Angular Momentum Simulations

Objectives

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

  1. Moment of Inertia
  2. Rolling Motion
  3. Massive Pulley

Do all three of these experiments. Keep detailed notes. Record your setup, procedure, data, and observations in your lab notes. Data will include tables and plots. In your memo, summarize your findings and discuss any conclusions you draw from your data.

Rolling Motion

In this portion of the experiment, we will investigate how the acceleration of a rolling object depends on its mass, radius, and mass distribution.

Open the Disk Rolling Down Incline simulation from The Physics Aviary. Play with the simulation to see how it works.

We will measure the acceleration using either the velocity graphs or the data points generated by the simulation. Be sure you understand how to measure the acceleration before you proceed to the next step.

When you are ready, proceed to the studies below.

Measure. The goal is to determine which factors have the strongest effect on the acceleration of the object. There are 3 to investigate: Outer Radius, Inner Radius, and Mass. This set of measurements will isolate one variable at a time.

  1. Outer Radius. First, we will investigate the acceleration of solid disks of different sizes. Set the Inner Radius to \(0\,\mathsf{mm}\), and set the mass to a value of your choice. Leave these parameters fixed. Measure the acceleration for three different values of Outer Radius: \(75\,\mathsf{mm}\), \(120\,\mathsf{mm}\), and one intermediate value of your choice.

  2. Inner Radius. Next, we will investigate the acceleration of hollow disks of different shapes. Set the Outer Radius to \(100\,\mathsf{mm}\), and set the mass to a value of your choice. Leave these parameters fixed. Measure the acceleration for three different values of Inner Radius: \(50\,\mathsf{mm}\), \(95\,\mathsf{mm}\), and one other value of your choice.

  3. Mass. Finally, we will investigate the acceleration of hollow disks of different masses. Set the Outer Radius to \(100\,\mathsf{mm}\), and set the Inner Radius to some value of your choice. Leave these parameters fixed. Measure the acceleration for three different values of Mass: \(25\,\mathsf{g}\), \(200\,\mathsf{g}\), and one intermediate value of your choice.

Make more measurements if you are interested. Try to observe how the acceleration depends on the size, shape, and mass of the disk.

Analyze.

  • Based on your measurements, which factor have the strongest influence on the acceleration of a rolling disk? Which have the least effect?
  • How should you design a wheel to have the fastest acceleration?

Critical Thinking. Solid rubber tires on cars would never go flat. Discuss the pros and cons of solid tires, based on your measurements and other factors, like cost, weight, etc.

Moment of Inertia

The moment of inertia of an object describes its distribution of mass. It affects how easy or difficult it is to make an object rotate, and it is part of the angular momentum of the object. For a point particle of mass \(m\) distance \(r\) from the axis of rotation, the moment of inertia is \(I = mr^2\). The first simulation today will allow you to explore the effect of changing the mass and location of a particle on the rotation of system.

Open the Person on a Rotating Platform simulation from oPhysics. This simulation holds the total angular momentum of the system constant as you adjust the size, mass, and location of the objects in it. Play with the simulation to see how it works, and then carry out the following studies.

Measure.

First, make the “Radius of Cylinder” as large as possible and the “Mass of Cylinder” as small as possible.

  • Set the mass of the person to \(100\,\mathsf{kg}\) and measure the angular velocity at 5 different settings of “Radius of Person’s Path”: \(0.5\,\mathsf{m}\), \(5.0\,\mathsf{m}\), and 3 intermediate values of your choice.

  • Set the radius of the person’s path to \(1\,\mathsf{m}\) and measure the angular velocity at 5 different settings of “Mass of Person”: \(50\,\mathsf{kg}\), \(200\,\mathsf{kg}\), and 3 intermediate values of your choice.

Analyze.

  • Plot your data for Angular Velocity vs. Radius.
  • Plot your data for Angular Velocity vs. Mass.
  • What relationships do you observe? Try to be as precise as possible — is it a linear relationship? exponential? power law? Use linear, semilog, and log-log plots to back up your conclusion.

Massive Pulley

The moment of inertia of a pulley affects the rate at which a mass will fall. In this final section of the lab, you will design your own set of measurements to investigate this effect.

Use the Moment of Inertia of Pulley simulation from The Physics Aviary to investigate the effect of a pulley on a falling mass. Remember: if we were to ignore the mass of the pulley, the acceleration of the mass would be \(9.8\,\mathsf{m/s^2}\).

Measure the acceleration of the hanging mass for different masses, sizes, and shapes of pulleys, and different hanging masses. Try to determine the how the acceleration depends on the pulley.

This section of the lab is intentionally open-ended. Real research in science and engineering almost never comes with instructions! Do your best, and discuss any challenges you encounter in your memo.