Circular Motion - Lab 4 (SPH4U) PDF

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This is an CIMP - SPH4U Document. This is a report following the APA format for Lab 4 - Circular Motion....

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Circular Motion - Lab 4 SPH4U CIMP 17 February 2022

Table of Contents

Purpose …………………………………………………………………………………………. 3 Introduction ………………………………………………………………………………….... 3-5 Hypothesis …………………………………………………………………………………...… 5 Materials …………………………………………………………………………………...…… 6 Procedure …………………………………………………………………………………..………. ● Part A ……………………………………………………………………………………. 7 ● Part B ……………………………………………………………………………………. 8 Data …………………………………………………………………………………..……… 9-16 Analysis …………………………………………………………………………………..…… 16 Conclusion ………………………………………………………………………………..…… 17 References ………………………………………………………………………………...…… 18 Appendices ……………………………………………………………………………...……… 19

Purpose: To experimentally replicate uniform circular motion and confirm the relationships between centripetal force and the frequency, mass, and radius of the moving object. Our goal is to compute the centripetal force operating on an item by measuring its period of rotation. The centripetal force will then be compared to an analogous force required to keep the item at the same radius.

Introduction: If you've ever been on an amusement park ride that follows a circular course, you might have felt a force known as “centripetal force”(Khan Academy, 2022) pushing you toward the ride. Whether it's the rear wall of a "Roundup" or "Rotor," the ride where the floor collapses from beneath your feet, or the seat belt of a "roller coaster," you're continually driven into the ride's center of curvature. If, as you most dread on such a trip, this force were abruptly eliminated, you would travel in a tangent to the circular course. That's what happens when you ride a roller coaster and go over a hill just before the seat belt kicks in. The image below shows the motion of a body in the presence of a centripetal force versus the motion of a body if the centripetal force were to abruptly cease. (Khan Academy, 2022)

Figure 1

Consider a ball linked to a thread and whirled around in a circle, as seen in Figure 2. The centripetal force is applied to the ball by the tension in the string, causing it to travel in a circular direction (Khan Academy, 2022). According to Newton's third law of action and reaction, the string pulls the ball toward the center of the circle, while the ball pushes outward on the string and so on your hand (Britannica, 2021). As a result, this outward force has no effect on the ball, despite the fact that it is usually and wrongly referred to as the centrifugal force acting on the ball (Britannica, 2021). When the centripetal force is removed, as seen in Figure 3 when the string snaps, the item goes in the direction of the velocity at the time. At that moment, this direction is tangential to the circle (Khan Academy, 2022). Top view of a ball on a string before and after the string breaks

Figure 2

Figure 3

With a few observations and calculations, you can estimate the centripetal force that keeps you in the ride (Khan Academy, 2022). The velocity of an item moving in circular motion at constant speed changes direction all the time. In addition, the item is accelerating towards the center of the circle. Uniform circular motion happens when an item has a constant speed and radius, whereas centripetal acceleration occurs when there is an instantaneous acceleration directed towards the center of the circle (Khan Academy, 2022). The amount of

centripetal acceleration is given by the formula: a = v2/r, where r is the radius and v is the constant speed (Khan Academy, 2022) (Britannica, 2021). As the radius increases, the direction changes more slowly, resulting in a lower acceleration. In this lab, we will identify which factors must be known in order to calculate the centripetal force that is necessary to maintain a mass traveling in a circular direction at a constant speed (Khan Academy, 2022).

Hypothesis: The connection between frequency and radius, mass, and centripetal force in this experiment is, 1.) When the centripetal force is constant, the radius is inversely proportional to the frequency of circular motion. 2.) When the centripetal force is constant, the mass is inversely proportional to the frequency of circular motion. 3.) When the radius and mass of the system are constant, the centripetal force in circular motion is exactly proportional to the frequency. The equation needed to understand these interactions is, Fc = 4π^2mrf^2

Materials: 1. Rubber Stopper 2. 6 Weights 3. String 4. Tube 5. Paper Clip 6. Meter Rule 7. Safety Glasses

Procedure: Part A : Force and Frequency 1. Attach a rubber stopper securely to one end of your string. It should be threaded through the tube. Swing the stopper above and around your head at a steady pace and radius while without moving your hand more than 2 cm to either left or the right. Ascertain that you are at ease and skilled with this step before moving on to the next step. 2. Place the equipment on the table and measure 75 cm from the stopper's center to the top of the hollow tube. Attach the clip at the bottom of the tube to secure this radius. 3. Place a 50 g mass on the opposite end of the string such that it is suspended by the string's loop. You should keep this clip in the same location throughout this experiment to verify that the radius of motion is consistent. 4. Swing the stopper in a horizontal circle over your head while holding the mass. Adjust the speed until there is no tension in the string. Allow the masses to fall away and adjust the rotation rate such that the reference clip is about 0.1 cm below the tube. Have your companion time how long it takes you to complete 10 revolutions. 5. Then, using varied masses, repeat the process and record the results in a table. Repeat the preceding procedures three more times, then compute the average of the three trials. Then, repeat the preceding stages with a mass of 350 grams, increasing 50 grams at a time. 6. Then, graph (# of Units of Force) vs. (Frequency) and (# of Units of Force) vs. (Frequency^2). Remember to add a best-fitting line or curve.

Part B : Radius and Frequency 1. Attach 100 g to the string's end. Connect the clip so that the radius of motion is 0.65m. As before, rotate one stopper at a steady pace while keeping the reference tape 0.1 cm below the glass tube. Calculate the time it takes to complete 10 revolutions. Repeat for 0.75 m, 0.65 m, and 0.55 m radii. Fill out the table below with your information. Repeat the process twice more for each radius. You should end up with three trails in all. 2. Then, make a graph showing radius vs. frequency2. Remember to add a best-fitting line or curve.

Data :

Time for 10 rotations Hanging Mass (g)

Trial 1 (secs)

Trial 2 (secs)

Trial 3 (secs)

50

7.78

9.72

9.56

150

7.41

8.18

7.16

200

6.00

6.41

6.44

250

6.10

5.63

5.75

300

5.47

4.75

5.28

350

4.87

5.56

4.69

This table shows the time needed for 10 rotations with the string having a radius of 75cm.

Time for 10 rotations Hanging Mass (g)

Trial 1 (secs)

Trial 2 (secs)

Trial 3 (secs)

50

7.34

7.25

7.28

150

6.28

7.06

6.75

200

5.53

6.06

6.16

250

4.85

5.31

5.00

300

4.87

4.72

4.78

350

4.22

4.09

4.91

This table shows the time needed for 10 rotations with the string having a radius of 65cm.

Time for 10 rotations Hanging Mass (g)

Trial 1 (secs)

Trial 2 (secs)

Trial 3 (secs)

50

7.41

8

7.44

150

6.41

6.35

6.22

200

5.5

5.91

5.1

250

4.4

5.56

4.84

300

4.07

4.03

4.81

350

4.09

4.09

4.25

This table shows the time needed for 10 rotations with the string having a radius of 55cm.

Average Time per Revolution (seconds)

Distance (m)

Velocity (m/s)

Frequency (Hz)

Frequency² (Hz)

Force Gravity

0.90

4.71

5.22

1.11

1.23

0.49

0.76

4.71

6.21

1.32

1.74

0.98

0.63

4.71

7.50

1.59

2.53

1.47

0.58

4.71

8.09

1.72

2.95

1.96

0.52

4.71

9.12

1.94

3.75

2.45

0.50

4.71

9.35

1.98

3.94

2.94

0.43

4.71

10.93

2.32

5.37

3.43

This table shows the calculated results of the average time per revolution, distance of the revolution, frequency, frequency² and force gravity with the radius at 0.75m.

Average Time per Revolution (seconds)

Distance (m)

Velocity (m/s)

Frequency (Hz)

Frequency² (Hz)

Force Gravity

0.73

4.08

5.60

1.37

1.88

0.49

0.67

4.08

6.10

1.49

2.23

0.98

0.59

4.08

6.90

1.69

2.86

1.47

0.51

4.08

8.08

1.98

3.92

1.96

0.48

4.08

8.53

2.09

4.36

2.45

0.44

4.08

9.27

2.27

5.15

2.94

0.39

4.08

10.56

2.59

6.69

3.43

This table shows the calculated results of the average time per revolution, distance of the revolution, frequency, frequency² and force gravity with the radius at 0.65m.

Average Time per Revolution (seconds)

Distance (m)

Velocity (m/s)

Frequency (Hz)

Frequency² (Hz)

Force Gravity

0.76

3.46

4.54

1.31

1.72

0.49

0.63

3.46

5.46

1.58

2.50

0.98

0.55

3.46

6.28

1.82

3.30

1.47

0.49

3.46

7.00

2.03

4.11

1.96

0.43

3.46

8.03

2.32

5.40

2.45

0.41

3.46

8.34

2.41

5.83

2.94

0.37

3.46

9.46

2.74

7.49

3.43

This table shows the calculated results of the average time per revolution, distance of the revolution, frequency, frequency² and force gravity with the radius at 0.55m.

This graph shows how the force of gravity and frequency correlate for 0.75m. This graph shows the force of gravity and frequency^2 correlate for 0.75m.

This graph shows how the force of gravity and frequency correlate for 0.65m.

This graph shows how the force of gravity and frequency^2 correlate for 0.65m.

This graph shows how the force of gravity and frequency correlate for 0.55m.

This graph shows how the force of gravity and frequency^2 correlate for 0.55m.

This graph shows how the different radius’ and frequency^2 correlate.

Analysis : Based on the graphs and data shown above, it can be seen that the calculated frequency of each changing radius increases as the radius reduces in length. Furthermore, it can be shown that each time another 50g of mass is added to the setup, the force of gravity increases as well. Observed from the calculated data of finding how much time each revolution takes for each varying radius (that of 0.75m, 0.65m and 0.55m), it shows that the longer the radius, the more time it takes to complete one revolution. This is because the rubber stopper requires more time to complete one round of a larger circumference. The setup with the radius of 0.75m is proved to have the highest set of velocity values compared to the setups with 0.65m and 0.55m.

In addition, looking at the graphs above, we see strong positive linear correlation between the force of gravity and the frequency meaning that as force of gravity increases, the frequency increases. As a result, we can say that the force of gravity and the frequency is directly proportional to each other.

Conclusion : In conclusion, the purpose of this lab is to replicate circular motion and verify the relationships between the centripetal force, mass , radius and frequency of the object and towards plotting the graphs of the relationships. When Centripetal force is constant, the radius and mass is inversely proportional to the frequency. We could also conclude that as radius is reduced the time taken it takes for 1 revolution reduces as well. The overall result supported our hypothesis that when the centripetal force is constant, the radius and mass is inversely proportional to the motion as seen on the graph. We could also conclude that the linearity of the velocity increased as the mass increased. It relates because an increased mass of hanging would result in the increase of downward force due to gravity which would require large centripeta l force to balance the forces out. In addition, with velocity increasing, it would also help balance the forces.

Aspect of the experiment that could be changed and improved is that when spinning the stopper, there should be a set height where the metal coin could go as we realized that sometimes it is either too low or way too high .Therefore, our results may not be accurate when the distance of the metal coin being spun is not constant. Furthermore,we did not note down the timer’s reflex to quickly start and stop the stopwatch to calculate any human errors in taking the time.

References : What is a centripetal force? (article). (n.d.). Khan Academy. Retrieved February 28, 2022, from https://www.khanacademy.org/science/physics/centripetal-force-and-gravitation/centripetal-force s/a/what-is-centripetal-force Britannica, T. Editors of Encyclopaedia (2021, July 23). Newton’s laws of motion. Encyclopedia Britannica. https://www.britannica.com/science/Newtons-laws-of-motion Britannica, T. Editors of Encyclopaedia (2021, March 30). centripetal acceleration. Encyclopedia Britannica. https://www.britannica.com/science/centripetal-acceleration Britannica, T. Editors of Encyclopaedia (2021, July 26). centrifugal force. Encyclopedia Britannica. https://www.britannica.com/science/centrifugal-force

Appendices: Appendix 1 The formula for Centripetal Force Fc = mv^2 / r

Mass (kg)

Velocity (m/s)

Radius (m)

0.50

5.22

0.75

Appendix 2 The formula for Frequency f = 1/T

Time (seconds) 0.90

Appendix 3

The formula for Frequency^2 f = 1/T^2

Time (seconds) 0.90

Appendix 4 The formula for Force Gravity F= mg

Mass (kg)

Gravity (m/s^2)

0.050

9.81...