# Rotational Motion

## Table of contents

## Activity 1

** Note 1:** Please do not tilt the ramp more than 10

^{o}. If you tilt it too much your hoops might slip and slide, which will ruin your data.

__ Note 2:__ Some students get results which they cannot easily explain for part of this experiment. Any time that you have doubts about your data, redo the experiment (maybe reduce the angle of the ramp as per the first note). If the results are the same you must then accept the results.

You have a ramp with a motion sensor, two hoops and a disk. Choose one of the hoops or the disk. Your first task is to figure out, experimentally, in what direction does the force of friction of the ramp on a rolling disk/hoop point when (i) the disk/hoop rolls ** downhill **and (ii) the disk/hoop rolls

**. Much of this experiment will be similar to the experiment involving carts rolling downhill to measure the acceleration due to gravity. This time you may assume you know the value of g=9.8 m/s**

__uphill__^{2}.

To make an experimental determination of the direction you should start with a prediction in the form of an “**IF **(hypothesis) **AND **(experiment) **THEN **(measurement)...” statement. For example:

“**IF **the friction points downhill when the hoop rolls downhill **AND **we measure the acceleration of the hoop as it rolls downhill **THEN **we should measure a downhill acceleration a > g sin(θ).”

When you have a prediction you should test it. If your measurement agrees with your prediction, you cannot say the hypothesis is correct but you can say it is plausible. If your measurement disagrees with your prediction then you can say your hypothesis is wrong.

Note: you do not lose marks for picking the “wrong” hypothesis. This is an experimental situation where you do not need to know the correct result, you just need to be able to find the correct result. In fact, disproving the hypothesis which is the opposite of your preferred hypothesis is a time-honoured method of demonstrating you are correct. So it may be wise to try to choose the wrong hypothesis and prove it is wrong.

Finally, justify your experimental result with a conceptual discussion of the forces. That is, **explain why static friction should point in the direction you observed**.

**Summary**: find the direction of friction on the hoop for both **uphill **and **downhill **motion. Explain this (perhaps surprising) result.

## Activity 2

In a previous class you were told that the moment of inertia of a hoop is given by \(I = \frac{1}{2} m (r_a^2 +r_b^2)\). Your task is to test this equation. Find the acceleration of the hoops and the disk as they roll downhill. You should find that \(a < g \sin\theta\). The amount by which they are less depends on the moment of inertia.

It can be shown that the acceleration of a hoop/disk rolling without slipping down a ramp is \(a = \frac{g \sin\theta}{1+X}\) where *X* is a dimensionless quantity related to the moment of inertia: \(X = \frac{I}{mr^2}\) for any round (outer radius r) object which is entirely rolling without slipping. (This does not work for a car, for example, but does work for a car’s wheel if it is rolling free from the rest of the car.) The Appendix can help you prove the equation for *a* if you wish to do so.

Now: measure X for a hoop and determine if it agrees with the theory within uncertainties.

**Summary**: measure X in two different ways and then describe quantitatively whether the two different measurements agree with each other.

## Activity 3

The two hoops are more or less the same size and shape, but quite different masses. The brass hoop and the brass disk are about the same size, but have quite different shapes. These choices were deliberate as it lets you control certain characteristics while changing others to see what effect an individual characteristic has. Use this information to answer the following two questions (experimentally):

- If all else is equal, which object wins a downhill race: the one with more mass or the one with less mass?
- If all else is equal, which object wins a downhill race: the one with even mass distribution (disk) or the one with an uneven mass distribution (hoop)?

## Appendix

The total kinetic energy of a rolling wheel is \(K = \frac{1}{2} m v^2 + \frac{1}{2}I\omega^2\).

For an object rolling without slipping, \(\omega = \frac{v}{r}\).

Energy conservation gives \(K = mg\Delta h = (mg\sin\theta)\Delta d\) where \(\Delta d\) is the distance measured along the slope of the ramp.

Relate this to the kinematic equation \(v_f^2=v_i^2+2a\Delta d\) to find the acceleration down the ramp.