**Lab 8: Rotational Motion**

### Learning Objective

To discover and understand the relationship of torque and rotational inertia to angular acceleration on the basis of both observations and theory.

**Introduction**

SOME BACKGROUND INFO ON ROTATIONAL MOTION WILL GO HERE!!

In this lab we will study the rotational motion of various objects using the rotational motion sensor. When you open the <filename> worksheet on the computer you will see that there is a space for an angular position vs time graph, an angular velocity vs time graph, and an angular acceleration vs time graph.

### Experimental Verification that τ = *I*α for a Rotating Disk

In the last session, you used the definition of rotational inertia, I, and spreadsheet calculations to determine a theoretical equation for the rotational inertia of a disk. This equation was given by

$$I = \frac{1}{2}MR^2$$

Does this equation adequately describe the rotational inertia of a rotating disk system? If so, then we should find that, if we apply a known torque, τ, to the disk system, its resulting angular acceleration, α, is actually related to the system’s rotational inertia, I, by the equation

τ = Iα

or

$$\alpha = \frac{\tau}{I}$$

The purpose of this experiment is to determine if, within the limits of experimental uncertainty, the measured angular acceleration of a rotating disk system is the same as its theoretical value. The theoretical value of angular acceleration can be calculated using theoretically determined values for the torque on the system and its rotational inertia.

The following apparatus will be available to you:

• A Rotating Cylinder System

• A 20 g or 50 g hanging mass (for applying torque)

• A clamp stand to mount the system on

• String

• A metre stick and a ruler

• A Motion Detection System

• A scale for determining mass

### Theoretical Calculations

You’ll need to take some basic measurements on the rotating cylinder system to determine theoretical values for *I* and τ. Values of rotational inertia calculated from the dimensions of a rotating object are theoretical because they purport to describe the resistance of an object to rotation. An experimental value is obtained by applying a known torque to the object and measuring the resultant angular acceleration.

### ✍ Activity 12-14: Theoretical Calculations

(a) Calculate the theoretical value of the rotational inertia of the stack of CDs using basic measurements of its radius and mass. Be sure to state units! If there’s a significant size hole in the centre, try to estimate the error involved in ignoring it.

For the CDs, ignoring the hole:

r_{d} = M_{d} =

I_{d} =

For the hole in the centre:

*r*_{h} = *M*_{h} =

I_{h} =

*I*_{CD} = *I*_{d} –* I*_{h} =

(b) Calculate the theoretical value of the rotational inertia of the spool using basic measurements of its radius and mass. Note: You’ll have to do a bit of estimation here. Be sure to state units.

rs = Ms =

Is =

(c) Calculate the theoretical value of the rotational inertia, I, of the whole system. Don’t forget to include the units. Note: By noting how small the rotational inertia of the spool is compared to that of the disk, you should be able to convince yourself that you can neglect the rotational inertia of the rotating axle in your calculations.

*I* =

(c) In preparation for calculating the torque on your system, summarize the measurements for the falling mass, m, and the radius of the spool in the space below. Don’t forget the units!

m = rs =

(d) Use the equation you derived in parts (b) and (c) of problem SP12-4 to calculate the theoretical value for the torque on the rotating system as a function of the magnitude of the hanging mass and the radius, rs, of the spool.

τ =

(e) Based on the values of torque and rotational inertia of the system, what is the theoretical value of the angular acceleration of the disk? What are the units?

αth =

### Experimental Measurement of Angular Acceleration

Devise a good way to measure the angular acceleration, α, of the CD stack caused by the torque applied by a hanging weight. The rotational motion sensor can measure the angle of rotation directly and has pulleys for a string from which to hang a mass.

You will need to take enough measurements to find a standard deviation for your measurement α. Can you see why it is desirable to make several runs for this experiment? Should you use a spread sheet?

*Note: The acceleration of the falling mass is not the gravitational acceleration, g.*

If you choose to use a graphical technique to find the acceleration be sure to include a copy of your graph and the equation that best fits the graph. Also show all the equations and data used in your calculations. Discuss the sources of uncertainties and errors and ways to reduce them.

### ✍ Activity 12-15: Experimental Write-up for Finding α

Describe your experiment in detail in the space below. Show your data and your calculations.