Phys lab report

Online Angular Momentum Lab

Objective

The object of this experiment is to observe and analyze one-dimensional rotational collisions between a disk and a bar rotating about a common axis. You will test the law of conservation of angular momentum and determine whether or not rotational kinetic energy is conserved during the collision.

Theory

Angular momentum is a useful quantity for analyzing the motion of objects over time because it is a “conserved” quantity:

~L = I~ω (1)

To understand this, consider the time derivative of angular momentum:

d~L

dt = I

d~ω

dt (2)

Recognizing that the term d~ω dt

on the right side of this expression is the angular acceleration ~α = dω

dt , and using Newton’s 3rd law for rotational motion ~τ = I~α we can say that:

d~L

dt = ~τ (3)

This implies that if no outside torques act on a system, the angular momentum of the system will not change in time. If there are no external torques, what would be the slope of equation (1)?

Think to yourself

Imagine you want to devise your own procedure for comparing the angular momentum of a set of objects before and after a collision. You have a rotation sensor, calipers and rulers,

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a scale, and two flat disks of different size. Think of a procedure for how these objects can be used to measure conservation of angular momentum. There is no “right” way to do this. Write a brief description of your proposed procedure. Angular momentum is only conserved if no external torques act on the system—how could you avoid introducing unwanted external torques?

Experiment

Someone conducted an experiment measuring angular momentum during a collision using a thin disk and a thin rectangular-shaped bar of aluminum. They also used a rotation sensor which measured angular velocity (radians/second) as a function of time. The disk was placed flat on a rotation sensor. The rotation sensor has a thin axle going straight through the center, and each object has a hole drilled through the center so that the disk and bar can be centered on the rotation sensor. The disk was spun for some time before the bar was dropped onto the center of the disk.

Here are the measurements that were taken for each object:

• Disk mass: 464 g

• Disk radius: 0.063 m

• Bar mass: 182 g

• Bar length: 0.29 m

• Bar width: 0.05 m

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Assume the scale has an uncertainty of 1g and the ruler has an uncertainty of 1mm.

1. What is the moment of inertia just before the collision? What is the moment of inertia just after the collision? The moment of inertia for a uniform disk of mass m and radius r rotating about its center of mass is Idisk =

1 2 mR2. The moment of inertia for

a uniform bar of mass m length L and width w rotating about its center of mass is Ibar =

1 12 m(L2 + w2)

Analysis

1. Download the data set from this experiment by going to the following link: https://tinyurl.com/AngularMomentumData. Using whatever tools you prefer (LoggerPro, Excel, Google Sheets, etc.) plot the angular velocity as a function of time.

2. When one stationary object is dropped onto another the “collision” is not instantaneous— it takes some time for the stationary object to spin up and the rotating object to slow down until they are moving at a common angular velocity. Just by looking at the graph, when does the collision start and end? How do you know?

3. Generate a procedure for comparing the angular momentum before and after the col- lision. Write a brief description of your procedure.

4. What are the sources of uncertainty in your measured values for the angular mo- mentum? Are you able to show that momentum is conserved within this uncertainty range?

5. Because there is friction between the objects as they collide we expect energy to not be conserved. In addition to comparing the angular momentum before and after the collision, determine the total rotational kinetic energy (KErot =

1 2 Iω2) of the system

before and after the collision. What percentage of the kinetic energy was lost in the collision?

Report

In addition to the standard elements of a well written lab report described in the introduction to this manual, your report must include:

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1. A detailed description of your procedure for the analysis

2. A graph of angular momentum before and after the collision and kinetic energy before and after the collision

3. A conclusion in which you address the following:

(a) Whether or not angular momentum was conserved.

(b) Whether or not the energy of the system was conserved.

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