Nicholas J. Giordano

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Chapter 9 Energy and Momentum of Rotational Motion

Introduction • There is kinetic energy associated with rotational • • • •

motion A work-energy theorem can be derived that relates torque and rotational kinetic energy Conservation of energy can be applied to situations that will include rotational kinetic energy Angular momentum is the rotational analog of linear momentum Angular momentum is conserved in many situations

Introduction

Kinetic Energy of Rotation • Remember, for a single

point particle of mass m moving with a linear speed v, the kinetic energy is KE = ½ m v² • Rotational motion is concerned with the extended object • Think of the object as composed of many small pieces • Each piece has the KE of a

point particle

Section 9.1

Kinetic Energy of Rotation, cont. • The total KE energy of the object can be found by

adding up all the kinetic energies of the small pieces • Assuming the pieces are part of a rigid object undergoing simple rotational motion, vi = ωri • Rearranging and solving, KErot = ½ I ω² • Remember, I is the moment of inertia •

Section 9.1

Total Kinetic Energy • Many objects undergo both translational and

rotational motion • The total kinetic energy has contributions from both the rotational motion and the translational motion • Using an axis that passes through the center of mass, the total kinetic energy is the sum of the contributions from the rotational and translational kinetic energies:

Section 9.1

Rolling Motion • Rolling objects have both translational and rotational

kinetic energy • The contributions from each type of kinetic energy can be calculated • For a typical wheel (disk), the rotational kinetic energy is one-half of the translational kinetic energy • • The rotational KE energy term, ½ I ω², simplifies to ¼ mv² • Note the result is independent of the radius of the wheel

Section 9.1

Torque and Rotational Kinetic Energy • Remember work done on

an object equals the change in the object’s translational kinetic energy • W = Fs = ΔKE • A similar relationship exists for torque and rotational kinetic energy

Section 9.1

Torque and KErot, cont. • Attach the mass to a light

rod so that it rotates •

• This is the work-energy

theorem for rotational motion

Section 9.1

Conservation of Energy and Rotational Motion • If all the forces that do work on an object are

conservative forces, the total mechanical energy of the object is conserved • KEi + PEi = KEf + PEf • The KE terms denote the total kinetic energy, including

any rotational KE • The PE terms depend on the forces involved

Section 9.2

Conservation of Energy and Rotational Motion, Example • A solid ball starts from

rest and rolls down a hill • The ball is a sphere

• What is its velocity at the

bottom of the hill? • The forces are: • Gravity • Does work on the ball • Is a conservative force

Section 9.2

Conservation of Energy and Rotational Motion, Example, cont. • Forces, cont. • Normal force • •

Perpendicular to the displacement Does no work

• Friction • • •

•

Does no work The point where the ball meets the surface does not slip Friction does not play a role in the conservation of energy condition when an object rolls without slipping The point of contact is at rest

Section 9.2

Conservation of Energy and Rotational Motion, Example, final • Applying conservation of energy and using the

relationships between linear and rotational velocities, the velocity at the bottom of the hill can be found • Nothing needs to be known about the hill and its

shape except for the initial height of the object

Section 9.2

Problem Solving Strategy – Conservation of Energy • Recognize the principle • Mechanical energy is conserved only if all the forces that act on the object are conservative forces • Sketch the problem • Collect the information concerning the initial and final states of the system • Identify the relationships • Find the initial and final kinetic and potential energies •

This will usually involve initial and final velocities and heights

• You will generally need to find the moment of inertia of

the object

Section 9.2

Problem Solving Strategy, Conservation of Energy, cont. • Solve • Solve for the unknown quantities • Use the conservation of energy condition • The kinetic energy terms are total kinetic energies • Check • Consider what your answer means • Does your answer make sense

Section 9.2

Angular Momentum • Remember, linear momentum was defined as • In the case of a single rotation axis that does not

change direction during the motion, the angular momentum is given by L=Iω • The scalar nature is due to our assumption of a

rotation axis that keeps a fixed direction

Section 9.3

Angular Momentum in Space • Angular momentum

applied to a particle moving freely through space • For a chosen pivot point, P, the particle will be moving tangent to the arc of radius r at any given moment • r is the distance between

P and the particle Section 9.3

Angular Momentum in Space, cont • The motion of the particle is the same as that of an

object that rotates along this circular arc • The angular velocity is ω = v┴ / r • The particle’s moment of inertia is I = m r² • Therefore, L = m r v┴

Section 9.3

Conservation of Angular Momentum • A rotating object will maintain its angular momentum

provided no external torques act on it • In this case, the total angular momentum of the object will be conserved • For a system of objects, if no external torques act on the system, the total angular momentum will be conserved

Section 9.3

Conservation of Linear Momentum Example: Skater • The skater has no

external torque acting on her • Assume the ice is

frictionless • The normal force and gravity do not produce torques

• Pulling her arms and legs

in decreases her moment of inertia Section 9.3

Skater Example, cont. • The moment of inertia decreases since the value of r

decreases • Each “piece” of the arms and legs of the skater is

closer to the axis of rotation

• Since her total angular momentum is conserved, her

angular velocity increases • The value of the final angular velocity depends on how

closely she can pull her arms and legs in line with the rest of her body

Section 9.3

Problem Solving Strategy – Conservation of Angular Momentum • Recognize the principle • If the external torque on a system is equal to zero, the angular momentum is conserved • Sketch the problem • Use the sketch to collect all the information concerning the initial and final states of the system • Identify the relationships • The system may be a single object or a collection of objects

Section 9.3

Problem Solving Strategy – Conservation of Ang. Momentum, cont. • Identify the relationships, cont. • Determine the initial and final angular velocities • Determine the initial and final moments of inertia • Apply any information concerning the initial and final mechanical energies •

This information will not always be available

• Solve • Solve for the unknowns using the principle of conservation of angular momentum • Check • Consider what your answer means • Does your answer make sense Section 9.3

Angular Momentum and Kinetic Energy • Even when angular momentum is conserved, kinetic

energy may not be conserved • In the skater example:

• Since the moment of inertia decreased, the KE

increased • The KE was not conserved • The skater does work to pull in her arms

Section 9.3

Angular Momentum and Planets • Kepler’s Laws of Planetary Motion • See chapter 5 for details about the laws • First Law • Planets follow elliptical orbits about the Sun • Second Law • A planet moving about its orbit sweeps out equal areas in equal times

Section 9.4

Planetary Motion, cont. • The two time intervals are

equal • Therefore, the areas swept out during those time intervals must be the same • The speed of the planet is less when it is farther from the Sun • Angular momentum is connected to Kepler’s Second Law Section 9.4

Planet’s Angular Momentum • The planet’s angular momentum is given by • r is the planet’s distance from the Sun

• As the planet moves around the Sun, r and v┴

change, but its angular momentum remain the same • The torque on the planet is zero • Gravity is the only force acting on the planet • Gravity is along the radial line •

So it is perpendicular to the displacement

Section 9.4

Kepler’s Second Law and Angular Momentum

Section 9.4

Area From Angular Momentum • Consider the points closest to and farthest from the

Sun • Points 1 and 2 in figure 9.16

• From Conservation of Angular Momentum:

• The areas are approximately triangular and given by:

Section 9.4

Energy and Planetary Motion • The gravitational force produces no torque on the • • • •

planet Gravity can, however, do work on the planet For a time interval Δt, Wby gravity = F d cos ϕ The work done by gravity causes the planet to speed up as it approaches the Sun Later in the orbit, the work is negative and the planet slows down

Section 9.4

Vector Nature of Rotational Motion • In many applications,

recognizing the vector nature of rotational quantities is very important • The right-hand rule provides a way to determine the direction • If the fingers of your right hand curl in the direction of motion of the edge of the object, your thumb will point in the direction of the rotational velocity vector Section 9.5

Vector Nature and a Gyroscope • The directions of other vector quantities are also

given by the right-hand rule • When angular momentum is conserved, both its magnitude and direction are conserved • This is the principle behind the gyroscope • There are many ways to build a gyroscope • One design uses a spinning wheel mounted on a

frame with an axis free to rotate

Section 9.5

Gyroscope • Because of the way the

wheel is mounted in the frame, the torque on the wheel is zero • Even when the frame is

moved or rotated

• The wheel’s angular

momentum is conserved • The orientation of the gyroscope provides a “direction finder” Section 9.5

Earth as a Gyroscope • The Earth acts as a

gyroscope as it spins on its axis • This is a “spin angular

momentum,” separate from its orbital angular momentum

• The spin angular

momentum points in the direction of the north pole

Section 9.5

Earth as a Gyroscope, cont. • There is no external torque on the Earth so its spin

angular momentum is conserved • The Earth’s spin axis is tilted about 23.5° from the perpendicular to the orbital axis • Since its spin angular momentum is conserved, the rotational axis remains tilted at a fixed angle with respect to the orbital plane • This produces seasons on the Earth

Section 9.5

Spinning Wheel • A rolling wheel is more stable than a stationary one • The increased stability is due to its angular

momentum • The angular momentum is directed along the axis of the wheel and so, ideally, there is no external torque on the wheel and it would remain in the same direction and never fall over • However, in reality there is some small external torque and so the wheel will eventually tip over • This torque is mainly from friction

Section 9.5

Precession • The external torque on a

rotating object can be substantial • This leads to an effect called precession • At rest, the device’s stability is very low • Its angular momentum adds to its stability, although the gravitational force acting on it is large Section 9.5

Precession, cont. • Assume the gravitational force acts on the center of

mass of the system • The center of mass of the system is close to the center

of the wheel

• The applied torque leads to a change in the angular

momentum of the system

Section 9.5

Direction of Torque • Another version of the right-

hand rule is used to find the direction of the torque • Start at the pivot point and place the fingers of your right hand along the rotational axis of the wheel • Point your fingers toward the

center of the wheel

• Curl your fingers in the

direction of the force • Your thumb points in the direction of the torque Section 9.5

Precession, final • The torque produced by the gravitational force is • • • • •

perpendicular to the rotation axis of the wheel The torque causes the wheel’s angular momentum to turn in the direction of the torque As the rotation axis turns, the direction of the torque also changes The torque and change in angular momentum are always perpendicular to the angular momentum The wheel plus axle rotate continuously about a vertical rotation axis This movement is called precession

Section 9.5

Rotating Cats • The cat’s initial angular

momentum is zero • The cat rotates by changing its shape • The head section and the back section rotate in opposite directions, maintaining zero angular momentum • A system with zero angular momentum can still rotate in interesting ways

Section 9.6

Rotating Cat, cont.

Section 9.6

Motorcycles • The system is the

motorcycle plus the rider • Two types of jumps can be considered • Frame maintains a fixed

angular orientation and lands on the rear wheel • The frame rotates during the jump and lands on both wheels

• In both jumps, the

system’s center of mass follows a parabolic trajectory

Section 9.6

Motorcycles, cont.

Section 9.6

Motorcycles, One-Wheel Landings • The motorcycle frame maintains a fixed angular

orientation throughout the jump • The angular velocity of the system is zero • The angular displacement of the motorcycle frame is zero

Motorcycles, Two-Wheel Landings • To cause the two-wheeled landing, the rider adjusts

the throttle so as to change the angular velocity of the rear wheel • So its angular momentum changes

• The total angular momentum is conserved • The change in the angular momentum of the rear

wheel is compensated for by changing the angular momentum of the rest of the system • The frame rotates and the motorcycle lands on both wheels

Section 9.6