Experiment 9 PDF

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Experiment Nine...

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Experiment 9: Centripetal Force Alexandra Tananbaum General Physics I: Laboratory Fall 2019: Section 005 Instructor: Kulchoakrungsun Partners: Libby Tuesday, September 17, 2019 Due date: Tuesday, September 24, 2019

INTRODUCTION The objective of this lab was to observe centripetal force, a special case of a particle moving in a circle with constant speed in which F and a are always perpendicular to v and point from the particle’s position towards the center of the circle. In this experiment, a hanging mass is rotated in a circle and the force on the mass is compared to the theoretical value. DESCRIPTION In addition to basic materials such as a metric ruler, meter stick, timer, weights and weight hangers, and a digital scale, this lab also requires a centripetal force apparatus which has a vertical shaft and bearing (that allows rotation) on a horizontal base. At the top of the shaft is a horizontal arm which supports a moveable slider and counter weight. A mass, M1 is suspended by string from the arm and attached to the shaft by a spring. At the base there is a vertical movable pointer that projects upwards towards M1. The shaft can be rotated so that the pointer on the base and M1 line up. To directly measure the centripetal force, the value of M2, a mass suspended across a pulley, can be adjusted so that the pointers again line up. THEORY In this experiment, we will be experimentally determining angular velocity, by using the ∆θ , or the change in angle divided by time. The angular velocity will be calculated equation ∆T 2π using . In order to determine the centripetal force, we can use the equation T 2 Fc =mr ω . Since the mass, radius, and velocity will be known from the experiment, it will be very easy to calculate the centripetal force. PROCEDURE The overall procedure of the experiment involved determining the centripetal force using different angular velocities. DATA/CALCULATIONS Mass 1: 0.5 kg Trial 1: Mass 2: 0.370 kg Distance to Axis of Rotation (Radius): 0.148 m Run #

# of Revolutions:

Time for Revolutions:

Angular Velocity = (2π)/[(Time for 10 revolutions)/(10 revolutions)] = 2π/T

Run 1

10

8.33 s

(2π) /[(8.33 s)/(10 rev)] = 7.54 radians/s

Run 2

10

8.29 s

(2π) /[(8.29 s)/(10 rev)] = 7.58 radians/s

Run 3

10

8.45 s

(2π) /[(8.45 s)/(10 rev)] = 7.44 radians/s

Run 4

10

8.25 s

(2π) /[(8.25 s)/(10 rev)] = 7.62 radians/s

Run 5

10

8.30 s

(2π) /[(8.30 s)/(10 rev)] = 7.57 radians/s

Average Experimental Angular Velocity: [(7.54 radians/s)+( 7.58 radians/s)+( 7.44 radians/s)+ ( 7.62 radians/s)+( 7.57 radians/s)]/5 = 7.55 radians/s Experimental Centripetal Force: (mass1)(radius)(angular velocity)2 =(0.5 kg)(0.148 m)(7.55 radians/s)2 = 4.21 N Theoretical Centripetal Force: (mass2)(g)= (0.370 kg)(9.81 m/s2) = 3.63 N Percent Error of FC: [( Experimental FC)-(Theoretical FC) ]/ (Theoretical FC) = (4.21 N-3.63 N)/ 3.63 N = 16.0% Trial 2: Mass 2: 0.380 kg Distance to Axis of Rotation (Radius): 0.151 m Run #

# of Revolutions:

Angular Velocity

Run 1

10

8.20 s

Angular Velocity = (2π)/[(Time for 10 revolutions)/(10 revolutions)] = 2π/T (2π) /[(8.20 s)/(10 rev)] = 7.66 radians/s

Run 2

10

8.21 s

(2π) /[(8.21 s)/(10 rev)] = 7.65 radians/s

Run 3

10

8.42 s

(2π) /[(8.42 s)/(10 rev)] = 7.46 radians/s

Run 4

10

8.10 s

(2π) /[(8.10 s)/(10 rev)] = 7.76 radians/s

Run 5

10

8.60 s

(2π) /[(8.60 s)/(10 rev)] = 7.31 radians/s

Average Experimental Angular Velocity: [(7.66 radians/s)+( 7.65 radians/s)+( 7.46 radians/s)+ ( 7.76 radians/s)+( 7.31 radians/s)]/5 = 7.57 radians/s Experimental Centripetal Force: (mass1)(radius)(angular velocity)2 =(0.5 kg)(0.151 m)(7.57 radians/s)2 = 4.33 N Theoretical Centripetal Force: (mass2)(g)= (0.380 kg)(9.81 m/s2) = 3.73 N

Percent Error of FC: [( Experimental FC)-(Theoretical FC) ]/ (Theoretical FC) = (4.33 N-3.73 N)/ 3.73 N = 16.1% Trial 3: Mass 2: 0.570 kg Distance to Axis of Rotation (Radius): 0.163 m Run #

# of Revolutions:

Time for Revolutions:

Run 1

10

7.10 s

Angular Velocity = (2π)/[(Time for 10 revolutions)/(10 revolutions)] = 2π/T (2π) /[(7.10 s)/(10 rev)] = 8.85 radians/s

Run 2

10

7.10 s

(2π) /[(7.10 s)/(10 rev)] = 8.85 radians/s

Run 3

10

7.38 s

(2π) /[(8.60 s)/(10 rev)] = 8.51 radians/s

Run 4

10

7.60 s

(2π) /[(8.60 s)/(10 rev)] = 8.27 radians/s

Run 5

10

7.71 s

(2π) /[(8.60 s)/(10 rev)] = 8.15 radians/s

Average Experimental Angular Velocity: [(8.85 radians/s)+( 8.85 radians/s)+( 8.51 radians/s)+ ( 8.27 radians/s)+( 8.15 radians/s)]/5 = 8.53 radians/s Experimental Centripetal Force: (mass1)(radius)(angular velocity)2 =(0.5 kg)(0.163 m)(8.53 radians/s)2 = 5.93 N Theoretical Centripetal Force: (mass2)(g)= (0.570 kg)(9.81 m/s2) = 5.59 N Percent Error of FC: [( Experimental FC)-(Theoretical FC) ]/ (Theoretical FC) = (5.93 N-5.59 N)/ 5.59 N = 6.1% Trial 4: Mass 2: 0.450 kg Distance to Axis of Rotation (Radius): 0.155 m Run #

# of Revolutions:

Time for Revolutions:

Run 1 Run 2 Run 3 Run

10

7.92 s

Angular Velocity = (2π)/[(Time for 10 revolutions)/(10 revolutions)] = 2π/T (2π) /[(7.92 s)/(10 rev)] = 7.93 radians/s

10

7.91 s

(2π) /[(7.91 s)/(10 rev)] = 7.94 radians/s

10

7.94 s

(2π) /[(7.94 s)/(10 rev)] = 7.91 radians/s

10

7.91 s

(2π) /[(7.91 s)/(10 rev)] = 7.94 radians/s

4 Run 5

10

8.10 s

(2π) /[(8.10 s)/(10 rev)] = 7.76 radians/s

Average Experimental Angular Velocity: [(7.93 radians/s)+( 7.94 radians/s)+( 7.91 radians/s)+ ( 7.94 radians/s)+( 7.76 radians/s)]/5 = 7.90 radians/s Experimental Centripetal Force: (mass1)(radius)(angular velocity)2 =(0.5 kg)(0.155 m)(7.90 radians/s)2 = 4.84 N Theoretical Centripetal Force: (mass2)(g)= (0.450 kg)(9.81 m/s2) = 4.41 N Percent Error of FC: [( Experimental FC)-(Theoretical FC) ]/ (Theoretical FC) = (4.84 N-4.41 N)/ 4.41 N = 9.8% Average Percent Error of Fc: ( 16.0% + 16.1% + 6.1% + 9.8%)/4 = 12% ERROR ANALYSIS The main error in this experiment was largely that angular velocity was not constant and the experimental values used in caclulting it were ridden with error. Since the weight was rotated by hand, it was hard to keep the weight moving a certain speed and distance the whole time, which undoubtedly led to error in the calculation of angular velocity. Furthermore, error resulted from friction; air friction, which increased the needed angular velocity to maintain a certain radius, compared to the theoretical angular velocity. Error also resulted from the friction of the string attached between the two weights, which resisted the tension in the string and thus impacted M2. These friction forces led the experimental centripetal force to be consistently greater than the theoretical, which is logical since the experimental force had to be greater to offset the pulling force of the string as well as the friction force. The experimental centripetal force was consistently less than the theoretical value, by an average value of 12%. Error due to inconsistent and inaccurate angular velocity was random and thus the overall effect on the average percent error was probably around zero (Sometimes the human hand spinning the weight went too fast and sometimes it went too slow, and therefore cancelled its overall error out). However since the error due to friction forces including air resistance was systematic, and consistently led to a greater experimental centripetal force, the error can be estimated from the average percent error between the theoretical and experimentally determined centripetal force values. QUESTIONS 1. What are the most significant causes of errors in this experiment? Explained above.

2. Are the significant errors random or systematic? Systematic errors (for the most part), as the error was largely a result of an imperfect system that affected all trials. It could also be argued that the errors were random, as they caused different results each time and were a result of human error; the experimenter could easily misjudge if the weight is circling over the pointer. 3. Does the data enable you to estimate any of these errors? Explain. Perhaps in a very limited manner; because the forces calculated from M2 and gravitational acceleration is lower in every trial (in comparison to the force calculated form M1), the weights were likely traveling in a circle that was smaller (/with a smaller radius) than needed with the pointer. That accounts for the smaller T, larger, w, and therefore, larger calculated F. This could be due to either the weight mass slowing down/at a lower velocity than expected, or also because of human error when judging whether the mass is circling over the pointer. 4. Does the data support Newton’s 2nd law when it’s applied to a mass going in a circular orbit at constant speed? Yes, despite these small errors, the observed and calculated forces for both M1 and M2 were in line with what was expected with Newton’s 2nd Law 5. What are all the forces that are occurring to M1 in rotation? Tension from the string Centripetal force Gravity Friction CONCLUSION In this experiment we effectively measured the centripetal force of an object in circular motion, and compared the theoretical force with the experimental force. Although the experiment was rife with systematic error, our results were consistent with each other, as the error remained constant throughout each trial of the experiment....