Experimental Investigation of Moment of Inertia: Mass, Radius, and Cylinder Type Effects

Moment of inertia, denoted as I, is a measure of an object's resistance to changes in rotational motion. It is influenced by factors such as mass, radius, and shape. Newton's 2nd Law, which relates net forces to accelerations, is employed to calculate moment of inertia. This law is applicable to both translational and rotational motion. The shape of an object also plays a crucial role in determining its moment of inertia.

Principles of Measurement: To measure moment of inertia, a specialized apparatus (TM 610) is utilized, as depicted in Figure 1.

The system involves drive weights subjected to gravitational force, inducing circular motion. The time taken for the drive weights to travel from a fixed height to the ground is recorded using a stopwatch. The measured time is then used to calculate the inertia of the object's motion. The following equations are employed:

1. Corrected Moment of Inertia: Icorrected​=Iexperiment​−Io​

Note:Note: corrected – moment of inertia calculated from experimental data but corrected with inherent moment of inertia.Icorrected​ – moment of inertia calculated from experimental data but corrected with inherent moment of inertia. experiment – moment of inertia calculated from experimental data.Iexperiment​ – moment of inertia calculated from experimental data. – Inherent moment of inertia for rotational axle and pipe, a constant with a value of 3.3×10−3 kg m2Io​ – Inherent moment of inertia for rotational axle and pipe, a constant with a value of 3.3×10−3 kg m2

2. Moment of Inertia Calculated from Experimental Data: experiment=…Iexperiment​=… (specific formula not provided)
3. Equations for Calculating Moment of Inertia Depending on Cylinder Type:
   • Solid Cylinder: Ix​=m⋅d2
   • Hollow Cylinder: Ix​=m⋅(D2+d2)

In essence, the experiment involves measuring the time of circular motion and utilizing it to determine the moment of inertia, correcting for inherent values.

Different equations are applied based on the type of cylinder being examined (solid or hollow).

Equipment:

1. Rotational motion inertia investigation apparatus (TM 610)
2. Digital stopwatch (or smartphone with stopwatch function)
3. Steel meter ruler
4. Set of weights (100g, 200g, and 400g)

Procedure:

Figure 3: Mounting the pin-point masses

A. Pin Point Mass: Moment of Inertia as a Function of Mass

1. Set up the TM 610 rotational motion inertia apparatus on a table.
2. Insert the thin-walled pipe into the TM 610 apparatus, placing it at the center of the rotational axle.
3. Fasten two 100g weights on both sides of the pipe.
4. Fix the effective radius (R) at 20cm.
5. Use a 100g drive mass for all experiments in this section.
6. Adjust the drive mass to a height of 40cm from the ground for each experiment. Measure the elapsed time (t) using the stopwatch, starting when the drive mass begins to drop and stopping when it reaches the ground.
7. Repeat the experiments three times at the same heights to minimize time measurement errors. After three trials, replace the two 100g weights with 200g and 400g weights.
8. Record the average time obtained in the results table. Calculate I_experiment and I_corrected. Plot graphs to identify the relationship between rotational inertia and mass.

B. Pin Point Mass: Moment of Inertia as a Function of Radius

1. Follow the procedure in Part A, fixing the pipe to the rotational axle.
2. Fix two 400g weights on the pipe, placing one on each side.
3. Set the effective radius (R) sequentially at 0.055m, 0.095m, 0.155m, and 0.245m.
4. Use a 100g drive mass for these experiments.
5. Adjust the drive mass to a height of 40cm for each experiment. Measure the elapsed time using the stopwatch.
6. Repeat the experiments three times at the same heights. After three trials, adjust the radius (R) as mentioned in step 3.
7. Record the average time in the results table. Calculate I_experiment and I_corrected. Plot graphs to identify the relationship between rotational inertia and radius.

C. Comparative Investigation: Solid Cylinder vs. Hollow Cylinder

1. Fasten the hollow cylinder to the rotational axle, removing the pipe.
2. Use a 100g drive mass for this set of experiments.
3. Adjust the drive mass to a height of 40cm, measuring the elapsed time with the stopwatch.
4. Repeat the experiments three times at the same heights. Then, replace the hollow cylinder with a solid cylinder and repeat the experiment.
5. Record the average time in the results table. Calculate I_experiment and I_corrected. Plot graphs to identify differences between the hollow and solid cylinders.

RESULTS
A. Pin Point Mass : Moment of inertia as a function of mass

B. Pin Point Mass : The moment of inertia as a function of radius

C. Comparative Investigation : Solid cylinder-hollow cylinder

Discussion:

1. The obtained results and graphs suggest that the moment of inertia of the rigid body is directly proportional to the weights attached to the thin pipe. The graph shows an increase in the time taken by the driver weights as the attached weights increase. This implies an increase in the moment of inertia as well. This is because the added weights make the thin pipe heavier, resulting in a harder rotation and longer travel time for the driver weights to reach the ground. Consequently, the moment of inertia of the rigid body increases.
2. Analyzing the results and graphs, we can infer that the moment of inertia of the rigid body is directly related to the effective radius. The graph illustrates an increase in the time taken by the driver weights as the effective radius grows. This signifies a corresponding increase in the moment of inertia. The reason behind this is that an increased effective radius makes the rotation of the thin pipe more challenging. Consequently, the time required for the driver weights to travel to the ground lengthens, leading to an increase in the moment of inertia of the rigid body.
3. When comparing the hollow and solid cylinders, significant differences in the moment of inertia readings are observed. The moment of inertia for the hollow cylinder is almost twice that of the solid cylinder, with readings of 0.0841 kgm² and 0.0424 kgm², respectively. However, these values differ significantly from the calculations using general equations (0.00323 kgm² for hollow and 0.0018 kgm² for solid). One reason for this difference is that the general equation assumes an even distribution of particles within the cylinder, while the experimental results indicate uneven distribution, leading to variations in the moment of inertia.

Precautions:

1. It is crucial to avoid systematic errors, such as zero errors, when measuring the height of the drive weights from the ground to ensure accurate readings.
2. Random errors, like reaction time, can be minimized by repeating experiments multiple times, allowing for the determination of the average travel time of the driver weights.

Conclusion: In conclusion, the moment of inertia of a rigid body is influenced by two variables: mass and effective radius. Both of these variables exhibit a direct proportionality to the moment of inertia, meaning it increases with an increase in mass or effective radius. Additionally, the hollow cylinder demonstrates a larger moment of inertia compared to the solid cylinder, with the former being approximately twice the latter. This discrepancy arises from the assumption in the general equation that all particles are evenly distributed, whereas the experimental results indicate uneven distribution, resulting in varying moment of inertia values. The general equation predicts a moment of inertia of 0.00323 kg m² for a hollow body and 0.0018 kg m² for a solid cylinder.