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Fundamentals of Physics. Chapter 12 Rolling, Torque & Angular Momentum Rolling The Kinetic Energy of Rolling Rolling as Pure Rotation The Forces of Rolling Friction & Rolling Rolling Down a Ramp The Yoyo Torque Revisited Angular Momentum Newton’s 2 nd Law in Angular Momentum Form

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fundamentals of physics
Fundamentals of Physics

Chapter 12 Rolling, Torque & Angular Momentum

  • Rolling
  • The Kinetic Energy of Rolling
        • Rolling as Pure Rotation
  • The Forces of Rolling
        • Friction & Rolling
        • Rolling Down a Ramp
  • The Yoyo
  • Torque Revisited
  • Angular Momentum
  • Newton’s 2nd Law in Angular Momentum Form
  • Angular Momentum of a System of Particles
  • Angular Momentum of a Rigid Body Rotating About a Fixed Axis
  • Conservation of Angular Momentum

Review & Summary

Questions

Exercises & Problems

Physics 2111 Fundamentals of Physics

rolling without slipping
Rolling without Slipping

Units: (m) (radians) (m)

Units: (m/s) (rad/s) (m)

Physics 2111 Fundamentals of Physics

rolling
Rolling

Physics 2111 Fundamentals of Physics

example value of x for rolling without slipping
Example: Value of x for rolling without slipping?

x = 0  no initial spin

Rolling without slipping:

Independent of m and F!

x > 2/5 R  topspin

Physics 2111 Fundamentals of Physics

the forces of rolling
The Forces of Rolling

Rolling with slipping initially

No Initial Rotation:

Topspin:

Frictional force accelerates the ball in the direction of motion:

increasing v

decreasing w

Frictional force reduces the speed &

increases the angular speed until:

Rolling without slipping

Physics 2111 Fundamentals of Physics

translation rotation
Translation & Rotation

pure rotation + pure translation = rolling motion

Add: Linear velocity due to rotation

Linear velocity of the center-of-mass

To get: Linear velocity of each point on a rolling wheel.

See p247 in HRW.

Physics 2111 Fundamentals of Physics

the kinetic energy of rolling
The Kinetic Energy of Rolling

A rolling object has two types of kinetic energy: a rotational KE due to its rotation about its center of mass and a translational KE due to the translation of its center of mass.

The instantaneous axis of rotation is at Point P:

Physics 2111 Fundamentals of Physics

the forces of rolling8
The Forces of Rolling

fs acts on the wheel at P, opposing its tending to slide.

Wheel is rotating about point P.

No work is done by fs!

The wheel does not slow down.

Friction & Rolling: friction is required!

No slipping (aka “smooth rolling”)

Physics 2111 Fundamentals of Physics

rolling down a ramp
Rolling down a Ramp

a> 0 counter clockwise

a< 0 down the ramp

Any body rolling down a ramp!

Physics 2111 Fundamentals of Physics

uniform ball rolling down
Uniform Ball Rolling Down

M = 6.00 kg

Radius R

q = 300

h = 1.20 m

v = ?

fs = ?

Conservation of energy:

Physics 2111 Fundamentals of Physics

hoop disk and sphere rolling down a ramp
Hoop, Disk and Sphere rolling down a ramp:

Physics 2111 Fundamentals of Physics

wheel has constant acceleration no slipping
Wheel has constant acceleration & no slipping

fS = ? I = ?

Physics 2111 Fundamentals of Physics

ball rolls down the hill
Ball rolls down the hill

x = ?

x

H = 6.0 m

h = 2.0 m

Energy:

Projectile:

Physics 2111 Fundamentals of Physics

loop the loop
Loop the Loop

Energy:

Newton:

Physics 2111 Fundamentals of Physics

torque revisited
Torque “Revisited”

Physics 2111 Fundamentals of Physics

angular momentum
Angular Momentum

Physics 2111 Fundamentals of Physics

all of the particles have the same momentum
All of the particles have the same momentum.

Largest Magnitude of Angular Momentum about O ???

1 and 3

2 and 3

Negative Angular Momentum???

5

Zero Angular Momentum???

Physics 2111 Fundamentals of Physics

angular momentum18
Angular Momentum
  • Angular momentum = ?

m = 2.0 kg

r = 3.0 m

v = 4.0 m/s

F = 2.0 N

out of the page

b) Torque = ?

out of the page

Physics 2111 Fundamentals of Physics

newton s 2 nd law in angular momentum form
Newton’s 2nd Law in Angular Momentum Form

Time derivative of the angular momentum:

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

Physics 2111 Fundamentals of Physics

checkpoint
Checkpoint

Largest Torque about Point O???

Positive Torque about Point O???

Zero Torque about Point O???

Physics 2111 Fundamentals of Physics

angular momentum of a system of particles
Angular Momentum of a System of Particles

Internal torques sum to zero:

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.

Physics 2111 Fundamentals of Physics

angular momentum about a fixed axis
Angular Momentum About a Fixed Axis

Physics 2111 Fundamentals of Physics

a rigid body rotating about a fixed axis
A Rigid Body Rotating About a Fixed Axis

w

Angular momentum of a mass element wrt the origin:

Component of angular momentum in z-direction:

Physics 2111 Fundamentals of Physics

a rigid body rotating about a fixed axis24
A Rigid Body Rotating About a Fixed Axis

w

Total angular momentum in z-direction:

Physics 2111 Fundamentals of Physics

checkpoint same radius same mass same force
Checkpoint : same radius same mass same force

Largest Angular Momentum about Center???

Newton’s 2nd Law:

All the angular momenta are the same.

Largest Angular Speed after a time interval???

Physics 2111 Fundamentals of Physics

conservation of angular momentum
Conservation of Angular Momentum

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

The Spinning Volunteer:

Spins faster!

Physics 2111 Fundamentals of Physics

conservation of angular momentum27
Conservation of Angular Momentum

The Diver:

If the component of the net external torque on a system along a certain axis is zero, then the component of angular momentum of the system along that axis remains constant, no matter what changes take place within the system.

Physics 2111 Fundamentals of Physics

conservation of angular momentum28
Conservation of Angular Momentum

The system = stool + person + spinning wheel

Physics 2111 Fundamentals of Physics

a train starts running on a wheel w
A train starts running on a wheel; w = ?

Physics 2111 Fundamentals of Physics

the sun goes out and then shrinks
The sun goes out and then shrinks!

Sun goes out

Shrinks size Earth

Period ?

The sun will become a white dwarf.

Physics 2111 Fundamentals of Physics