Lab 7 - Angular Momentum

The moment of inertia of the rotating object is unknown. In this part of the lab, you will apply an external torque to the rotating object and measure the angular acceleration of the rotation by means of the equation \(\sum \tau = I \alpha\).

1. Remove all 4 masses from the arms of the pulley. Set up the pulley so that it is approximately 90 cm above the surface of the table. Put a 500g mass on the end of the string such that the mass is approximately 50cm above the table.

2. Measure the angular distance that the pulley travels when the mass moves from 50 cm above the table to the surface of the table.

   \(\Delta \theta:~\rule{3cm}{0.25mm}~\)

3. Let the mass fall. Measure the time needed for the mass to fall this distance 5 times and take the average of these measurements for your final time.

   Trial	\(\qquad \qquad t~(s) \qquad \qquad\)
   1
   2
   3
   4
   5

4. Calculate the average \(\mathbb{E}[t]\) and the standard deviation \(\sigma^2 = \mathbb{E}[t^2] - \mathbb{E}[t]^2\)

   \(t_{\textnormal{ave}}:~\rule{3cm}{0.25mm}~\) \(\sigma_t:~\rule{3cm}{0.25mm}~\)

5. Use rotational kinematics to calculate the angular acceleration of the rotating mass. Recall kinematics requires constant acceleration. Is this a reasonable assumption? Where could external torque be found and is this torque linear?

   \(\alpha:~\rule{3cm}{0.25mm}~\)

6. Measure the radius of the pulley from the center of rotation to the string’s point of final contact. Draw a Free Body Diagram and write out Newtons Laws for Force and Torque:

   \[\sum F_x = m a_x\] \[\sum F_y = m a_y\] \[\sum \tau_z = I \alpha\]

   \(I_p:~\rule{3cm}{0.25mm}~\) \(\tau :~\rule{3cm}{0.25mm}~\)

   These are only approximations to the actual values (assumption that \(\alpha\) is small and the tension is assumed to be the hanging weight)

Lets now estimate the theoretical value of the moment of inertia. In order to perform this calculation, some assumptions must be made. Assume that the entire center pulley is made out of a solid chunk of aluminum of uniform density (ignore the center section with the ball bearing).

1. Using the moment of inertia of a disk (\(I_{\text{disk}}= \frac12 MR^2\)), calculate the moment of inertia of the center rotating section. This calculation involves summing the moment of inertia of a number of individual disks, each computed with the above formula. There are 12 separate disks on the pulley which will need to all be summed. The density of aluminum is \(\rho_{\text{Al}} = 2700 kg/m^3\).

   \(I_{\text{Al}}:~\rule{3cm}{0.25mm}~\)

2. Now calculate the moment of inertia of the 4 extended rods. This rotation is not taking place at the center of mass of the object, thus we will use the parallel axis theorem to calculate this moment of inertia:

   \(I = I_{CM} + MD^2\)

   The center-of-mass moment-of-inertia of a rod was calculated in lecture, so the first term is simply \(I_{CM} = 1/12 ~MR^2\). To find the distance \(D\), measure the distance from the center of mass of the rods to the center of rotation of the entire object. Also measure the mass of one of the metal rods.

   \(D:~\rule{3cm}{0.25mm}~\) \(M:~\rule{3cm}{0.25mm}~\)

   \(I_{\text{rod}}:~\rule{3cm}{0.25mm}~\)

   Finally, add all of the moments above to find the overall estimate of the moment of inertia:

   \(I = I_{\text{Al}} +4 I_{\text{rod}}:~\rule{3cm}{0.25mm}~\)

   How does this value compare to the value you experimentally determined in part 1?

Now you will add movable masses into the system. Each of the rods requires a mass, and these masses will change the moment of inertia of the system.

1. Measure the mass of the weights which attach to the 4 rods. Add masses at the end of the rods so they are flush with the end of the rods. Measure the distance from the center of rotation (center of the pulley) to their center of mass.

   \(M:~\rule{3cm}{0.25mm}~\) \(R:~\rule{3cm}{0.25mm}~\)

1. These new moments of inertia can now be summed and added to the original pulley moment of inertia (\(I_p\)) find a total moment of inertia of the system. Remember the moment of inertia of a mass \(M\) rotating a distance \(R\) from the rotation axis is \(I_m = M~R^2\). Calculate the total moment of inertia:

   \[ I = I_p + \sum I_m \]

   \(I_{out}:~\rule{3cm}{0.25mm}~\)

2. Using this moment of inertia, find the time needed for the mass to fall 50 cm with the masses added to the rotational system. Then measure the time and calculate the percent error between the calculated and measured values.

   \(\textnormal{Calculated }t_{out}:~\rule{3cm}{0.25mm}~\)

   \(t_{out,1}:~\rule{3cm}{0.25mm}~\) \(t_{out,2}:~\rule{3cm}{0.25mm}~\) \(t_{out,3}:~\rule{3cm}{0.25mm}~\) \(t_{out,ave}:~\rule{3cm}{0.25mm}~\)

   Percent error:\(~\rule{3cm}{0.25mm}~\)

Now you will modify the rotational distance which the masses are located, calculate the moment of inertia, and finally predict the time required for this new system to fall the required distance.

1. Add masses at the end of the rods so they are at their minimum distance. Measure the distance from the center of rotation (center of the pulley) to their center of mass.

   \(R:~\rule{3cm}{0.25mm}~\)

2. Calculate this new moment of inertia when the masses are at their minimum distance.

   \[ I = I_p + \sum I_m \]

   \(I_{in}:~\rule{3cm}{0.25mm}~\)

3. Find the time needed for the mass to fall 50 cm with the masses added to the rotational system. Then measure the time and calculate the percent error between the calculated and measured values.

   \(\textnormal{Calculated }t_{in}:~\rule{3cm}{0.25mm}~\)

   \(\textnormal{Measured }t_{in,1}:~\rule{3cm}{0.25mm}~\) \(t_{in,2}:~\rule{3cm}{0.25mm}~\) \(t_{in,3}:~\rule{3cm}{0.25mm}~\) \(t_{in,ave}:~\rule{3cm}{0.25mm}~\)

   Percent error:\(~\rule{3cm}{0.25mm}~\)