Chapter 12 Lecture

FOR SCIENTISTS AND ENGINEERS A STRATEGIC APPROACH 4/E PHYSICS

RANDALL D. KNIGHT

Chapter 12 Lecture

© 2017 Pearson Education, Inc.


Chapter 12 Rotation of a Rigid Body

IN THIS CHAPTER, you will learn to understand and apply the physics of rotation.

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Chapter 12 Preview

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Chapter 12 Reading Questions

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When floating in space with no external forces acting on it, an object tends to rotate about its

A. Axle. B. Center of mass. C. Edge. D. Geometrical center. E. Pivot point.

Reading Question 12.1

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When floating in space with no external forces acting on it, an object tends to rotate about its

A. Axle. B. Center of mass. C. Edge. D. Geometrical center. E. Pivot point.


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A rotating object has some rotational kinetic energy. If its angular speed is doubled, but nothing else changes, the rotational kinetic energy

A. Increases by a factor of 4. B. Doubles. C. Does not change. D. Halves. E. Decreases by a factor of 4.


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A rotating object has some rotational kinetic energy. If its angular speed is doubled, but nothing else changes, the rotational kinetic energy

A. Increases by a factor of 4. B. Doubles. C. Does not change. D. Halves. E. Decreases by a factor of 4.


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A single particle has a mass, m, and it is at a distance, r, away from the origin. The moment of inertia of this particle about the origin is

A. mr B. m2r C. m2r2

D. m2r4

E. mr2


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A single particle has a mass, m, and it is at a distance, r, away from the origin. The moment of inertia of this particle about the origin is

A. mr B. m2r C. m2r2

D. m2r4

E. mr2


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When discussing torque, the “line of action” is

A. The line connecting the pivot with the point where the force acts.

B. The line along which the force acts. C. The line that passes through the center of

mass. D. The line along which motion occurs. E. The axis around which the object rotates.


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When discussing torque, the “line of action” is

A. The line connecting the pivot with the point where the force acts.

B. The line along which the force acts. C. The line that passes through the center of

mass. D. The line along which motion occurs. E. The axis around which the object rotates.


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A new way of multiplying two vectors is introduced in this chapter. What is it called?

A. The dot product. B. The scalar product. C. The tensor product. D. The cross product. E. The angular product.


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A new way of multiplying two vectors is introduced in this chapter. What is it called?

A. The dot product. B. The scalar product. C. The tensor product. D. The cross product. E. The angular product.


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Moment of inertia is

A. The rotational equivalent of mass. B. The point at which all forces appear to act. C. The time at which inertia occurs. D. An alternative term for moment arm.


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Moment of inertia is

A. The rotational equivalent of mass. B. The point at which all forces appear to act. C. The time at which inertia occurs. D. An alternative term for moment arm.


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A rigid body is in equilibrium if


A. B. τnet = 0 C. Neither A nor B D. Either A or B E. Both A and B

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A rigid body is in equilibrium if


A. B. τnet = 0 C. Neither A nor B D. Either A or B E. Both A and B

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Chapter 12 Content, Examples, and QuickCheck Questions

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The Rigid-Body Model

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Three Basic Types of Motion of a Rigid Body

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Rotational Motion Review Recall that angular

velocity is

All points on a rotating rigid body have the same ω and the same α.

If the rotation is speeding up or slowing down, its angular acceleration is

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Rotational Motion Review Rotational kinematics for constant angular acceleration

The signs of angular velocity and angular acceleration.

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Two coins rotate on a turntable. Coin B is twice as far from the axis as coin A.

QuickCheck 12.1

A. The angular velocity of A is twice that of B.

B. The angular velocity of A equals that of B.

C. The angular velocity of A is half that of B.

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Two coins rotate on a turntable. Coin B is twice as far from the axis as coin A.

QuickCheck 12.1

A. The angular velocity of A is twice that of B.

B. The angular velocity of A equals that of B.

C. The angular velocity of A is half that of B.

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The fan blade is speeding up. What are the signs of ω and α?

A. ω is positive and α is positive. B. ω is positive and α is negative. C. ω is negative and α is positive. D. ω is negative and α is negative.

QuickCheck 12.2

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The fan blade is speeding up. What are the signs of ω and α?

A. ω is positive and α is positive. B. ω is positive and α is negative. C. ω is negative and α is positive. D. ω is negative and α is negative.

QuickCheck 12.2

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Rotation About the Center of Mass

An unconstrained object (i.e., one not on an axle) on which there is no net force rotates about a point called the center of mass.

The center of mass remains motionless while every other point in the object undergoes circular motion around it.

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Finding the Center of Mass

Consider an object made of particles. Particle i has mass mi. The center-of-mass is at

Calculating center of mass is much like calculating your grade-point average. Particles of higher mass count more than particles

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Example 12.1 The Center of Mass of a Barbell

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Finding the Center of Mass by Integration

Divide a solid object into many small cells of mass Δm.

As Δm → 0 and is replaced by dm, the sums become

Before these can be integrated: dm must be replaced by

expressions using dx and dy. Integration limits must be established.

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Example 12.2 The Center of Mass of a Rod

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A baseball bat is cut in half at its center of mass. Which end is heavier?

A. The handle end (left end) B. The hitting end (right end) C. The two ends weigh the same.

QuickCheck 12.3

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A baseball bat is cut in half at its center of mass. Which end is heavier?

A. The handle end (left end) B. The hitting end (right end) C. The two ends weigh the same.

QuickCheck 12.3

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Rotational Energy

A rotating object has kinetic energy because all particles in the object are in motion. The kinetic energy due to

rotation is called rotational kinetic energy. Adding up the individual

kinetic energies, and using vi = riω:

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Define the object’s moment of inertia:

The units of moment of inertia are kg m2. Moment of inertia depends on the axis of

rotation. Mass farther from the rotation axis contributes more

to the moment of inertia than mass nearer the axis. This is not a new form of energy, merely the familiar

kinetic energy of motion written in a new way.

Then the rotational kinetic energy is simply

Rotational Energy

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Calculating the Moment of Inertia

As we did for center of mass, divide a solid object into many small cells of mass Δm and let Δ m → 0. The moment of inertia sum becomes

The procedure is much like calculating the center of mass. One rarely needs to do this integral because

moments of inertia of common shapes are tabulated.

where r is the distance from the rotation axis.

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Moments of Inertia

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Consequences of Moment of Inertia

Easier to spin up

Harder to spin up

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Which dumbbell has the larger moment of inertia about the midpoint of the rod? The connecting rod is massless.

QuickCheck 12.4

A. Dumbbell A. B. Dumbbell B. C. Their moments of inertia are the same.

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Which dumbbell has the larger moment of inertia about the midpoint of the rod? The connecting rod is massless.

QuickCheck 12.4

A. Dumbbell A. B. Dumbbell B. C. Their moments of inertia are the same.

Distance from the axis is more important than mass.

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The Parallel-Axis Theorem You do sometimes need to know the moment of

inertia about an axis in an unusual position. You can find it if you know the moment of inertia

about a parallel axis through the center of mass.

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Example 12.4 The Speed of a Rotating Rod

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The four forces shown have the same strength. Which force would be most effective in opening the door?

A. Force F1 B. Force F2 C. Force F3 D. Force F4 E. Either F1 or F3

QuickCheck 12.5

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The four forces shown have the same strength. Which force would be most effective in opening the door?

A. Force F1 B. Force F2 C. Force F3 D. Force F4 E. Either F1 or F3

QuickCheck 12.5

Your intuition likely led you to choose F1. The reason is that F1 exerts the largest torque about the hinge.

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Torque Torque measures the “effectiveness” of the force at

causing an object to rotate about a pivot. Torque is the rotational equivalent of force. On a bicycle, your foot exerts a torque that rotates the

crank.

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The effectiveness of a force at causing a rotation is called torque. Torque is the rotational equivalent of force. We say that a torque is exerted about the pivot point.

Torque

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Mathematically, we define torque τ (Greek tau) as

The ability of a force to cause a rotation depends on 1. the magnitude F of the force. 2. the distance r from the point of application to the pivot. 3. the angle at which the force is applied.

SI units of torque are N m. English units are foot-

pounds.

Torque

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Torque

Torque has a sign.

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Moment Arm

In practice, torque is often calculated using the moment arm or lever arm.

The torque is |τ| = dF. But this is only the absolute value. You have to provide the sign of τ, based on which direction the object would rotate.

What is the sign of τ here?

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Which third force on the wheel, applied at point P, will make the net torque zero?

QuickCheck 12.6

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Which third force on the wheel, applied at point P, will make the net torque zero?

QuickCheck 12.6

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Example 12.8 Applying a Torque



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The figure shows the forces acting on the crankset of a bicycle. axle exerts a force on

the axle to balance the other forces and keep . The net torque about

the axle is the sum of the torques due to the applied forces:

Net Torque

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Gravitational Torque

The torque due to gravity is found by treating the object as if all its mass is concentrated at the center of mass.

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Rotational Dynamics

What does a torque do? For linear motion, a net force causes an object to

accelerate. For rotation, a net torque causes an object to

have angular acceleration.

In the absence of a net torque (τnet = 0), the object either does not rotate (ω = 0) or rotates with constant angular velocity (ω = constant).

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Analogies Between Linear and Rotational Dynamics

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A student gives a quick push to a puck that can rotate in a horizontal circle on a frictionless table. After the push has ended, the puck’s angular speed

QuickCheck 12.7

A. Steadily increases. B. Increases for awhile, then holds steady. C. Holds steady. D. Decreases for awhile, then holds steady. E. Steadily decreases.

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A student gives a quick push to a puck that can rotate in a horizontal circle on a frictionless table. After the push has ended, the puck’s angular speed

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A. Steadily increases. B. Increases for awhile, then holds steady. C. Holds steady. D. Decreases for awhile, then holds steady. E. Steadily decreases.

A torque changes the angular velocity. With no torque, the angular velocity stays the same. This is Newton’s first law for rotation.

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Problem-Solving Strategy: Rotational Dynamics Problems

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Example 12.11 Starting an Airplane Engine



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Constraints Due to Ropes and Pulleys

A rope passes over a pulley and is connected to an object in linear motion.

The rope does not slip as the pulley rotates.

Tangential velocity and acceleration of the rim of the pulley must match the motion of the object:

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The Constant-Torque Model

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Static Equilibrium

A rigid body is in static equilibrium if there is no net force and no net torque.

An important branch of engineering called statics analyzes buildings, dams, bridges, and other structures in total static equilibrium.

For a rigid body in total equilibrium, there is no net torque about any point.

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The Static Equilbrium Model

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Example 12.15 Will the Ladder Slip?

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Balance and Stability

Stability depends on the position of the center of mass.

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This dancer balances en pointe by having her center of mass directly over her toes, her base of support.

Balance and Stability

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Which object is in static equilibrium?

QuickCheck 12.8

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Which object is in static equilibrium?

QuickCheck 12.8

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What does the scale read? A. 500 N B. 1000 N C. 2000 N D. 4000 N

Answering this requires reasoning, not calculating.

QuickCheck 12.9

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What does the scale read? A. 500 N B. 1000 N C. 2000 N D. 4000 N

QuickCheck 12.9

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Rolling Without Slipping

Rolling is a combination of rotation and translation. For an object that rolls without slipping, the translation of the center of mass is related to the angular velocity by

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Taking the time derivative of position of a particle i in a rolling object:

The velocity of the particle is the velocity of the center of mass of the whole object plus the velocity of particle i relative to the center of mass.


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The figure below shows how the velocity vectors at the top, center, and bottom of a rolling wheel are found. vtop = 2vcm vbottom = 0 The point on the bottom of a rolling object is

instantaneously at rest.


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A wheel rolls without slipping. Which is the correct velocity vector for point P on the wheel?

QuickCheck 12.10

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A wheel rolls without slipping. Which is the correct velocity vector for point P on the wheel?

QuickCheck 12.10

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Kinetic Energy of Rolling

The kinetic energy of a rolling object is

In other words, the rolling motion of a rigid body can be described as a translation of the center of mass (with kinetic energy Kcm) plus a rotation about the center of mass (with kinetic energy Krot).

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One-dimensional motion uses a scalar velocity v and force F.

A more general understanding of motion requires vectors and .

Similarly, a more general description of rotational motion requires us to replace the scalars ω and τ with the vector quantities and .

Doing so will lead us to the concept of angular momentum.

The Vector Description of Rotational Motion

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The magnitude of the angular velocity vector is ω.

The angular velocity vector points along the axis of rotation in the direction given by the right-hand rule as illustrated.

The Angular Velocity Vector

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If vectors and have angle α between them, their cross product is the vector:

The Cross Product of Two Vectors

The dot product is one way to multiply two vectors, giving a scalar. A different way to multiple two vectors, giving a vector, is called the cross product.

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The Right-Hand Rule

Note that . Instead, .

The cross product is perpendicular to the plane of and . The right-hand rule for the direction comes in several forms. Try them all to see which works best for you.

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Example 12.16 Calculating a Cross Product

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The Torque Vector

We earlier defined torque τ = rFsinϕ. r and F are the magnitudes of vectors, so this is a really a cross product:

A tire wrench exerts a torque on the lug nuts.

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Angular Momentum of a Particle

A particle of mass m is moving. The particle’s momentum vector makes an angle β with the position vector.

We define the particle’s angular momentum vector relative to the origin to be

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Torque causes a particle’s angular momentum to change. This is the rotational equivalent of and is a general statement of Newton’s second law for rotation.

If you take the time derivative of and use the definition of the torque vector (see book for details), you find

Angular Momentum of a Particle

Why this definition?

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Angular Momentum of a Rigid Body

on a fixed axle, or about an axis of

symmetry

For a rigid body, we can add the angular momenta of all the particles forming the object. If the object rotates

then it can be shown that

And it’s still the case that

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Analogies Between Linear and Angular Momentum and Energy

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Conservation of Angular Momentum

An isolated system that experiences no net torque has

and thus the angular momentum vector is a constant:

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As an ice skater spins, external torque is small, so her angular momentum is almost constant.

Conservation of Angular Momentum

By drawing in her arms, the skater reduces her moment of inertia I.

To conserve angular momentum, her angular speed ω must increase.

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Example 12.19 Two Interacting Disks

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Two buckets spin around in a horizontal circle on frictionless bearings. Suddenly, it starts to rain. As a result,

A. The buckets speed up because the potential energy of the rain is transformed into kinetic energy.

B. The buckets continue to rotate at constant angular velocity because the rain is falling vertically while the buckets move in a horizontal plane.

C. The buckets slow down because the angular momentum of the bucket + rain system is conserved.

D. The buckets continue to rotate at constant angular velocity because the total mechanical energy of the bucket + rain system is conserved.

E. None of the above.

QuickCheck 12.11

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Two buckets spin around in a horizontal circle on frictionless bearings. Suddenly, it starts to rain. As a result,

A. The buckets speed up because the potential energy of the rain is transformed into kinetic energy.

B. The buckets continue to rotate at constant angular velocity because the rain is falling vertically while the buckets move in a horizontal plane.

C. The buckets slow down because the angular momentum of the bucket + rain system is conserved.

D. The buckets continue to rotate at constant angular velocity because the total mechanical energy of the bucket + rain system is conserved.

E. None of the above.

QuickCheck 12.11

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Advanced Topic: Precession of a Gyroscope Consider a horizontal

gyroscope, with the disk spinning in a vertical plane, that is supported at only one end of its axle, as shown.

You would expect it to simply fall over—but it doesn’t.

Instead, the axle remains horizontal, parallel to the ground, while the entire gyroscope slowly rotates in a horizontal plane.

This steady change in the orientation of the rotation axis is called precession, and we say that the gyroscope precesses about its point of support.

The precession frequency Ω is much less that the disk’s rotation frequency ω.

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Gravity on a Nonspinning Gyroscope Shown is a nonspinning

gyroscope. When it is released, the net

torque is entirely gravitational torque.

Initially, the angular momentum is zero.

Gravity acts to increase the angular momentum gradually in the direction of the torque, which is the -direction.

This causes the gyroscope to rotate around x and fall.

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Gravity on a Spinning Gyroscope Shown is a gyroscope initially

spinning around the z-axis. Initially, gravity acts to

increase the angular momentum slightly in the direction of the torque, which is the -direction.

This causes the gyroscopes angular momentum to shift slightly in the horizontal plane.

The gravitational torque vector is always perpendicular to the axle, so dL is always perpendicular to L.

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Advanced Topic: Precession of a Gyroscope The precession frequency of a gyroscope, in rad/s, is

Here M is the mass of the gyroscope, I is its moment of inertia, and d is the horizontal distance of the center of mass from the support point.

The angular velocity of the spinning gyroscope is assumed to be much larger than the precession frequency: ω >> Ω

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Example 12.20 A Precessing Gyroscope

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Chapter 12 Summary Slides

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General Principles

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General Principles

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Important Concepts

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Important Concepts

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Important Concepts

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Important Concepts

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Applications

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Applications

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Applications

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Chapter 12 Lecture

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