PES 1150 post 11

Moment of Inertia Name: Objective The purpose of this lab was to determine how to measure the moments of inertia with a rotating body and how the moment of inertia changes with mass and location. The other purpose of this lab was to recognize what rotational inertia was and how it changes with the size, shape, and mass distribution of an object. Data and Calculations Part I: Moment of Inertia of the apparatus (I o ) 1.) Measure the hanging mass and the radius of the pulley. Hanging Mass (scale) = 0.1426 kg Pulley radius = 0.01872 m

P E S 1 1 5 0 - G E N E R A L P H Y S I C S L A B I 2.) Remove all masses from the rotating arm to get the Moment of inertia of just the bare arm system . Retain a sample graph of the experiment with all the needed curve fit information. 3.) Average at least 3 drops to improve the value of your final measurement. Trial Angular acceleration (rad/s 2 ) 1 2.20 2 2.24 3 2.26 4 2.25 Average 2.2375 (2.24) 4.) From this data calculate the Moment of inertia of the bare arm (I o ). Moment of Inertia - 2

P E S 1 1 5 0 - G E N E R A L P H Y S I C S L A B I 5.) From this data calculate the Moment of inertia of the point-mass system (I 12 ). Experiment Total Inertia of the system ( point- masses + arm ) (kg · m 2 ) Inertia of the point-masses I 12 (kg · m 2 ) 1 0.03318 0.02154 2 0.03020 0.01856 3 0.02771 0.01607 4 0.02580 0.01406 Moment of Inertia - 4 𝐼 1, 𝑇 = ℎ 𝑚 1( 𝑟 1) 2 𝐼 2, 𝑇 = ℎ 𝑚 2( 𝑟 2) 2 𝐼 1, 𝑇 =( ℎ 0.2703kg)( 0.20m) 2 =0.010812kgm 2 𝐼 2, 𝑇 =( ℎ 0.2682kg)(0.20m) 2 = 0.010728kgm 2 I=0.010812 kgm 2 +0.010728kgm 2 = 0.02154 kgm 2 𝐼𝑇𝑜𝑡 , 𝑇 = ℎ 𝐼𝑜 + 𝐼 1, 𝑇 + ℎ 𝐼 2, 𝑇 ℎ 𝐼𝑇𝑜𝑡 , 𝑇 = ℎ 0.01164 kgm 2 + 0.010812kgm 2 + 0.010728kgm 2 𝐼𝑇𝑜𝑡 , 𝑇 =0.03318kgm ℎ 2

P E S 1 1 5 0 - G E N E R A L P H Y S I C S L A B I 5 0.02417 0.01253 6 0.02312 0.01148 Results and Questions Part I: Moment of Inertia of the apparatus (I o ) 1.) Restate the experimental value for the rotational platform I o : 2.) Would doubling the hanging mass change the value of I o ? What will change? Doubling the hanging mass would change the value of the I o. The mass doubling would double the I o because the equation is I o = r ❑ ( mg ) − ( mr 2 ) . 3.) How would decreasing the radius of the pulley change the measurement of I o ? What other values would be affected. Will the final value of the moment of Inertia change? If the radius of the pulley was decreased than the measurement of I o would also decrease. The final value of the moment of Inertia would change and would be lower when the radius was decreased. 4.) Would you classify the moment of inertia as an intrinsic property of an object or something that can be changed with the modification of other variables? The moment of inertia would be something that can be changed with the modification of other variables. Some of the variables which change the moment of inertia were the angular acceleration, the mass, and the radius. Part II: Moment of Inertia of point-masses (I 12 ) 1.) The experimental value of the moment of inertia of the point-mass system is a combination of the inertia of each point mass plus the apparatus (I o ). I 12 total = I 12 + I o (Experimental) Moment of Inertia - 5 I o (arm) = 0.01164 kg · m 2

P E S 1 1 5 0 - G E N E R A L P H Y S I C S L A B I 1 0.010812 0.010728 0.03318 4.) Compare the Experimental value to the Theoretical value for the experiment with a percent difference calculation. Experimen t Total Experimental Inertia I 12 exp (kg · m 2 ) Total Theoretical Inertia I 12 th (kg · m 2 ) Percent Difference (%) 1 0.03858 0.03318 15.1 Conclusion The objective of this lab was to learn how to obtain the moment of inertia of a moving apparatus and how this changed with different masses and locations. The second objective of this lab was to determine the rotational inertia and how it differed with size, shape, and mass distribution of an object. The first part of this lab was to determine the moment of inertia of a moving apparatus. This was done by measuring the radius of the drum/the pulley of the apparatus then adding a hanging mass on a string which allowed the spinning apparatus to move. The hanging mass was measured on a scale. This was completed four times. The angular acceleration was found from the slope of the angular velocity vs time graph, the average of the four runs was taken to get the angular acceleration. Finally, the moment of inertia was found by pulling in the data into the equation I o = r ❑ ( mg ) − ( mr 2 ) . This gave the I o a value of 0.01164kgm 2 . The second part of this lab had the same set up but two masses were placed on the apparatus. The masses were measured on a scale. The masses were placed at different radii for the two masses for the six trials. The theoretical and experimental yield was found for each run. The inertia of the point masses were found Moment of Inertia - 7 𝐼𝑇𝑜𝑡 , 𝑇 = ℎ 𝐼𝑜 + 𝐼 1, 𝑇 + ℎ 𝐼 2, 𝑇 ℎ =0.010812 kgm 2 + 0.010728 kgm 2 + 0.01164 kgm 2 =0.03318 kgm 2 % difference = | value 1 − value 2 | 1 2 ( value 1 + value 2 ) × 100% % difference = | 0.03858 − 0.03318 | 1 2 ( 0.03858 + 0.03318 ) × 100%

P E S 1 1 5 0 - G E N E R A L P H Y S I C S L A B I using the equation I point-mass = m R 2 . The total theoretical moment of inertia was found by adding together the two-point masses moment of inertia and the moment of inertia of the arm which was found in the first part of the lab. The experimental moment inertia for the system was found by using the I o = r ❑ ( mg ) − ( mr 2 ) . Finally, a percent difference was taken of the experimental and theoretical values which had a value of 15.1%. This percent value indicated that the theoretical value was not an accurate value and did not verify the idea of moment of inertia being found two ways. For the experiment and the data to be accurate the percent difference was expected to be 1.5%. This experiment did not verify that the moment of inertia could be measured by adding the inertias of the point mass with the inertia of the arm or by using the angular acceleration with the equation I o = r ❑ ( mg ) − ( mr 2 ) . However, this experiment did verify the idea that the moment of inertia will change when the radius of the masses differs. Not only did this data change the total theoretical moment of inertia of the system but the data followed a pattern. When the radius of the first was held constant but the radius of the second mass was decreased than the angular acceleration was increased. With this increase of the angular acceleration, the total rotational/moment of inertia was decreased. This experiment had multiple experimental errors which lead to this inaccurate data. Some errors could be the processing of the computer or if the data was not measured accurately or precisely by the student. If the computers could not process the data correctly than this could change the experimental moment of inertia. The equation for the experimental moment of inertia was dependent on the angular acceleration which was being measured by the computer. If this data was not processed accurately by the computer than the experimental value would change which would also make the percent difference greater. The other possible error was if the mass of the point masses and the hanging mass or radius of the point masses and the pulley were not measured precisely by the study the moment of inertia would be wrong and cause the higher percent difference. Moment of Inertia - 8