**Lab 8: Rotational Motion**

**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:

• Bryan Smith with his Happy Piano LP record

• 50 g hanging mass (for applying torque)

• A clamp stand to mount the system on

• Thread

• 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 Bryan Smith with his Happy Piano LP record 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 record disk, ignoring the hole:

r_{d} =

M_{d} =

I_{d} =

You’ll have to measure the mass of the disk because different disks can have different masses.

For the hole in the centre:

*r*_{h} =

*M*_{h} =

I_{h} =

*I*_{disc} =

*I*_{d} –* I*_{h} =

(b) Calculate the theoretical value of the rotational inertia of the pulley system on the rotational motion sensor using basic measurements of its radius and mass. *Note: You’ll have to do a bit of estimation here (explain your method and any estimates you make). Be sure to state units.*

r_{s} =

*M*_{s} =

I_{s} =

(c) Calculate the theoretical value of the rotational inertia, $I$, of the whole system. Don’t forget to include the units. Compare the moment of inertia of the Brian Smith with his Happy Piano LP with your estimate of the moment of inertia of the pulley system. From this comparison, justify whether or not you can ignore the pulley in your calculations in this lab.

*I* =

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

*m* =

*r*_{s} =

(d) Use the equation you derived in parts (b) and (c) of the prelab problem 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, $r_s$, of the spool.

τ =

(e) Now mount the Bryan Smith with his Happy Piano LP onto the pulley system with the screw provided. 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?

α_{th} =

### Experimental

Measurement of Angular Acceleration

With the *Bryan Smith with his Happy Piano* LP record mounted onto the spool of the rotational motion sensor, you are going to measure the “angular position”, $\theta$, and the angular velocity, $\omega$, of this object. You are going to apply a torque to the system by hanging a weight onto one of the pulleys on the motion sensor by the thread.

Cut approximately one meter of thread and make a small loop at one end of the thread piece so that you can hang the 50g weight on it later. The other end of the thread piece goes through the hole in the pulley system on the motion sensor as shown in Figure 1. Tie the thread piece to the pulley through the small hole in the pulley as shown.

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.

### ✍Experimental Write-up for Finding α

Describe your experiment in detail in your labbook. Show your data and your calculations.

Compare your experimental results for a to your theoretical calculation of a for the rotating system. Present this comparison with a neat summary of your data and calculated results.

### ✍Comparing Theory with Experiment

(a) Summarize the theoretical and experimental values of angular acceleration along with the standard deviation for the experimental value.

α_{th} =

α_{exp }= σ_{exp} =

(b) Do theory and experiment agree within the limits of experimental uncertainty?

### Kick it up a notch.

Ideas for refining the experiment are

- Adjust the string length so that the weight does not hit the floor, but instead partially winds back up. With care the string will unwind and rewind on the same pulley many times. Each time a little energy is lost but it can keep going for at least a couple of minutes. Thus you can record many instances of constant angular velocity: going up, coming down, clockwise and counterclockwise. Measure the angular acceleration for each one and take the average and standard deviation.
- Explore whether the angular acceleration is different going up and coming down.
- Does the the angular acceleration depend on whether the disk is rotating clockwise or counterclockwise?
- How much energy is lost in each cycle? Can you fit an exponentially dying curve to the amplitude of the angle vs time graph or the angular velocity vs time graph?
- The the angle vs time graph looks very similar to a sine curve with an exponential envelope $\theta(t)=\Theta\sin(\omega t +\phi)e^{-\lambda t}$. Is it in theory? Is it experimentally?