Chapter 11Chapter 11Angular MomentumAngular Momentum

Rolling, Torque, and Angular MomentumRolling, Torque, and Angular Momentum

I. Rolling- Kinetic energy- Forces

II. Torque

III. Angular momentum - Definition

IV. Newton’s second law in angular form

V. Angular momentum - System of particles - Rigid body - Conservation

I. RollingI. Rolling

- Rotation + Translation combinedRotation + Translation combined.

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dt

dsRs

Smooth rolling motionSmooth rolling motion

Example:Example: bicycle’s wheel. bicycle’s wheel.

The motion of any round body rolling smoothly over a surface can be The motion of any round body rolling smoothly over a surface can be separated into purely rotational and purely translational motions.separated into purely rotational and purely translational motions.

- Pure rotation.Pure rotation.

Rotation axis Rotation axis through point where wheel contacts ground. through point where wheel contacts ground.

Angular speed about P (Angular speed about P (ωω)) = Angular speed ( = Angular speed (ωω))about O for stationary observer.about O for stationary observer.

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- Kinetic energy of rolling.- Kinetic energy of rolling.

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Instantaneous velocity vectors Instantaneous velocity vectors = sum of translational = sum of translational and rotational motions.and rotational motions.

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A rolling object has two types of kinetic energy A rolling object has two types of kinetic energy Rotational:Rotational: (about its COM).(about its COM). (translation of its COM).(translation of its COM). Translational: Translational:

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2

1 COMI

2

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- Forces of rolling.- Forces of rolling.

(a)(a) Rolling at constant speedRolling at constant speed no sliding at P no sliding at P no friction.no friction.

(b) Rolling with acceleration(b) Rolling with acceleration sliding at P sliding at P friction force opposed to sliding.friction force opposed to sliding.

Static frictionStatic friction wheel does not slide wheel does not slide smoothsmooth rolling motionrolling motion aaCOM COM = = α Rα R SlidingSliding

Increasing accelerationIncreasing acceleration

ExampleExample11: : wheels of a car moving forward while its tires are spinning wheels of a car moving forward while its tires are spinning

madly, leaving behind black stripes on the road madly, leaving behind black stripes on the road rolling with slipping = rolling with slipping = skidding skidding Icy pavements.Icy pavements.Anti-block braking systems are designed to ensure that tires roll without Anti-block braking systems are designed to ensure that tires roll without slipping during braking.slipping during braking.

ExampleExample22:: ball rolling smoothly down a ramp. (No slipping). ball rolling smoothly down a ramp. (No slipping).

1.1. Frictional force causes the rotation. Without Frictional force causes the rotation. Without friction the ball will not roll down the ramp, friction the ball will not roll down the ramp, will just slide.will just slide.

Sliding tendency

2.2. Rolling without sliding Rolling without sliding the point of contact the point of contact between the sphere and the surface is at restbetween the sphere and the surface is at rest the frictional force is the static frictional force.the frictional force is the static frictional force.

3. Work done by frictional force = 0 Work done by frictional force = 0 the point the point of contact is at rest (static friction).of contact is at rest (static friction).

Example:Example: ball rolling smoothly down a ramp. ball rolling smoothly down a ramp.

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Note:Note: Do not assume f Do not assume fss = f = fs,maxs,max . The only f . The only fss requirement is that its requirement is that its

magnitude is just right for the body to roll smoothly down the ramp, magnitude is just right for the body to roll smoothly down the ramp, without sliding.without sliding.

Newton’s second law in angular formNewton’s second law in angular form Rotation about center of massRotation about center of mass

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Linear acceleration of a body rolling along a Linear acceleration of a body rolling along a incline planeincline plane

- Yo-yo- Yo-yo

Potential energy (mgh)Potential energy (mgh) kinetic energy: translational kinetic energy: translational (0.5mv(0.5mv22

COMCOM) and rotational (0.5 I) and rotational (0.5 ICOMCOMωω22))

Analogous to body rolling down a ramp:Analogous to body rolling down a ramp:

- Yo-yo rolls down a string at an angle - Yo-yo rolls down a string at an angle θθ =90º with =90º with the horizontal.the horizontal.

- Yo-yo rolls on an axle of radius R- Yo-yo rolls on an axle of radius R00..

- Yo-yo is slowed by the tension on it from the - Yo-yo is slowed by the tension on it from the string.string.

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- Vector quantity- Vector quantity. Fr

Direction:Direction: right hand rule.right hand rule.

Magnitude:Magnitude: FrFrFrFr )sin(sin

Torque is calculated with respect to (about) a point. Changing the point can Torque is calculated with respect to (about) a point. Changing the point can change the torque’s magnitude and direction.change the torque’s magnitude and direction.

II. TorqueII. Torque

III. Angular momentumIII. Angular momentum

- Vector quantity.- Vector quantity. )( vrmprl

Direction:Direction: right hand rule.right hand rule.

Magnitude:Magnitude: vmrprprprvmrvmrprl )sin(sinsin

l positive l positive counterclockwise counterclockwisel negative l negative clockwise clockwise

Direction of l is always perpendicular to plane formed Direction of l is always perpendicular to plane formed by r and p.by r and p.

Units:Units: kg mkg m22/s/s

IV. Newton’s second law in angular formIV. Newton’s second law in angular form

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The vector sum of all torques acting on a particle is equal to the time rate The vector sum of all torques acting on a particle is equal to the time rate of change of the angular momentum of that particle.of change of the angular momentum of that particle.

Proof:Proof:

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V. Angular momentumV. Angular momentum

- System of particles:- System of particles:

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Includes internal torques (due to forces between particles within system) Includes internal torques (due to forces between particles within system) and external torques (due to forces on the particles from bodies outside and external torques (due to forces on the particles from bodies outside system).system).

Forces inside system Forces inside system third law force pairs third law force pairs torque torqueintint sum =0 sum =0 The The

only torques that can change the angular momentum of a system are the only torques that can change the angular momentum of a system are the external torques acting on a system.external torques acting on a system.

The net external torque acting on a system of particles is equal to the time The net external torque acting on a system of particles is equal to the time rate of change of the system’s total angular momentum L.rate of change of the system’s total angular momentum L.

- Rigid body - Rigid body (rotating about a fixed axis with constant angular speed (rotating about a fixed axis with constant angular speed ωω):):

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IL Rotational inertia of a rigid body about a fixed axisRotational inertia of a rigid body about a fixed axis

- Conservation of angular momentum:Conservation of angular momentum:

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Newton’s second lawNewton’s second law

If no net external torque acts on the system If no net external torque acts on the system (isolated system)(isolated system)

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0

Law of conservation of angular momentum:Law of conservation of angular momentum: )( systemisolatedLL fi

If the net external torque acting on a system is zero, the angular If the net external torque acting on a system is zero, the angular momentum of the system remains constant, no matter what changes take momentum of the system remains constant, no matter what changes take place within the system.place within the system.

Net angular momentum at time tNet angular momentum at time tii = Net angular momentum at later time t = Net angular momentum at later time tff

If the component of the net external torque on a system along a certain If the component of the net external torque on a system along a certain axis is zero, the component of the angular momentum of the system axis is zero, the component of the angular momentum of the system along that axis cannot change, no matter what changes take place within along that axis cannot change, no matter what changes take place within the system.the system.

This conservation law holds not only within the frame of Newton’s This conservation law holds not only within the frame of Newton’s mechanics but also for relativistic particles (speeds close to light) and mechanics but also for relativistic particles (speeds close to light) and subatomic particles.subatomic particles.

ffii II

( I( Ii,fi,f, , ωωi,fi,f refer to rotational inertia and angular speed before and after refer to rotational inertia and angular speed before and after

the redistribution of mass about the rotational axis ).the redistribution of mass about the rotational axis ).

Examples:Examples:

IIff < I < Iii (mass closer to rotation axis) (mass closer to rotation axis)

Torque ext =0 Torque ext =0 I Iiiωωii = I = Iff ωωff

ωωff > > ωωii

Spinning volunteerSpinning volunteer

A solid cylinder of radius A solid cylinder of radius 15 cm15 cm and mass and mass 3.0 kg3.0 kg rolls rolls without slipping at a constant speed of without slipping at a constant speed of 1.6 m/s1.6 m/s. (a) What is . (a) What is its angular momentum about its symmetry axis? (b) What is its angular momentum about its symmetry axis? (b) What is its rotational kinetic energy? (c) What is its total kinetic its rotational kinetic energy? (c) What is its total kinetic

energy? ( energy? ( II cylinder= ) cylinder= )

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rotational inertia for cylinder . The angular rotational inertia for cylinder . The angular

momentum about the symmetry axis ismomentum about the symmetry axis is

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A solid cylinder of radius A solid cylinder of radius 15 cm15 cm and mass and mass 3.0 kg3.0 kg rolls rolls without slipping at a constant speed of without slipping at a constant speed of 1.6 m/s1.6 m/s. (a) . (a) What is its angular momentum about its symmetry axis? What is its angular momentum about its symmetry axis? (b) What is its rotational kinetic energy? (c) What is its (b) What is its rotational kinetic energy? (c) What is its total kinetic energy? ( total kinetic energy? ( II cylinder= ) cylinder= )2

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A light rigid rod A light rigid rod 1.00 m1.00 m in in length joins two particles, with length joins two particles, with masses masses 4.00 kg4.00 kg and and 3.00 kg3.00 kg, , at its ends. The combination at its ends. The combination rotates in the rotates in the xyxy plane about a plane about a pivot through the center of the pivot through the center of the rod. Determine the angular rod. Determine the angular momentum of the system momentum of the system about the origin when the about the origin when the speed of each particle is speed of each particle is 5.00 5.00 m/s.m/s.

4.00 kg 5.00 m s 0.500 m 3.00 kg 5.00 m s 0.500 m

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A conical pendulum consists of a A conical pendulum consists of a bob of mass bob of mass mm in motion in a in motion in a circular path in a horizontal plane circular path in a horizontal plane as shown. During the motion, the as shown. During the motion, the supporting wire of length supporting wire of length maintains the constant angle maintains the constant angle with the vertical. Show that the with the vertical. Show that the magnitude of the angular magnitude of the angular momentum of the bob about the momentum of the bob about the center of the circle iscenter of the circle is

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The position vector of a particle of mass 2.00 kg is The position vector of a particle of mass 2.00 kg is

given as a function of time by . given as a function of time by .

Determine the angular momentum of the particle about Determine the angular momentum of the particle about

the origin, as a function of time.the origin, as a function of time.

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A uniform solid disk is set into rotation with an angular speed A uniform solid disk is set into rotation with an angular speed ii about an axis through its center. While still rotating at this about an axis through its center. While still rotating at this speed, the disk is placed into contact with a horizontal speed, the disk is placed into contact with a horizontal surface and released as in the Figure. (a) What is the surface and released as in the Figure. (a) What is the angular speed of the disk once pure rolling takes place? (b) angular speed of the disk once pure rolling takes place? (b) Find the fractional loss in kinetic energy from the time the Find the fractional loss in kinetic energy from the time the disk is released until pure rolling occurs. (disk is released until pure rolling occurs. (HintHint: Consider : Consider torques about the center of mass.)torques about the center of mass.)

(a)(a) The net torque is The net torque is zero at the point of contact, zero at the point of contact, so the angular momentum so the angular momentum before and after the collision before and after the collision must be equal.must be equal.

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