Physics 218, Lecture XX 1

Physics 218Lecture 20 Dr. David Toback

Physics 218, Lecture XX 2

Angular Quantities

• Position Angle • Velocity Angular Velocity • Acceleration Angular Acceleration • Force Torque Today we’ll finish:

– Mass

– Momentum

– Energy

Physics 218, Lecture XX 3

Physics 218, Lecture XX 4

Angular Quantities

• Position Angle • Velocity Angular Velocity • Acceleration Angular Acceleration • Force Torque Today we’ll finish:

– Mass

– Momentum

– Energy

Physics 218, Lecture XX 5

Calculating Moments of Inertia

dmR I

mR I

2

2

Physics 218, Lecture XX 6

Calculate the Moment of Inertia

A pulley has mass M, uniform density, radius R, and rotates around its fixed axis

Calculate its moment of inertia

R

Physics 218, Lecture XX 7

Angular Quantities

• Position Angle • Velocity Angular Velocity • Acceleration Angular Acceleration • Force Torque Today we’ll finish:

– Mass Moment of Inertia – Momentum

– Energy

Physics 218, Lecture XX 8

Angular Momentum

Define the Angular Momentum L

L = I

Is this a vector?

dt

dLI

dt

dL

dt

Ld

dt

Id

dt

dII

)()(

Physics 218, Lecture XX 9

Newton’s Laws

• Momentum vs. Angular Momentum

p=mv L=I

• Newton’s Laws

F=dp/dt = dL/dt

Physics 218, Lecture XX 10

Conservation of Angular Momentum

Const L 0 if

dt

dL

By Newton’s laws, the angular momentum of a body can change, but the angular momentum for a system cannot

change Conservation of Angular Momentum

Physics 218, Lecture XX 11

Ice Skater

• This one you’ve seen on TV

• Try this at home in a chair that rotates

• Get yourself spinning with your arms and legs stretched out, then pull them in

Physics 218, Lecture XX 12

Clutch Design

Model a car clutch as two plates, each with radius R, and masses MA and MB

Assume IPlate = ½MR2

Plate A spins with speed 1 and Plate B is at rest.

We close them and they spin together

Find the final angular velocity of the system

Physics 218, Lecture XX 13

Angular Quantities

• Position Angle • Velocity Angular Velocity • Acceleration Angular Acceleration • Force Torque Today we’ll finish:

– Mass Moment of Inertia – Momentum Angular Momentum L

– Energy

Physics 218, Lecture XX 14

Rotational Kinetic Energy

KE = ½I2

Conservation of Energy must take into account

rotational kinetic energy

Physics 218, Lecture XX 15

Rotating Rod

A rod of mass uniform density, mass m and length l pivots at a hinge. It starts at rest at a right angle and is let go

What is when it reaches the bottom?

What is the velocity of the tip at the bottom?

Physics 218, Lecture XX 16

Rotation and Translation• Objects can both Rotate

and Translate

• Both have energy separately

KE = ½ mv2 + ½I2

• Always true when you have rolling without slipping

Physics 218, Lecture XX 17

Rolling Down an Incline

A solid ball of mass m and radius R rolls without slipping down a plane with height h.

What will be the final speed of the ball?

What would be the speed if the ball didn’t roll? Note:

Isphere = 2/5MR2

Physics 218, Lecture XX 18

Person on a Disk

A person with mass m stands on the edge of a disk with radius R and moment ½MR2. Neither is moving.

The person then starts moving on the disk with speed V.

Find the angular velocity of the disk

Physics 218, Lecture XX 19

Same Problem: Forces

Same problem but with Forces

Physics 218, Lecture XX 20

Next Time

• Chapter 11– It’s a review of Chapter 10, but more

sophisticated

– Math, Torque, Angular Momentum, Energy

• The material will not be on the 3rd exam, but will help with the exam. It will all be on the final.

• Exam 3 is Tuesday November 26th

Physics 218, Lecture XX 21

Physics 218, Lecture XX 22

Hollow Cylinder

Consider a hollow cylinder with uniform density, inner radius R1, outer radius R2 and total Mass M.

Find the moment of Inertia

Physics 218, Lecture XX 23

Parallel-Axis Theorem

• Quick Trick for calculating Moments

• I = Icm + Mh2

• Example