# Rotational Moment of Inertia

#### Due: Thursday, March 29, at 4:00 PM.

You are familiar with Newton's Second Law for translation:

```          F  =  m * a
```

```          torque = (moment of inertia) * (angular acceleration)
```

This week, you will measure the moment of inertia of a big hoop experimentally, using this equation. Because the hoop is a relatively simple geometric shape, it is also possible to calculate its moment of inertia theoretically. The goal of this week's experiment is to see if the two values match to within the uncertainties; and, if not, explain the discrepancy.

#### Theoretical moment of inertia

For simple solid objects, one can calculate the moment of inertia from the mass, size, and shape. Have one member of each lab group concentrate on doing this for your hoop.

Measure the mass of the hoop M, and its uncertainty. Assume that the hoop is perfectly circular and of uniform thickness. Measure its inner and outer diameters. Calculate its inner and outer radii, R(inner) and R(outer). Find an equation for the moment of inertia of a hoop in your textbook. Use it and your measurements to calculate the moment of inertia of your hoop, and the uncertainty therein.

#### Dynamic moment of inertia

You can apply a constant torque to the hub holding your hoop by hanging weights from a string which is wrapped around the hub. As the weights fall, they cause the hub to spin. You may assume that the torque applied by the weights is

```       torque  =  (mass of weight) * g * (radius of hub)
```
In reality, the torque is a little bit smaller than this. Why?

Due to this torque, the hub and hoop spin. Their angular acceleration is

```       angular accel  =  (torque) / (moment of inertia)
```

The weight falls downwards with a linear acceleration which is related to the angular acceleration of the hub via:

```       linear accel   =  (angular accel) * (radius of hub)
```

Derive an equation which relates the linear acceleration of the weight to its mass. Express it in the form:
```             linear accel  =  (something) * (mass of weight)
```

Perform a set of experiments to measure this acceleration. Place at least 4 different masses on the hanging weight (anywhere from 100 to 500 grams). For each mass, measure the time it takes for the weight to fall all the way to the padded box. Make three trials for each mass. Calculate the average acceleration for each mass, and its uncertainty.

Now, you can use your data to determine the moment of inertia.

Derive an equation which relates the linear acceleration Make a graph with the weight's mass on the x-axis, and the acceleration of the weight on the y-axis. Find the slope of the line on the graph, and its uncertainty. Use the slope to calculate the moment of inertia, and its uncertainty.

#### What must I hand in this week?

This week, I want you to hand in all your work before the lab period ends.

I expect

• A neat table of all your measurements, including headings and all appropriate units and uncertainties
• The theoretical calculation of the moment of inertia of the hoop, and its uncertainty.
• A derivation of an equation like this:
```             linear accel  =  (something) * (mass of weight)
```
where you must describe the algebraic terms inside the "something". One of them should be moment of inertia.
• A graph, showing acceleration vs. hanging mass. Be careful to express everything in SI units. Remember, I think a good graph
• covers nearly the entire extent of a sheet of graph paper with data points
• has labels on each axis, which include units
• places errorbars on each plotted datum
• has a title at the top which is not just a rehash of the axes, but describes in some way the experiment; that is, not "Displacement versus mass", but something like "A test of Hooke's Law"
• A paragraph which compares formally the moment of inertia of the hoop derived theoretically and dynamically. Are they consistent with each other? If not, try to explain why not.
• One paragraph which mentions sources of error in the experiment. You must state the single largest source of error and its magnitude as a percentage.

I will deduct a full letter grade from any report which includes the phrase "human error."