Rotation and Angular Momentum

Objectives

The mathematical and physical description of rotating objects is fascinating and full of surprises. In this lab, you will carry out a variety of experiments to investigate many of the phenomena associated with rotation. There are three experiments:

  1. Rolling Objects
  2. Moment of Inertia
  3. Gyroscope

Do all of these experiments. For each, observe & describe the system as accurately as you can. The goal this week to to make detailed qualitative observations. Record your observations in your notes. Make diagrams and sketches. Try to interpret what you observe using the concepts of torque, angular momentum, and moment of inertia when you summarize your findings in this week’s memo.

Key Ideas

This section briefly describes the fundamental quantities in rotating systems.

Angular Momentum: For a single particle, angular momentum is defined as \[\vec{L} = \vec{r} \times \vec{p}\] Angular momentum is a vector that depends on both the position and momentum of a particle. It points in a direction perpendicular to both. To find the direction of the angular momentum vector, curl the fingers of your right hand in the direction of rotation. Your thumb will point in the direction of the angular momentum vector.

Conservation of Angular Momentum: Like energy and momentum, angular momentum is a quantity that is conserved in all interactions. This concept is the key to understanding many of the phenomena in today’s lab.

Moment of Inertia: For a collection of particles like a wheel or pulley, we describe the collective rotational motion with two quantities: the moment of inertia \(I\) and the angular velocity \(\vec{\omega}\). \[\vec{L} = I \vec{\omega}\] The angular velocity describes both the direction and speed of rotation. The moment of inertia depends on how the mass of an object is distributed about the rotation axis. The moment of inertia of a single particle rotating around an axis distance \(R\) away is \[I = mR^2.\] The total moment of inertia of an object is the sum of contributions from all of the particles that comprise it.

Torque: Just as a force is required to change the momentum of an object, a torque is required to change the angular momentum of an object. Torque is defined as a moment of force: \[\vec{\tau} = \dfrac{\Delta \vec{L}}{\Delta t} \qquad \qquad \qquad \qquad \vec{\tau} = \vec{r} \times \vec{F}\]

You are not expected to be an expert on these topics, but try to use these concepts in describing the phenomena you observe in today’s lab.

Rolling Objects

For this experiment, we will use a ramp and a collection of spherical or cylindrical objects. The objects have different masses and mass distributions. Some are hollow and some are solid.

Make a set of observations to determine what factors affect how quickly an object rolls down a ramp. Compare pairs of objects side by side and try to answer the following questions. Answer them with experimental observations — not speculation or theorizing.

  • How does the radius of an object affect its acceleration down the ramp?
  • How does the mass of an object affect its acceleration down the ramp?
  • How does the shape of an object affect its acceleration down the ramp?
  • Which factors have the strongest influence on the acceleration?

Rotating Platform

This set of experiments will allow you to experience torque and angular momentum directly. As you work through these activities, observe and describe the effects and your sensations. No measurements are required. Try instead to explain the phenomena in terms of torque, angular momentum, angular velocity, and moment of inertia.

  • Step on the platform without causing it to rotate. While you are standing on the platform, try to make yourself rotate. What happens? Why?

Moment of Inertia

Stand on the rotating platform. Spin yourself or have your partner spin you — slowly at first. Raise and lower your arms as you spin.

  • What happens to your rotational speed? What quantity are you changing, and why does this affect your motion?

Now grab a pair of \(1\,\mathsf{kg}\) weights — one in each hand. Raise and lower your arms as you are spinning.

  • What happens to your rotational speed this time? Why does a little extra mass make such a difference?

Bicycle Wheel

  • Hold the bicycle wheel. Give it a spin. Try to rotate the spinning wheel. What happens? Why?

Next, (carefully) stand on the rotating platform. Have your partner hand you the bicycle wheel. Start with both the bicycle wheel and yourself at rest.

  • What happens as you or your partner make the bicycle wheel spin?
  • What happens as you change the orientation the spinning bicycle wheel?

Draw some simple sketches that illustrate the effects you observed. Try to illustrate the angular momentum vectors of yourself and the bicycle wheel.

Gyroscope

Each table has a pair of gyroscopes. Play with a gyroscope for a while. The fascinating behavior of a gyroscope is a consequence of conservation of angular momentum.

Observation

  • Observe and describe the behavior of a spinning gyroscope. Describe how it moves or something interesting you can do with it.
  • Draw a simple sketch that depicts the angular momentum vector of a spinning gyroscope.

Precession

If you set a gyroscope spinning, tilt it at an angle, and set it on a table or a post, the gyroscope undergoes precession. It will maintain a constant angle as it rotates, without falling over. Make some detailed qualitative observations of this phenomenon.

  • How does the rate of precession depend on the tilt angle? The rate at which the gyroscope is spinning?
  • If two gyroscopes are spinning in opposite directions, will they also precess in the opposite directions?

Further Exploration

The following activities are not required. If you have the time and the inclination, you may carry out these investigations for extra credit. You are also free to use any of the materials available in the lab to carry out your own investigations. You need to keep a record of your work (setup, procedure, data, analysis) to get credit for your efforts.

Gyroscope Analysis: Draw a free body diagram of a precessing gyroscope. Indicate all forces acting on it. Draw another diagram that shows the angular momentum vector and the net torque on the gyroscope. Use your diagrams to explain why the spinning gyroscope precesses instead of falling over.

Quantitative Study: Make quantitative measurements of any of the phenomena you investigate in this lab. Record and analyze numerical data. For example, measure the acceleration of an object rolling down a ramp, or the rate of precession of a gyroscope, then analyze your findings.

Moment of Inertia: Design and carry out an experiment to measure the moment of inertia of a pulley in the lab.

Mathematical Modeling: Make a quantitative mathematical model of your experience on the rotating platform. Estimate your moment of inertia and the changes when you raise and lower your arms. Are the predictions of this model consistent with your experience? If not, try to improve the model.