Physics Lab 7_ Rotation

Lab 7: Rotation Lab Date:4/8/2023 Lab Section #: _005____________

Purdue University Northwest – PHYS 15200 – Spring 2023 Abstract: In this lab, we aimed to determine how things such as inertia, torque, and angular acceleration are understood using a rotational inertia demonstrator. The rotating object comprises four pulleys of different radius, and four thin rods extending radially from the pulley. A "moveable mass" can be attached to each of the rods using a set-screw. As we noticed from the labs, as the masses are concentrated away from the rotational axis or center, the mass begins to carry more of the momentum, we saw that as angular acceleration decreases, inertia increases. Also when the mass is closer to the center, momentum is conserved and the object spins faster vs. as farther away the object spins slower. Although outside of this lab, moment of inertia does also determine the amount of kinetic energy that an object possesses from its rotational motion. The larger the momentum of inertia, the more energy is required to make the object spin faster or slower. Understanding the momentum of inertia is crucial in calculating the energy requirements and efficiency of various mechanical systems that involve rotational motion. In conclusion, by manipulating the demonstrator and observing its motion along with the various concepts it presents, we can develop a deeper understanding of these concepts and relate these to real-world scenarios. Introduction: The primary objectives of this lab experiment are: 1. To determine the moment of inertia of the spinning object 2. To modify the moment of inertia of the spinning object and moveable mass system and observe any changes in behavior 3. To compare the measured moment of inertia to a theoretical calculation 4. To adjust the torque on the spinning object and observe any changes in behavior. In this lab experiment, we will be using the Rotational Inertia Demonstrator to enhance our understanding of concepts such as moment of inertia, torque, and angular acceleration. The device consists of a rotating object that is affixed to a hub with low friction. The rotating object comprises four pulleys of different radius, and four thin rods extending radially from the pulley. A "moveable mass" can be attached to each of the rods using a set-screw. When we hang a "hanging mass" from a string wrapped around the pulley, the hanging mass produces tension in the string, which generates a torque on the system, resulting in the entire apparatus rotating. The torque can be adjusted by altering the suspended mass or the pulley that the string is wrapped around. Additionally, the moment of inertia of the system can be adjusted by moving the moveable masses.

followed each time. For the third experiment, the moveable masses were moved to the ends of each rod, and the distance between the rotation axis and the moveable mass was measured. Then, five trials were conducted using the smallest pulley and the times were recorded. Next, the string was wrapped around the second smallest pulley and five trials were conducted again. Data was recorded and compiled after each experiment. Various calculations were also done and are shown in the results and analysis section. Data: Setup: Measurement Symbol Value and measured unit Converted Value and SI unit Mass of one moveable mass Mm 185.6g 0.1856kg Mass of hanging mass Mh 200.2g 0.2002kg Diameter of smallest pulley D1 3.978cm 0.03978m Radius of smallest pulley R1 1.989cm 0.01989m Diameter of 2nd smallest pulley D2 5.556cm 0.05556m Radius of 2nd smallest pulley R2 2.778cm 0.02778m The weight values are recorded using a scale with a precision of 0.0g, and the length measurements are measured using a vernier caliper with a precision of 0.000cm. The conversion calculations are shown in the results and analysis section. Experiment One: Vertical Displacement of hanging mass = 83.8cm = 0.838m

This is the distance between the bottom of the hanging mass and the bottom of the table, measured using a tape measure with a precision of 0.0cm. The conversion is shown in the results and analysis section. Trial Time (s) 1 4.65 2 5.31 3 5.15 4 5.12 5 5.22 6 5.40 7 5.19 8 5.01 9 5.18 10 5.15 These are the time values recorded on the stopwatch for each trial in experiment one. Experiment Two: Distance between axis of rotation and center of moveable mass = 11.9cm = 0.119m This is the distance between the axis of rotation and the center of the moveable mass, measured using a tape measure with a precision of 0.0cm. The conversion is shown in the results and analysis section.

4. 9.86 13.56 5. 9.65 13.59 This table shows the times recorded for the five trials done on the two different pulleys. Results and Analysis: Conversions: Example Mass Conversion for Mm = 185.6g * (1kg / 1000g) = 0.1856kg Example length or distance conversion for D1 = 3.978cm * (1m / 100cm) = 0.03978m Experiment One: Trial Time (s) 1 4.65 2 5.31 3 5.15 4 5.12 5 5.22 6 5.40 7 5.19 8 5.01 9 5.18 10 5.15 Mean 5.14 St. Dev 0.201

St. Dev of Mean 0.0636 This table shows the values for the mean, standard deviation, and standard deviation of the mean for the first experiment. Calculations are shown below. Calculations: Mean = (4.65 + 5.31 + 5.15 + 5.12 + 5.22 + 5.40 + 5.19 + 5.01 + 5.18 + 5.15) / 10 = 5.14 St. Dev = (1/9 * (4.65 - 5.14)^2 + (5.31 - 5.14)^2 + (5.15 - 5.14)^2 + (5.12 - 5.14)^2 + (5.22 - 5.14)^2 + (5.40 - 5.14)^2 + (5.19 - 5.14)^2 + (5.01 - 5.14)^2 + (5.18 - 5.14)^2 + (5.15 - 5.14)^2))^(½) = 0.201 St. Dev of Mean = 0.201 / (10)^(½) = 0.0636 Calculation of acceleration of hanging mass: To find the acceleration in this experiment, one of the constant acceleration equations can be used. Here is the equation: X = Xo + Vo * t + ½ * a * t^2 In this equation, X is the final position, Xo is the initial position, Vo is the initial velocity, a is the acceleration, and t is the time. X - Xo will be the displacement we measured and time will be the mean time from the ten trials. a = (2 * (X - Xo - Vo * t)) / t^2 a = (2 * (0.838m)) / (5.14s)^2 = 0.0634m/s^2 Calculation of uncertainty in acceleration: Ua = a * ((Ux / x)^2 + (Ut / t)^2)^(½) In this equation, Ua represents the uncertainty of acceleration, Ux represents the uncertainty of the displacement, Ut represents the uncertainty of time, t represents time, a represents acceleration, and x represents displacement. Ux is assumed to be 0.5cm and Ut is assumed to be the standard deviation of the mean of the time previously calculated from the ten trials. Ua = 0.0634m/s^2 * ((0.005m / 0.838m)^2 + (0.0636 / 5.14)^2)^(½) = 8.71e-4

R * T = I * ang I = (R * T) / ang In this equation, the net torque is coming from the tension on the pulley. So, Nettor, or net torque, is just T tor, or the torque from the pulley. Since the torque due to tension force is equal to the radius of the pulley times the tension force, T tor can be replaced with R * T. As a result, I, the moment of inertia, can be solved for. I = (R * T) / ang I = (0.01989m * 1.95N) / 3.19rad/s^2 = 0.01216kg m^2 Experiment Two: Distance between axis of rotation and movable mass (m) Avg measured time (s) Calculated acceleration (m/s^2) Calculated angular acceleration (rad/s^2) Calculated tension in rope (N) Calculated torque (Nm) Measured moment of Inertia (kg m^2) 0.119 6.89 0.0353 1.78 1.96 0.0389 0.0219 0.169 8.23 0.0247 1.24 1.96 0.0390 0.0313 0.219 9.67 0.0179 0.901 1.96 0.0390 0.0433 0.269 11.56 0.0125 0.631 1.96 0.0390 0.0619 0.319 13.43 0.00929 0.467 1.96 0.0390 0.0835 The average time, calculated acceleration, calculated angular acceleration, calculated tension, and measured moment of inertia are all calculated the same way as in experiment one shown above. Calculating torque: The torque is calculated using this equation: Tor = R * F In this equation, Tor is the torque, R is the radius of the smallest pulley, and F is the tension in the rope T.

Tor = R * F Tor = 0.01989m * 1.96N = 0.0389Nm Calculate the theoretical moment of inertia of the entire spinning system: Distance between axis of rotation and movable mass (m) Measured moment of inertia (kg m^2) Theoretical moment of inertia (kg m^2) 0.119 0.0219 0.0227 0.169 0.0313 0.0334 0.219 0.0433 0.0478 0.269 0.0619 0.0659 0.319 0.0835 0.0877 The equation used to calculate the theoretical moment of inertia is as follows: I theory = I pulley+rod + 4 * Mm * r^2 In this equation, I pulley+rod is the moment of inertia calculated in experiment one, Mm is the mass of the moveable mass, and r is the distance between the axis of rotation and the moveable mass. I theory = I pulley+rod + 4 * Mm * r^2 I theory = 0.01216kg m^2 + 4 * 0.1856kg * (0.119)^2 I theory = 0.0227kg m^2 Experiment three: Pulley radius (m) Average Measured Time (s) Calculated Acceleration (m/s^2) Calculated Angular accelerated rad/s^2) Calculated Tension in rope (N) Calculated Torque (Nm) 0.01989 13.63 0.009022 0.4536 1.96 0.0390

the masses outward increases the moment of inertia. As a result, the object spins slower, which causes the angular acceleration to decrease. C) In Experiment 2, as you moved the masses outward along the rod, what trend did you observe in your moment of inertia measurements? Did the moment of inertia increase, decrease, or stay the same? Explain why this happened using physics concepts. Moving the masses outward along the rod increases the rotational inertia of the system, which decreases the angular acceleration of the system. So, the moment of inertia increases as the masses are moved outward. Again, this is because there is more area that the mass of the system has to cover using the same momentum, which causes the system to decrease in speed. D) In Experiment 2, as you moved the masses outward along the rod, what trend did you observe in your tension measurements? Did the tension in the string increase, decrease, or stay the same? Explain why this happened using physics concepts. According to the data, the tension measurements remained the same for all of the trials. This value remained the same because the net force acting on the weight needs to be zero. The force of gravity acting on the hanging mass is constant for all trials since it is the g constant times the weight of the mass. Therefore, for the net force to be zero, the force of tension in the rope would remain constant as well. E) In Experiment 2, did your theoretical moment of inertia match your measured value? Do you observe the same trend as you moved the moveable mass in your theoretical moment of inertia as you measured? The theoretical moment of inertia was very close to the measured values. According to the table, the theoretical values were slightly higher, but the difference between the values seemed to be around 0.0010kg m^2 to 0.0040kg m^2. The same trend was also observed in the theoretical values as in the measured values, since the values also increased as the mass was moved outwards.

F) In Experiment 3, what trend did you observe in the measured average time when you moved to the larger radius pulley? Explain why this happened using physics concepts. When we moved to the second smallest pulley, the average time decreased when compared to the same weight configuration on the smallest pulley. This is because as the radius increases, the amount of torque is greater. This causes the system to spin faster. G) In Experiment 3, what trend did you observe in the angular acceleration when you moved to the larger radius pulley? Explain why this happened using physics concepts. When the larger radius pulley is used, the angular acceleration increases. This is because there is more torque, which is causing the object to spin faster. H) In Experiment 3, what trend did you observe in the torque when you moved to the larger radius pulley? Explain why this happened using physics concepts Torque equals the R vector times the force times the angle between these two vectors. Since the pulley with the larger radius is being used, the torque increases. The data acquired shows that the torque increases when the second smallest pulley is used.