answer the same questions in #6, (a) – (e).

8) You have a thin infinite

cylindrical shell of radius 8 cm and a uniform surface charge density of – 12 μC/m2. Inside the shell is an infinite wire with a linear charge density of 15 μC/m. The wire is running along the central axis of the cylinder. (a) What is the discontinuity of the E-field at the surface of the shell? Also, find the magnitude and direction of the total electric field at: (b) r = 4 cm, and (c) r = 13 cm.

9) You have a thin infinite

cylindrical shell of radius 15 cm and a uniform surface charge density of 10 μC/m2. Inside the shell is an infinite solid cylinder of radius 5 cm with a volume charge density of 95 μC/m3. The solid cylinder is running along the central axis of the cylindrical shell. (a) What is the discontinuity of the E-field at the surface of the shell? (b) What is the discontinuity of the E-field at the surface of the solid cylinder. Also, find the magnitude and direction of the total electric field at: (c) r = 4 cm, (d) r = 11 cm, and (e) r = 20 cm.

–

–

G L

+

10) You have a thick spherical shell of outer diameter 20 cm and inner diameter 12 cm. The shell has a total charge of – 28 μC spread uniformly throughout the object. Find the magnitude and direction of the total electric field at: (a) r = 6 cm, (b) r = 15 cm, and (c) r = 24 cm.

11) You have a thick cylindrical shell

of outer diameter 20 cm and inner diameter 12 cm. The shell has a uniform volume charge density of 180 μC/m3. Find the magnitude and direction of the total electric field at: (a) r = 6 cm, (b) r = 15 cm, and (c) r = 24 cm.

12)

You have an thin infinite sheet of charge with surface charge density of 8 μC/m2. Parallel to this you have a slab of charge that is 3 cm thick and has a volume charge density of – 40 μC/m3. Find that magnitude and direction of the total electric field at: (a) point A which is 2.5 cm to the left of the sheet, (b) point B which is 4.5 cm to the right of the sheet, and (c) point C which is 1 cm to the left of the right edge of the slab.

13)

You have an infinite slab of charge that is 5 cm thick and has a volume charge density of 700 μC/m3. 10 cm to the right of this is a point charge of – 6 μC. Find that magnitude and direction of the total electric field at: (a) point A which is 2.5 cm to the left of the right edge of the slab, (b) point B which is 6 cm to the right of the slab, and (c) point C which is 4 cm to the right of the point charge.

14) You have two infinite sheets of charge with equal surface charge magnitudes of 11 μC/m2 but opposite signs. Find the magnitude and direction of the total electric field, (a) to the right of the sheets, (b) in between the sheets, and (c) to the left of the sheets.

15)

A hydrogen molecule (diatomic hydrogen) can be modeled incredibly accurately by placing two protons (each with charge +e) inside a spherical volume charge density which represents the “electron cloud” around the nuclei. Assume the “cloud” has a radius, R, and a net charge of –2e (one electron from each hydrogen atom) and is uniformly spread throughout the volume. Assume that the two protons are equidistant from the center of the sphere a distance, d. Calculate, d, so that the protons each have a net force of zero. The result is darn close to the real thing. [This is actually a lot easier than you think. Start with a Free-Body Diagram on one proton and then do F = ma.]

ANSWERS:

NOTE: Units for 1 – 4

are CmN 2 1) a) 0 for F/B,  0.375 for L/R,  0.65 for T/B

b) & c) 0 2) a) – 1.81 x 10–8

b) – 9.05 x 10–9 3) 7.54 x 104 4) 5.66 x 104 5) a) 9.99 x 106 N/C, OUTWARD [O] b) 7.99 x 105 N/C

INWARD [I] 6) a) 6.29 x 106 N/C

b) 0 c) 9.04 x 106 N/C, I d) 7.63 x 106 N/C, I e) 8.36 x 105 N/C, O 7) a) 6.78 x 106 N/C

b) 0 c) 4.99 x 105 N/C, O d) 7.03 x 106 N/C, O e) 6.67 x 106 N/C, O

8) a) 1.36 x 106 N/C b) 6.74 x 106 N/C, O c) 1.24 x 106 N/C, O 9) a) 1.13 x 106 N/C b) 0 c) 2.15 x 105 N/C, O d) 1.22 x 105 N/C, O e) 9.15 x 105 N/C, O 10) a) 0 b) 2.94 x 106 N/C, I c) 4.37 x 106 N/C, I 11) a) 0 b) 5.49 x 105 N/C, O c) 1.09 x 106 N/C, O 12) a) 3.84 x 105 N/C, L b) 5.20 x 105 N/C, R c) 4.30 x 105 N/C, R 13) a) 3.84 x 105 N/C, R b) 3.57 x 107 N/C, R c) 3.18 x 105 N/C, L 14) a) 0 b) 1.24 x 106 N/C, R c) 0 15) 0.794R

10 cm

A B C

10 cm

A B –

C

R

+ +

d d

Physics 226 Fall 2013

Problem Set #6

NOTE: Any answers of zero must have some kind of justification. 1) You have a cylindrical metal shell of

inner radius 6 cm and outer radius 9 cm. The shell has no net charge. Inside the shell is a line of charge of linear density of – 7 μC/m. Find the magnitude and direction of the electric field at (a) r = 3 cm, (b) r = 7 cm, and (c) r = 13 cm. Also, calculate the surface charge density of the shell on (d) the inner surface and (e) the outer surface.

2) You have a uniformly charged

sphere of radius 5 cm and volume charge density of – 7 mC/m3. It is surrounded by a metal spherical shell with inner radius of 10 cm and outer radius of 15 cm. The shell has a net charge 8 μC. (a) Calculate the total charge on the sphere. Find the magnitude and direction of the electric field at (b) r = 13 cm and (c) r = 18 cm. Also, calculate the surface charge density of the shell on (d) the inner surface and (e) the outer surface.

3) Two 2 cm thick infinite slabs of metal are positioned as

shown in the diagram. Slab B has no net charge but Slab A has an excess charge of 5 μC for each square meter. The infinite plane at the origin has a surface charge density of – 8 μC/m2. Find the magnitude and direction of the electric field at (a) x = 2 cm, and (b) x = 4 cm. Also, calculate the surface charge density on (c) the left edge of A, (d) the right edge of A, and (e) the left edge of B.

4) A positive charge of 16 nC is nailed down with a #6 brad.

Point M is located 7 mm away from the charge and point G is 18 mm away. (a) Calculate the electric potential at Point M. (b) If you put a proton at point M, what electric potential energy does it have? (c) You release the

proton from rest and it moves to Point G. Through what potential difference does it move? (d) Determine the velocity of the proton at point G.

5)

All the charges above are multiples of “q” where

q = 1μC. The horizontal and vertical distances between the charges are 25 cm. Find the magnitude of the net electric potential at point P.

6) Use the same charge distribution as in problem #5 but

change all odd-multiple charges to the opposite sign. Find the magnitude of the net electric potential at point P.

7) A parallel plate setup has a distance

between the plates of 5 cm. An electron is place very near the negative plate and released from rest. By the time it reaches the positive plate it has a velocity of 8 x 106 m/s. (a) As the electron moves between the plates what is the net work done on the charge? (b) What is the potential difference that the electron moves through? (c) What is the magnitude and direction of the electric field in between the plates?

3 cm 5 cm 8 cm 0 10 cm

A B

– 8q

– 4q

+9q

+9q

– 5q

+6q +6q

+2q

P

+ M

G

–

8)

A uniform line of charge with density, λ, and length, L is positioned so that its left end is at the origin. See diagram above. (a) Determine an equation (using integration) for the magnitude of the total electric potential at point P a distance, d, away from the origin. (b) Calculate the magnitude of the electric potential at P if d = 2 m, L = 1 m, and λ = – 5 μC/m. c) Using the equation you derived in part a), calculate the equation for the electric field at point P. It should agree with the result we got in Lecture Example #19.

9) You have a thin spherical shell

of radius 10 cm with a uni- form surface charge density of 11 μC/m2. Centered inside the sphere is a point charge of – 4 μC. Using integration, find the magnitude of the total electric potential at: (a) r = 16 cm and (b) r = 7 cm.

10) You have a uniformly

charged sphere of radius 5 cm and volume charge density of 6 mC/m3. It is surrounded by a metal spherical shell with inner radius of 10 cm and outer radius of 15 cm. The shell has no net charge. Find the magnitude of the electric potential at (a) r = 20 cm, (b) r = 12 cm, and (c) r = 8 cm.

11) Use the same physical situation with the exception

of changing the inner sphere to a solid metal with a surface charge density of 9 μC/m2 and giving the shell a net charge of – 3 μC. Find magnitude of the electric potential at (a) r = 20 cm, (b) r = 12 cm, (c) r = 8 cm, and (d) r = 2 cm.

12) CSUF Staff Physicist & Sauvé Dude, Steve

Mahrley, designs a lab experiment that consists of a vertical rod with a fixed bead of charge Q = 1.25 x 10–6 C at the bottom. See diagram. Another bead that is free to slide on the rod without friction has a mass of 25 g and charge, q. Steve releases the movable bead from rest 95 cm above the fixed bead and it gets no closer than 12 cm to the fixed bead. (a) Calculate the charge, q, on the movable bead. Steve then pushes the movable bead down to 8 cm above Q. He releases it from rest. (b) What is the maximum height that the bead reaches?

13)

d

P

0 – – – – –

+

L 20 cm

You have two metal spheres each of diameter 30 cm that are space 20 cm apart. One sphere has a net charge of 15 μC and the other – 15 μC. A proton is placed very close to the surface of the positive sphere and is release from rest. With what speed does it hit the other sphere?

14) A thin spherical shell of radius, R, is centered at the

origin. It has a surface charge density of 2.6 C/m2. A point in space is a distance, r, from the origin. The point in space has an electric potential of 200 V and an electric field strength of 150 V/m, both because of the sphere. (a) Explain why it is impossible for r < R. (b) Determine the radius, R, of the sphere.

– 4 μC 12 μC 15) – – +

6 cm The two charges above are fixed and cannot move. Find a

point in space where the total electric potential will equal zero.

ANSWERS:

1) a) 4.20 x 106 N/C, I

b) 0 c) 9.68 x 105 N/C, I d) 1.86 x 10–5 C/m2 e) – 1.24 x 10–5 C/m2 2) a) – 3.67 x 10–6 C b) 0 c) 1.20 x 106 N/C, O d) 2.92 x 10–5 C/m2 e) 1.73 x 10–6 C/m2 3) a) 7.35 x 105 N/C, L

b) 0 c) 6.5 x 10–6 C/m2 d) – 1.5 x 10–6 C/m2 e) 1.5 x 10–6 C/m2 f) – 1.5 x 10–6 C/m2 4) a) 2.06 x 104 V b) 3.29 x 10–15 J c) – 1.26 x 104 V d) 4.91 x 105 m/s 5) 5.02 x 105 V

6) – 7.87 x 104 V 7) a) 2.92 x 10-17 J b) 182.2 V c) 3644 N/C

8) a)   

   

d Ldlnk

b) – 1.83 x 104 V 9) a) – 1.47 x 105 V b) – 3.90 x 105 V 10) a) 1.41 x 105 V b) 1.88 x 105 V c) 2.59 x 105 V 11) a) – 8.37 x 104 V b) – 1.12 x 105 V c) – 8.62 x 104 V d) – 9900 V 12) a) 2.48 x 10–6 C b) 1.42 m 13) 1.4 x 107 m/s 14) 2.86 m 15) 1.5 cm

q

Q

Physics 226 Fall 2013

Problem Set #7 1) You have a parallel plate capacitor of plate separation

0.1 mm that is filled with a dielectric of neoprene rubber. The area of each plate is 1.8 cm2. (a) Calculate the capacitance of the capacitor. The capacitor is charged by taking electrons from one plate and depositing them on the other plate. You repeat this process until the potential difference between the plates is 350 V. (b) How many electrons have been transferred in order to accomplish this?

2) A capacitor with ruby mica has an effective electric field

between the plates of 4600 V/m. The plates of the capacitor are separated by a distance of 4 mm. 50 mJ of energy is stored in the electric field. (a) What is the capacitance of the capacitor? (b) Calculate the energy density in between the plates.

3) A capacitor with a dielectric of paper is charged to 0.5 mC.

The plates of the capacitor are separated by a distance of 8 mm. 40 mJ of energy is stored in the electric field. (a) What is the strength of the effective electric field? (b) Calculate the energy density in between the plates.

4) A capacitor of 10 μF is charged by connecting it to a

battery of 20 V. The battery is removed and you pull the plates apart so that you triple the distance between them. How much work do you do to pull the plates apart?

5) The flash on a disposable camera contains a capacitor

of 65 F. The capacitor has a charge of 0.6 m C stored on it. (a) Determine the energy that is used to produce a flash of light. (b) Assuming that the flash lasts for 6 ms, find the power of the flash. (Think back to 225.)

6) A spherical shell conductor of

radius B encloses another spherical shell conductor of radius A. They are charged with opposites signs but same magnitude, q. (a) Using integration, derive an equation for the capacitance of this spherical capacitor. (b) Calculate the capacitance if A = 45 mm and B = 50 mm. (c) If q = 40 μC, what is the energy density in between the shells?

7) You attach a battery of 15 V to an air-filled capacitor of 5 μF and let it charge up. (a) If the plate separation is 3 mm, what is the energy density in between the plates? You then remove the battery and attach the capacitor to a different uncharged capacitor of 2 μF. (b) What is the amount of charge on each capacitor after they come to equilibrium?

8) You attach a 100 pF capacitor to a battery of 10 V. You

attach a 250 pF battery to 7 V. You remove both of the batteries and attach the positive plate of one capacitor to the positive plate of the other. After they come to equilibrium, find the potential difference across each capacitor.

9) Do problem #8 but when you attach the capacitors

together attach the opposite sign plates instead of the same sign plates.

10)

Determine the equivalent capacitance between points A and B for the capacitors shown in the circuit above.

11)

Determine the equivalent capacitance between points A and B for the capacitors shown in the circuit above.

12) Design a circuit that has an equivalent capacitance of

1.50 μF using at least one of each of the follow capacitors: a 1 μF, a 2 μF, and a 6 μF. [You must also show where your A and B terminals are located.]

A

20 F

4 F

4 F

6 F

12 F

B

30 F

A 12 F

18 F 6 F

20 F

B

12 F 75 F

13) The two capacitors above both have plates that are

squares of sides 3 cm. The plate separation is 1.2 cm for both. Between each of the capacitor plates are two different dielectrics of neoprene rubber and Bakelite. Everything is drawn to scale. Find the capacitance of each capacitor. (HINT: Think series and parallel.)

14) The plates of an air-filled capacitor have area, A, and are

separated by a distance, d. The capacitor is charged by a battery of voltage, V. Three things are going to change: (1) The plates of the capacitor are pulled apart so that the distance between the plates triples. (2) The area of the plates increase by a factor of 6. (3) The voltage of the battery decreases by a factor of 4. Determine expressions in terms of A, d, and/or V for (a) the new capacitance, (b) the new charge, and (c) the new energy density.

15)

A massless bar of length, L, is hanging from a string that is attached 1/3 of the length of the bar from the right end. A block of mass, M, is hung from the right end. The left end of the bar has an air-filled massless capacitor of plate area, A, and plate separation, d. Find an expression for the potential difference between the plates so that this system is in equilibrium. (HINT: You will

need the equation dx dU

F  from 225.)

ANSWERS:

(a) (b)

1) a) 1.067 x 10–10 F

b) 2.34 x 1011 e–

2) a) 2.95 x 10–4 F b) 5.05 x 10–4 J/m3 3) a) 2 x 104 V/m

b) 6.7 x 10–3 J/m3 4) 4 x 10–3 J 5) a) 2.8 x 10–3 J b) 0.467 W

6) a)   

  

 

AB AB4C o

b) 5.01 x 10–11 F c) 1.125 x 105 J/m3 7) a) 1.11 x 10–4 J/m3 b) 2.14 x 10–5 C, 5.36 x 10–5 C

8) 7.86 V 9) 2.14 V 10) 4 μF 11) 9 μF 13) a) 3.85 pF b) 3.76 pF

14) a) d

A2 C o

 

b) d2 AV

Q  o

c) 2 2

o

d288 Vu  

15) A

Mg dV

o 

M

Physics 226 Fall 2013

Problem Set #8 1) Analyze the circuit below using a QCV chart. You must

show appropriate work for full credit. 2) Analyze the circuit below using a QCV chart. You must

show appropriate work for full credit. 3) Analyze the circuit below using a QCV chart. You must

show appropriate work for full credit. 4)

An Oppo Digital Blu-Ray player [DMP-95] (Yes, I am an audiophile.) has a power cable which has a metal that allows 9 x 1019 electrons per cubic millimeter. On average, the cable passes 1 x 1022 electrons every hour. The electrons passing through the player have a drift velocity of 4.5 μm/s. (a) What current does the Oppo draw? (b) Calculate the diameter of the cable?

5) The Large Hadron Collider at CERN creates proton beams which collide together resulting in pictures like the one at the right. Some of these beams can have a radius of 1.1 mm with a current of 1.5 mA. The kinetic energy of each proton in this beam is 2.5 MeV. (a) Calculate the number density of the protons in the beam. (b) If the beam is aimed at a metal target, how many protons would strike the screen in 1 minute?

C1 = 8 μF C2 = 15 μF

20 V

C3 = 30 μF

6)

Two copper wires are soldered together. Wire #1 has a radius of 0.7 mm. Wire #2 has a radius of 1.2 mm. Copper has a number density of 8.47 x 1028 e–/m3. The drift velocity in Wire #1 is 0.72 mm/s. If you want the current to remain the same in both, what is the drift velocity in Wire #2?

7) A nichrome cable has a current of 140 A running through

it. Between two points on the cable that are 0.22 m apart, there is a potential difference of 0.036 V (a) Calculate the diameter of the cable. (b) How much heat energy does this part of the wire emit in 1 minute?

8) A “Rockstar” toaster uses a

tungsten heating element (wire). The wire has a diameter of 1.2 mm. When the toaster is turned on at 20 C, the initial current is 1.6 A. (a) What is the current density in the wire? (b) A few seconds later, the toaster heats up and the current is 1.20 A. What is the temperature of the wire? (c) If the toaster is plugged into a standard wall outlet in Kankakee, Illinois, what is the rate that energy is dissipated from the heating element?

9) Skid runs a 10 mile line of copper cable out to his shack in

the sticks so he can have electricity to play Lord of the Rings Online. At 20ºC the resistance of the cable is 12 . At 50ºC the cable emits 1.5 kJ every second. (a) What is the resistance of the cable at 50ºC? (b) What is the current running through the cable at 50ºC? (c) Calculate the current density at 50ºC.

C1 = 18 F

Wire #1 Wire #2 C2 = 6 μF

C3 = 4 μF

C4 = 30 μF 25 V

C1 = 5 F C2 = 4 μF

15 V C3 = 1 μF

C4 = 12 μF

10) A modern hair dryer uses a nichrome heating element that typically is 30-gauge wire around 40 cm in length. The gauge rating on a wire refers to its diameter. In this case, 30-gauge wire has a diameter of 0.254 mm. Nichrome has a number density of 7.94 x 1028 e–/m3. If the drift velocity of the electrons in the wire is 18.7 mm/s, what is the voltage that the hair dryer is plugged into?

11) Before LCD, LED, Plasma,

and (the latest) OLED TVs, there were CRT (Cathod-Ray Tube) TVs. Inside these TVs were electron guns that shot an electron beam of diameter 0.5 mm and current density of 244 A/m2 onto the inside of a glass screen which was coated with phosphor. How many electrons would hit the phosphor every minute?

12)

Determine the equivalent resistance between points A and B for the resistors shown in the circuit above.

13)

Determine the equivalent resistance between points A and B for the resistors shown in the circuit above.

14)

Determine the equivalent resistance between points A and B for the resistors shown in the circuit above.

15) Design a circuit that has an equivalent resistance of

1.00  using at least one of each of the follow resistors: a 1 , a 2 , and a 6 . [You must also show where your A and B terminals are located.]

ANSWERS:

NOTE: Some of these answers are minimal since there are checks that you can do to verify your answers.

A

27 

B 54 

8 

30 

16 

14 

10 

30 

B

18 

96 

6 

32  18 

60  A

A

20 

30 

B

30 

7 

50 

12 

45 

60 

1) CEQ = 18 μF 8) a) 1.415 x 106 A/m2 2) CEQ = 6 μF b) 94.1ºC 3) CEQ = 2 μF c) 144 W

9) a) 13.4  4) a) 0.444 A b) 2.96 mm b) 10.58 A

c) 5.14 x 105 A/m2 5) a) 1.13 x 1014 p+/m3 b) 5.63 x 1017 p+ 10) 95.0 V 6) 0.262 mm/s 11) 1.8 x 1016 e–

7) a) 0.033 m 12) 4  b) 302 J 13) 14  14) 22 

Physics 226 Fall 2013

Problem Set #9

NOTE: You can only use circuit tricks on 9 – 11 but not on any others. 1) Analyze the following circuit using a VIR chart. 2) Swap the location of the battery and R1 in the circuit from

problem #1. Analyze the circuit using a VIR chart. 3) Analyze the following circuit using a VIR chart. 4) The battery in this problem has an internal resistance of

0.15 . (a) Analyze the following circuit using a VIR chart. (b) Is this circuit well designed? Discuss, explain.

5) Analyze the following circuit using a VIR chart.

6) Analyze the following circuit using a VIR chart. 7) The battery in this problem has an internal resistance of

1 . (a) Analyze the following circuit using a VIR chart. (b) Is this circuit well designed? Discuss, explain.

8) A load of 3.5  is connected across a 12 V battery. You

measure a voltage of 9.5 V across the terminals of the battery. (a) Find the internal resistance of the battery. (b) Is this circuit well designed? Discuss, explain.

9) Analyze the circuit from problem

#5 using a VIR chart. You are using only the diagram in #5, not the values. New values are given at the right. You may use a circuit trick for this circuit, but only for ONE value.

10) Analyze the circuit from problem

#6 using a VIR chart. You are using only the diagram in #6, not the values. New values are given at the right. You may use a circuit trick for this circuit, but only for ONE value.

R1

20 V

R2 R3 R4

R5

Given: R1 = 12  R2 = 3  R3 = 8  R4 = 36  R5 = 15 

50 V

R1 Given: R1 = 28  R2 = 6  R3 = 84  R4 = 7  R5 = 54 

R3

R2

R4

R5

55 V

R1 Given: R1 = 18  R2 = 32  R3 = 15  R4 = 21  R5 = 42  R6 = 30  R7 = 52 

R3

R2

R4 R5

R6 R7

R1

VB

R2

R3 R4

Given: VB = 60 V V2 = 50 V

I1 = 2 A I4 = 3 A

R3 = 8 

R1

VB

R2 R3

R4

R5

Given: V5 = 32 V

I2 = 0.4 A I4 = 0.5 A

R1 = 36  R6 R3 = 60  R4 = 36  R6 = 32 

R1

VB

Given: VB = 32 V

R2 I1 = 4 A R3 R3 = 12 

R4 R4 = 8 

Given: VB = 63 V R1 = 8  R2 = 20  R3 = 35  R4 = 49 

Given: VB = 75 V R1 = 16  R2 = 40  R3 = 48  R4 = 24  R5 = 8  R6 = 24 

11) Analyze the following circuit using a VIR chart. 12) Using the information you are

given for the circuit at the right, answer the following. (a) Determine the magnitude and direction of the current in the circuit. (b) Determine which point, A or B, is at a higher potential.

13) Calculate the unknown currents I1, I2, and I3 for the circuit

below.

14) Calculate the unknown currents I1, I2, and I3 for the circuit below.

Given: 15) Calculate the unknown currents I1, I2, and I3 for the circuit

below.

ANSWERS:

NOTE: These answers are minimal since there are checks that you can do to verify your answers.

R1 R2

R3 R4

R5 R6

I1 8 V VB = 50 V

R1 = 9  R2 = 4  R3 = 18  R4 = 4  R5 = 7  R6 = 12 

B

A

17 V

13  7 

5 

11 

23 V

6 

1 

10 V

25 V

3 

5 

7 

I1

I2

I3

4 

9 

10 

4  7 

I2

6 

I3 22 V

3  10 V I1

4 

4  25 V

2  5 

I2 I3

20 V 4 

7) REQ = 8  1) REQ = 2  8) a) 0.923  2) REQ = 11.48  9) REQ = 21  3) REQ = 25  10) REQ = 25  4) REQ = 12.15  11) REQ = 20  5) REQ = 12  12) a) 1.11 A 6) REQ = 40 

Gas Properties Simulation Activity

In this activity you’ll use the Gas Properties PhET Simulation

(https://phet.colorado.edu/en/simulation/gas-properties) to explore and explain the relationships

between energy, pressure, volume, temperature, particle mass, number, and speed.

This activity has 5 modules:

○ Explore the Simulation

○ Kinetic Energy and Speed

○ Kinetic Molecular Theory of Gases

○ Relationships between Gas Variables

○ Pressure and Mixtures of Gases

You will get the most out of the activity if you do the exploration first! The rest of the sections

can be worked in any order; you could work on any sections where you want to deepen your

conceptual understanding.

Part I: Explore the Simulation

Take about five minutes to explore the sim. Note at least two relationships that you observe and

find interesting.

Part II: Kinetic Energy and Speed

Sketch and compare the distributions for kinetic energy and speed at two different temperatures

in the table below. Record your temperatures (T1 and T2), set Volume as a Constant Parameter,

and use roughly the same number of particles for each experiment (aim for ~100-200). Use the

T2 temperature to examine a mixture of particles.

Tips:

T1 = __________K The Species Information and Energy Histograms tools will help.

T2 = __________K The system is dynamic so the distributions will fluctuate.

Sketch the average or most common distribution that you see.

“Heavy” Particles Only “Light” Particles Only Heavy + Light Mixture

# of particles

(~100-200)

Kinetic

Energy

Distribution

sketch for T1

Speed

Distribution

sketch for T1

Kinetic

Energy

Distribution

sketch for T2

Speed

Distribution

sketch for T2

1. Compare the kinetic energy distributions for the heavy vs. light particles at the same

temperature. Are these the same or different? What about the speed distributions?

2. Compare the kinetic energy distributions for the heavy vs. light particles at different

temperatures. Are these the same or different? What about the speed distributions?

3. Compare the kinetic energy distributions for the mixture to those of the heavy-only and light-

only gases at the same temperature. Are these the same or different? What about the speed

distributions?

4. Summarize your observations about the relationships between molecular mass (heavy vs.

light), kinetic energy, particle speed, and temperature.

Part III: Kinetic Molecular Theory (KMT) of Gases

Our fundamental understanding of “ideal” gases makes the following 4 assumptions.

Describe how each of these assumptions is (or is not!) represented in the simulation.

Assumption of KMT Representation in Simulation

1. Gas particles are separated by

relatively large distances.

2. Gas molecules are constantly in

random motion and undergo

elastic collisions (like billiard

balls) with each other and the

walls of the container.

3. Gas molecules are not attracted

or repulsed by each other.

4. The average kinetic energy of

gas molecules in a sample is

proportional to temperature (in K).

Part IV: Relationships Between Gas Variables

Scientists in the late 1800’s noted relationships between many of the state variables related to

gases (pressure, volume, temperature), and the number of gas particles in the sample being

studied. They knew that it was easier to study relationships if they varied only two parameters at

a time and “fixed” (held constant) the others. Use the simulation to explore these relationships.

Variables Constant Parameters Relationship Proportionality

(see hint below)

pressure, volume directly proportional

or

inversely proportional

volume, temperature directly proportional

or

inversely proportional

volume, number of

gas particles

directly proportional

or

inversely proportional

Hint: A pair of variables is directly proportional when they vary in the same way (one increases

and the other also increases). A pair of variables is inversely proportional when they vary in

opposite ways (one increases and the other decreases). Label each of your relationships in the

table above as directly or inversely proportional.

Part V: Pressure and Mixtures of Gases

The atmosphere is composed of many gases in different ratios, and all of them contribute to the

total atmospheric pressure. Use the simulation to explore this relationship by testing

combinations of heavy and light gases.

For each Test #, record your measurement and the make the prediction before moving on to the

next row of the table.

Test

#

Pressure

Measurement

Pressure Prediction

(greater than, equal to, less than, twice as much, half as much, etc)

1 100 Light particles =

Pressure for 100 Heavy Particles will be __________________

the pressure from Test #1.

2 100 Heavy particles =

Pressure for 200 Heavy particles will be __________________

the pressure from Test #2.

3 200 Heavy particles = Pressure for 100 Light AND 100 Heavy particles will be

__________________ the pressure from Test #3

4 100 Heavy + 100

Light particles =

Pressure for 200 Heavy AND 100 Light particles will be

__________________ the pressure from Test #4.

5 200 Heavy + 100

Light particles =

Pressure for 150 Heavy AND 50 Light particles will be

_________________ the pressure from Test #5.

6 150 Heavy + 50 Light

particles =

Write your own prediction:

1. For Test 6 (150 Heavy + 50 Light particles), what is the pressure contribution from the heavy

particles (Pheavy)? How did you figure this out?

2. What is the pressure contribution from the light particles (Plight)? How did you figure this

out?

3. For each test above, calculate the mole fraction of each gas (number of particles of that type /

total particles). Find a relationship between the mole fraction and the pressure contribution of

each type of gas.

4. The atmosphere is composed of about 78% nitrogen, 21% oxygen, and 1% argon. Typical

atmospheric pressure in Boulder, Colorado is about 0.83 atm. What is the pressure contributed

by each gas?

• Objectives (5 points) – in one or two sentences describe the purpose of the lab. What physical quantities are you measuring? What physical principles/laws are you investigating?

• Procedure (5 points) – this section should contain a brief description of the main steps and the significant details of the experiment.

• Experimental data (15 points) – your data should be tabulated neatly in this section. Your tables should have clear headings and contain units. All the clearly labeled plots (Figure 1, etc.) produced during lab must be attached to the report. The scales on the figures should be chosen appropriately so that the data to be presented will cover most part of the graph paper.