PHYS111Q : Labs

9 Rotational Motion

Introduction

**Figure 7:** Rotational motion apparatus.
$\includegraphics{rot-fig1.eps}$

Figure 7 shows a diagram of the apparatus you will use to measure rotational inertia. The object of interest is mounted on a rotational motion detector so that it rotates about the axis of interest. A torque is applied to the system via a mass hanging from a thread wrapped around a spool of radius . The moment of inertia of the object of interest can then be deduced from the resulting angular acceleration of the system as follows. The net torque $\tau$ exerted on the system is related to its resulting angular acceleration $\alpha$ by the rotational version of Newton's second law,

$\begin{displaymath} I \alpha = \tau_{net} \end{displaymath}$

(36)

where

is the rotational inertia of the system. Since the tension

in the string is the only force, other than a small frictional force, acting on the system, and this tension is applied tangent to the spool, the net torque is given by

$\begin{displaymath} \tau_{net} = T r_s \end{displaymath}$

(37)

Newton's second law for the hanging mass is

$\begin{displaymath} ma = mg - T \end{displaymath}$

(38)

where we have chosen a coordinate system in which downward motion is positive for convenience. The string is not slipping against the spool, so the angular and linear accelerations are related via

$\begin{displaymath} a = r_s \alpha \end{displaymath}$

(39)

Combining Eq.'s 36-39, we find

$\begin{displaymath} I = mr_s^2 \left(\frac{g}{r_s \alpha} - 1 \right) \end{displaymath}$

(40)

The theoretical rotational of inertias of the objects you will be studying are

$\displaystyle I_\mathrm{mass}$	$\textstyle =$	$\displaystyle Mr^2$
$\displaystyle I_{rod}$	$\textstyle =$	$\displaystyle \frac{1}{12} M L^2$
$\displaystyle I_{disk}$	$\textstyle =$	$\displaystyle \frac{1}{2}MR^2$
$\displaystyle I_{ring}$	$\textstyle =$	$\displaystyle \frac{1}{2}M(R_\mathrm{in}^2 + R_\mathrm{out}^2)$	(41)

where

is the radius of the orbit of the mass, $R_\mathrm{in}$ and $R_\mathrm{out}$ are the inner and outer radii of the ring, and

is the length of the rod. Rotational inertias combine (like masses) by simple scalar addition, so the predicted rotational inertia of the rod-mass system is given by

$\begin{displaymath} I_\mathrm{rod-mass} = \frac{1}{12} m_\mathrm{rod} L^2 + m_A r_A^2 + m_B r_B^2 \end{displaymath}$

(42)

According to parallel axis theorem, if we know the rotational inertia $I_\mathrm{CM}$ of an object of mass rotating about an axis through its center of mass, then its rotational inertia about an axis parallel to this original axis and a distance from it is given by

$\begin{displaymath} I = I_\mathrm{CM} + Mh^2 \end{displaymath}$

(43)

Experiments

Please be gentle with the equipment - instead of stopping it from spinning abruptly, slow it down gradually.

Moment of Inertia of a Rod-mass System

Measure the masses of the hanging mass, the rod, and the two brass cylinders.
Use the Vernier caliper to measure the radius of the spool. Be sure to measure the radius of the level the thread actually winds around. This should be the middle level. (You need 0.1 mm precision here. If you've never used a Vernier caliper, ask for help.)
Measure the length of the rod, and the distances from the center of the rod to the center of each of the masses.
Run Logger Pro and load the file RotationalMotion.cmbl provided to you in advance of lab.
Mount the rod and masses on the rotational motion detector. (The spool must be oriented with its largest radius on top.)
Rotate the system to wind the thread onto the spool, begin acquiring data with Logger Pro, and release the system from rest.
Collect several clean measurements.

Moment of Inertia of a Ring

Measure the masses of the ring and the aluminum disk.
Measure the inner and outer radii of the ring and the radius of the disk.
Mount the aluminum disk on the rotational motion detector. (The spool must be oriented with its smallest radius on top.)
Collect several clean measurements of (a) the aluminum disk alone and (b) the aluminum disk and the ring together.

Testing the Parallel Axis Theorem

Mount the rod-mass system to one of the two threaded holes on the outer rim of the aluminum disk.
Measure the radius of the aluminum disk. (The threaded holes are at a distance from the axis of rotation equal to the radius of the disk.)
Collect several clean measurements of the system.

Analysis

We will assume that the plastic pulley and spool have negligible moments of inertia. We will also treat the brass cylinders as point masses.

Measured Rotational Inertias

Devise a method of extracting a ``best value'' of the angular acceleration of each of the systems you studied, along with the associated uncertainty. Discuss your method with your instructor before carrying it out.
Set up a spreadsheet calculation using Eq. 40 to find the measured moment of inertia of each system.
The uncertainty in the measured angular acceleration dominates these results, so the percentage uncertainty of the moment of inertia is the same is the percentage uncertainty of the associated angular acceleration.
Subtract the measured moment of inertia of the aluminum disk from the combined disk and ring value to find the moment of inertia of the ring. (Follow the addition/subtraction rule to find the associated uncertainty.)

Theoretical Predictions

Use your measured masses and dimensions to calculate theoretical predictions of the moments of inertia for each system via Eq.'s 41 and the parallel axis theorem (Eq. 43).
The uncertainties in your length measurements are dominant here. For each theoretical prediction, use as its uncertainty $2 \times$ the percentage uncertainty in the least precise length used in the prediction.

Group Assignment

Show your instructor your spreadsheet and discuss your response to question 4 below.

Individual Assignment

Hand in a hard copy of your spreadsheet and answers to the following questions

Combine Eq.'s 36-39 to obtain Eq. 40.
Describe your method of extracting angular accelerations and uncertainties from your measurements.
Explain why you used the factor of in finding the uncertainties in the theoretical rotational inertias.
In a table, summarize your measured results and theoretical predictions, with uncertainties, and discuss the compatibility of your measurements with the theoretical predictions.