Experiment 3

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Experiment 3: Rotation • Learning Goals

After you finish this lab, you will be able to:

1. Describe and experience the conservation of angular momentum in complicated systems (e.g. when the axis of rotation is not fixed).

2. Get an intuitive understanding of torque and angular momentum as vector quantities.

3. Observe and explain the phenomenon of precession.

Introduction: Please read all of this BEFORE you come to lab.

Background and introduction

• In this lab you will explore conservation of angular momentum and precession. Rotational motion is different from linear motion in ways that can be confusing and counter-intuitive. The exercises and questions in this lab will help you get a feel for rotational motion and angular momentum.

Moment of inertia

• The rotational analogue of mass is called moment of inertia, or rotational inertia. It characterizes the motion of a rigid object that rotates. The moment of inertia of a very small object that has a mass m and rotates around an axis a distance raway is given by I = mr2 . The moment of inertia of a large object rotating around a given axis can be found by imagining the object being composed of small pieces, finding the contribution that each piece makes to the moment of inertia, and then adding up the separate contributions:

I = miri2

i∑

The moment of inertia of rigid bodies depends not only on the mass of the object, but on how that mass is distributed and the location of the axis of rotation. The formulae for calculating moment of inertia for various objects with uniform mass density can be found on the Internet and in textbooks.

Torque and angular momentum

When a force is applied on an extended object, three things can happen: the object remains static, the object moves, or the object deforms. One type of motion is rotational motion. The way a force applied to an object results in rotational motion is through a quantity called torque. Torque is the rotational analogue of force, and it depends not just on the force applied (

F ) but also on the displacement vector

R

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between the axis of rotation and the point of application of that force. The vector torque is defined as:

τ =R×F

The magnitude of torque can be calculated as: τ = RF sinθ where θ is the angle between the vectors

R and

F . Since torque is a vector, it also has a direction. The direction of the

torque vector is determined by the right-hand rule, shown by the handy (!) summary below:

HAND – make sure you use your RIGHT hand

HAND – point your right hand in the direction of R

FINGERS – rotate your arm such that you can bend or curl your fingers so that they point in the direction of

F

THUMB – stick your thumb out; it points in the direction of the torque τ .

Note that the torque vector is perpendicular to both the force vector and the displacement vector.

Just as we have Newton’s Second Law for linear motion, relating the net force on an object with the rate of change of the object’s linear momentum with time:

F = d

pdt∑

we have Newton’s Second Law for rotational motion, relating the net torque on an object with the rate of change of the object’s angular momentum with time:

τ = d

Ldt∑

The angular momentum of an object depends on the object’s moment of inertia, as well as the object’s rotational or angular velocity:

L = I ω

Some of the strangeness of rotational motion arises from the fact that both the angular velocity and the moment of inertia can change—so when you change the angular momentum, it’s not simply a change in the angular velocity. In contrast, with linear momentum

p = mv we’re usually dealing with objects whose mass doesn’t change.

In addition, the fact that torque is perpendicular to force leads to counter-intuitive behavior like precession…

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Precession

Motion in which a torque changes the direction of the rotation axis is called precession. An example of precession is a spinning top, or a gyroscope. If the top is tilted so that it does not stand vertically, and it is not spinning, it will immediately fall towards the ground when released due to the pull of gravity. However, if the top is spinning, it does not immediately fall to the ground but instead it precesses—it moves sideways. How can we explain this?

First we note that the force of gravity acting on the center of mass of the off-center top will provide a torque on the top. We also note that Newton’s Second Law for rotational motion is a vector equation:

τ = d

Ldt∑

Newton’s Second Law for rotational motion tells us that the direction of the net torque will be the same direction as the change in the angular momentum vector.

Consider the picture shown at right. The bottom tip of the top is fixed; it provides the axis of rotation. The direction of the torque due to the gravitational force will be out of the page (use the right hand rule). If the top is not spinning, its initial angular momentum is zero. An instant later, it will have an angular momentum pointing in the same direction as the torque due to gravity, which is out of the page. This means that the top’s center of mass rotates counterclockwise around the pivot: it falls down towards the floor.

However, if the top is spinning, its initial angular momentum is not zero: it is pointing along the direction of the axis of rotation (up and to the left in the picture). An instant later, the angular momentum will point in a direction such that the difference between the final angular momentum and the initial angular momentum is in the direction of the net torque. Another way of saying this is that the direction of the angular momentum will change in the direction of the net torque, which is out of the page in this case. The final angular momentum will point diagonally (not vertical nor horizontal) and slightly out of the page. If we repeat this analysis for the following instant in time, and the one after that, and the one after that, we see that the resulting motion of the top is for its center of mass to move in a horizontal circle (around the vertical axis). This is precession.

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We can use a similar analysis to figure out in which way we need to push or pull a rotating object to make it do what we want it to do. If we want the direction of the axis of rotation to change in a particular way, we need to apply a torque in the direction of the desired change. This means that we need to apply a force in a direction and at a location such that the torque has the desired direction.

Since torque is perpendicular to both applied force and distance from the axis of rotation to the point of application of the force, the force we apply must be perpendicular to the direction of the desired change! This is counter-intuitive, but it really works out this way! You will explore this remarkable behavior in lab this week.

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Experiment 3: Lab Activity • Follow along in this lab activity. Wherever you see a question highlighted in red, be sure

to answer that question, or paste in some data, or a graph, or whatever is being asked. Your lab report will be incomplete if any of these questions remains unanswered.

Who are you? Take a picture of your lab group with Photo Both and paste it below along with your names.

• Materials:

You will be sharing materials with another group for this lab. Except for the computer, there is one of the following for every two tables.

- Stool on rotating platform

Use this four-legged stool that sits on a rotating platform with low friction to explore angular momentum phenomena, and to measure your moment of inertia.

- Bicycle wheel

This bicycle wheel has a heavy weighted rim and handles along the axis of rotation so you can hold on to it while it spins. Use it with the spinning stool to explore (and experience!) conservation of angular momentum and precession.

- Gyroscope

Use this gyroscope on a gimbal mount that allows it to rotate freely along two axes of rotation (horizontal and vertical) to explore the phenomenon of precession.

• Make some predictions!

Before you actually do the lab, you should make some predictions based on what you’ve learned in class.

• Imagine you are sitting on a rotating chair holding two dumbbells, one in each hand, with your arms stretched to the side. Your friend gives you a push, and you and the chair start

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rotating. You then bring your arms in, close to your body. Describe what you think will happen next.

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• Imagine you are sitting at rest on a rotating chair holding a bicycle wheel by the axle. The wheel is vertical, and it is spinning so that the top edge is moving away from you as shown in the picture. What do you think will happen if you turn the bicycle wheel to the left into the horizontal position? What do you think will happen if you turn the wheel back to being vertical?

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• Imagine you are once again sitting at rest on a rotating chair holding a bicycle wheel by the axle. This time the wheel is horizontal and it is spinning. The chair is not rotating. What do you think will happen if you turn the wheel by 180 degrees, so that it is horizontal in the opposite direction? What do you think will happen if you turn the wheel back to its original orientation?

A:

Part 1: Rotation and angular momentum

In this part of the lab you will work with a rotating stool and explore various aspects of rotation and angular momentum. The stool is mounted on a ball bearing with low friction, so that it does not immediately slow down its rotation. When using the bike wheel, be careful of scarves or long hair.

Sitting on a chair with two dumbbells, one in each hand, and arms stretched to the side, give yourself a push and start rotating. Slowly bring your arms close to your body. What happens? Everyone in your group should do this so you can each experience it!

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Start with the chair at rest and the bicycle wheel vertical. The wheel should be spinning so that the top edge is moving away from you. Turn the wheel to the left or to the right so that it is now horizontal. What happens?

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Now start with the chair at rest and the spinning bicycle wheel horizontal. Slowly turn the wheel by 180 degrees until it is horizontal with the opposite orientation. What happens?

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What happens if you turn the wheel back to the initial orientation?

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How do your experiences compare with your predictions from above?

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Part 2: Gyroscopes

In this part of the lab, you will explore gyroscopes and the phenomenon of precession.

First, hold a stationary (non-spinning) bicycle wheel from one handle as shown in the photo, such that the wheel is vertical and the handle is horizontal. How difficult is it to tilt the wheel upwards, over your head?

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What do you think would happen if the bicycle wheel were spinning?

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Now spin the wheel as fast as you can and hold it from one handle, such that the wheel is vertical and the handle is horizontal. Describe what happens when you try to tilt the wheel upwards. Is this what you would have expected?. Everyone should try it out – you really need to feel it to believe it.

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Disk Example: How to predict Precession.

Here is an example of the strange motion of spinning objects, called precession. You experienced the effects of precession when lifting the spinning wheel

The disk to the right spins at >500rpm about its own axle, as shown in the picture. The axle-disk system can pivot at the “joint” in the figure.

Here’s how to predict the direction of precession of the disk.

To help you visualize the setup see this animation: https://youtu.be/a5Sgqeur50M.

Here’s a 3-step process for predicting direction of precession of ANY spinning object when a torque is applied on it:

1. Identifythedirectionoftheangularvelocityvector𝝎ofthespinningobjectusingtheright-handruleinthefigurebelow.(𝝎andtheangularmomentumvector𝑳 are in the same direction)

2. Findthedirectionofthetorque𝝉appliedtothespinningobjectusingtheright-handrule,asshowninthefigurebelow.Rememberthatthemomentarm

R isavectorthatpoints

fromthepivottothepointofapplicationoftheforce(gravityinthiscase).

Up

Down

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3. ThespinningobjectwillmovesuchthattheTIPoftheangularmomentum𝑳(or𝜔)vectorbeginstomovealongthedirectionofthetorquevector(seedrawingbelow).Afterashortperiodoftimethefinal𝜔canbefoundbyaddingtheinitial𝜔toasmallvectorinthedirectionofthetorque.

EACH MEMBER of the group must (individually) use the 3 steps above to figure out if the disk precesses up, down, in direction 1, or in direction 2. How many people in your group predicted precession in each of the following directions? (it’s ok if you disagree here, you’ll discuss in the next step)

a.) Up:

b.) Down:

c.) Direction1:

d.) Direction2:

Compare each of your 3 steps with the other members of the group. Which of the 3 steps gave your group the most trouble and why?

Come to a unanimous decision about which direction you think the disk will precess. Which direction will the disk precess?

Now, you will test your prediction using the wheel. Before you do though, put a string through the hole in the handle (where the string meets the handle is like the above joint/pivot). Hold the string vertical and the wheel vertical, then let go of the wheel. What happens if the wheel is not spinning?

1

2

Up

Down

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A:

Describe what happens if the wheel is spinning in the same direction as the disk (from the above example) and you let go. Were you right about its direction of precession?

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What happens if the wheel (or disk) is spinning in the other direction?

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• For the rest of the lab you will be using a gyroscope. Get a gyroscope from your TF. IMPORTANT: Be very careful with the gyroscope. When you are finished, put the gyroscope back in the wooden box. If you drop it on the floor, it will be damaged, and you will have to exchange the ball bearings!

The gyroscope comes with a gimbal mount. This mount allows the gyroscope to freely rotate along two axes of rotation (horizontal and vertical). First, you will explore the basic behavior of the gyroscope, as shown in the picture in the materials section.

Spin the gyroscope with the motor and place it on the mount. With the spinning disk horizontal, touch the U-shaped gimbal and try rotating it around the vertical axis. What happens? What do you think should happen?

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Now tilt the gyroscope so that the disk is not spinning horizontally. Touch the edge of the U-shape gimbal and try rotating it around the vertical axis. What happens? Can you explain what happens based on the angular velocity of and torque on the gyroscope? (hint: pivot is now the center and the force is given by your hand)

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Wait for the gyroscope to stop spinning (this might take a minute or two). Spin it up again to maximum speed using the motor, and place it on the mount. Attach the long rod and orient the gyroscope such that the rod is neither vertical nor horizontal. Now the gyroscope is clearly out of balance. Via the lever arm, gravity will cause a torque. This torque sets the gyroscope into a rotation around the vertical axis, called precession. How

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can you change the orientation of the gyroscope (the angle the rod makes with the vertical) without pushing/pulling on the rod?

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Change the orientation of the gyroscope a little bit and watch what happens to the precession speed. Change the orientation a few more times, observing what happens to the precession speed after each orientation change. Does the precession speed depend on the gyroscope orientation?

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What would happen if you attached a weight to the end of the lever arm? Why?

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• Conclusion

What is the most important thing you learned in lab today?

What aspect of the lab was the most confusing to you today?