PHYS111Q : Labs

10 Rotational Energy and Angular Momentum

Introduction

In the process of adding rotational motion to our models of kinematics and dynamics, we have introduced the concepts of rotational kinetic energy and angular momentum. We must include rotational kinetic energy in order to apply the principle of conservation of energy to systems involving rotational motion. Angular momentum is a new (to us) quantity that is conserved in a similar way to linear momentum. That is, the total angular momentum of an isolated system is conserved even when energy is not.

Energy of a Mass-pulley System

**Figure 8:** A hanging mass-pulley system.
$\scalebox{0.9}{ \includegraphics{rotEL-fig1.eps} }$

Figure 8 shows a diagram of a system of two hanging masses and connected by a string running over a pulley with moment of inertia . If the hanging masses are not equal, then the system will accelerate when released from rest. The pulley has rotational inertia, so the kinetic energy of the system is a combination of linear and rotational terms. The kinetic and potential energies of the system are expressed as

$\displaystyle K$	$\textstyle =$	$\displaystyle \frac{1}{2} m_A v^2 + \frac{1}{2} m_B v^2 + \frac{1}{2} I \omega^2$
$\displaystyle U$	$\textstyle =$	$\displaystyle m_A g y_A + m_B g y_B$	(44)

The hanging masses are connected by a string that does not stretch. If mass 1 moves upward a distance $\Delta y_1 = d$ as shown in Fig. 8, then mass 2 moves the same distance downward $\Delta y_B = -d$ . A convenient coordinate system for describing changes in the vertical positions of the masses (for calculating potential energy, for example) is therefore

$\begin{displaymath} y_A = -y_B \end{displaymath}$

(45)

Furthermore, the string does not slip over the pulley, so the distance

is related to the angular position of the pulley in radians via

$\begin{displaymath} d = R \theta \end{displaymath}$

(46)

and the angular and linear velocities are related via

$\begin{displaymath} v = R \omega \end{displaymath}$

(47)

With Eq.'s 45-47, the energy expressions of Eq.'s 44 become

$\displaystyle K$	$\textstyle =$	$\displaystyle \frac{1}{2} \left( m_A R^2 + m_B R^2 + I \right) \omega^2$	(48)
$\displaystyle U$	$\textstyle =$	$\displaystyle (m_A - m_B ) g R \theta$	(49)

If no work is done by forces other than gravity, and energy is conserved, then the total mechanical energy of the system is constant. All of the quantities in Eq.'s 48 and 49 can be readily measured. You will have an opportunity to test conservation of energy for this system using a rotational motion detector and Logger Pro.

A Rotational Collision

**Figure 9:** A perfectly inelastic rotational collision.
$\scalebox{0.7}{ \includegraphics{rotEL-fig2.eps} }$

Fig. 9 shows a perfectly inelastic (``sticky'') rotational collision in which an initially stationary object with moment of inertia is dropped onto another object with moment of inertia initially spinning with angular velocity $\omega_{A1}$ . After the collision, both objects spin together with angular velocity $\omega_2$ .

The total angular momentum of an isolated system is conserved. The angular momentum of an object with moment of inertia and angular velocity $\omega$ is given by

$\begin{displaymath} L = I \omega \end{displaymath}$

(50)

Conservation of angular momentum for a two-body rotational collision is expressed

$\begin{displaymath} L_{A1} + L_{B1} = L_{A2} + L_{B2} \end{displaymath}$

(51)

For the perfectly inelastic rotational collision shown in Fig. 9, Eq. 51 becomes

$\begin{displaymath} I_A \omega_{A1} = (I_A + I_B) \omega_2 \end{displaymath}$

(52)

You will test conservation of angular momentum for perfectly inelastic rotational collisions between the ring and disk you considered in Lab 9.

Experiments

Note : In order to analyze these measurements, you will need the moments of inertia of the aluminum disk and the heavy ring with uncertainties, which you measured as part of Lab 9.

Energy of a Mass-pulley System

Use the Vernier caliper to measure the diameter of the largest spool on the rotational motion sensor (RMS).
Mount the RMS on the horizontal bar, away from the table so that the masses can hang freely from the pulley. The aluminum disk should be mounted on the RMS.
Hang the two 50 g mass hangers over the largest spool of the RMS using the thread provided.
Run Logger Pro and load the file RotationalEnergy.cmbl.
Add a 10 g mass to one of the mass hangers, and measure the motion of the system as it accelerates. The system will slow down and stop on its own once the heavier mass hits the floor.
Collect measurements of five clean drops. (Reject drops during which the masses collide as they pass one another.)

A Rotational Collision

Mount the RMS on the vertical bar so that its rotational axis is aligned vertically.
Measure the mass of the ring.
Run Logger Pro and load the file RotationalMotion.cmbl.
Start acquiring RMS data with Logger Pro, and start the aluminum disk spinning very rapidly ( $\omega_1 > 60$ rad/s).
Carefully hold the ring, smooth-side-down, centered over the disk. Gently drop the ring onto the disk from a height of no more than a few millimeters.

Figure 10: A typical collision in which the ring lands a distance off-center.

$\scalebox{0.7}{ \includegraphics{rotEL-fig3.eps} }$
It is very unlikely that the ring will be centered on the disk after the collision. Instead, it will usually end up with its center of mass some distance from the axis of rotation of the system as shown in Fig. 10. You will need to correct the moment of inertia of the ring for this offset using the parallel axis theorem. The distance is difficult to measure directly, but it is related to the shortest and longest distances, and between the outside edges of the ring and disk by

$\begin{displaymath} h = \frac{b-a}{2} \end{displaymath}$ (53)

After you have your angular velocity data, and before you remove the ring from the disk, measure and as precisely as you can by using the metal rod that emerges from the end of the Vernier caliper.
Collect data for at least five rotational collisions. Make sure that you have angular velocity data both before and after each collision.

**Figure 10:** A typical collision in which the ring lands a distance off-center.
$\scalebox{0.7}{ \includegraphics{rotEL-fig3.eps} }$

Analysis

Energy of a Mass-pulley System

Note : The aluminum disk provides rotational inertia, but does not act as the actual pulley, so in the analysis of your measurements, the radius in Eq.'s 46-49 is the radius of the largest spool of the rotational motion detector, not the radius of the aluminum disk.

Put your measured , , , and values into the calculation of the kinetic energy in Logger Pro. Click on Data -> Column Options -> Kinetic energy to bring up a Calculated Column Options widow. Choose the Column Definition tab. In the ``Equation'' box, you should see
1/2*((0.05+0.06)*0.02^2 + 0.0001)*"Angular Velocity"^2
which follows the form of Eq. 48 with g, g, kgm and cm. Replace the values of and with your measured values, and click Ok.
Put your measured , , , and values into the calculation of the potential energy in Logger Pro. Click on Data -> Column Options -> Potential energy to bring up a Calculated Column Options widow. Choose the Column Definition tab. In the ``Equation'' box, you should see
(0.05-0.06)*9.8*0.02*abs("Angle")
which follows the form of Eq. 49 with g, g, kgm and cm. Replace the value of with your measured value, and click Ok.
Devise and carry out a strategy to determine the average rate at which the mechanical energy of this system is converted into nonconservative forms, along with its uncertainty.

A Rotational Collision

For each collision, devise and carry out a strategy for determining $\omega_{A1}$ and $\omega_2$ from your measurements using Logger Pro.
Use Excel to calculate and for all of your collision measurements. In calculating , we must correct for the fact that the ring was off-center using the parallel axis theorem, so

$\begin{displaymath} L_2 = (I_A + I_B + m_\mathrm{ring} h^2) \omega_2 \end{displaymath}$ (54)
Calculate the percentage change in angular momentum for each collision. Then, calculate the average percentage change in momentum and the standard deviation of the mean.

Group Assignment

Show your work to your instructor and discuss preliminary answers to the questions below.

Individual Assignment

Hand in a printout of your spreadsheet and answers to the following questions.

Report your result with uncertainty for the average rate at which the mechanical energy of the mass-pulley system is converted into nonconservative forms of energy. Is your result significant? Explain.
Give the average percentage change in angular momentum, and its uncertainty (the SDOM), for the rotational collisions you observed.
Are your measurements of rotational collisions compatible with with the law of conservation of angular momentum? Explain.