Rotational Motion

Angular Quantities

• Angular Displacement

• Angular Speed

• Angular Acceleration

if

t

dt

d

2

2

dt

d

dt

d t

Linear to Angular• Angle to Distance

• Angular to Linear Velocity

• Angular to Linear Acceleration

rs

rvt

rat

r

s

Kinematics• For each kinematic equation there is an

angular analog.

vvv 021 02

1

atvv 0 t 0 2

21

0 atvtxx 221

0 tt

020

2 2 xxavv 020

2 2

Bike Ride• A bicycle wheel with radius 0.91 m is

spinning at 2.2 revolutions per second. If the wheel stops in 50 m, what is the average angular deceleration?

Torque• Torque - Force’s effectiveness in altering

rotation motion. (Also called the moment of the force.)

F

r

Pivot

Fr

sinrF

Units (N·m)

Torque depends on Distance from pivot Magnitude of applied force Direction of applied force

Diving• A 85 kg man stands on a diving board. If

he is 2.5 m from the pivot point, what torque is he exerting on the board?

Moment of Inertia• From Newton’s 2nd Law

maF

rmr

2 mr

I

Moment of Inertia

2mrI

2 iitotal rmI

One particle

Multiple particles

Weighted Bar• What is the moment of inertia for the object

illustrated below if it spins about its center of mass?

2.4 m

8 kg 2 kg

0.6 m 0.8 m

Continuous Media

• For Continuous Media the moment of inertia is

• Ex. What is the moment of inertia for the washer about its axis if the inner radius is 1.0 cm and the outer radius is 3.0 cm and the mass is 20 g?

dmrI 2

a

b

Atwood’s Machine

• Two masses are attached to a string and placed over a pulley of mass 2.0 kg and radius 15 cm.

• What is the acceleration of the masses if m1 = 3.0 kg and m2 = 2.0 kg?

m1

m2

Parallel-Axis Theorem

• If the moment of inertia is know w.r.t. the center of mass, then by shifting the pivot point the new moment of inertia is

• From the washer in the previous problem what is the moment of inertia about the edge of the washer?

2MhII cm h

New Pivot

CM

Rotational Energy• Rotational equivalent to kinetic energy

221 IKrot

Pivot

221 IK total 22

21 MRIK cmtotal

2

212

21 RMIK cmtotal

2212

21 MvIK cmtotal transrottotal KKK

Race Down the Slope

• A block and a ball move down an inclined plane. The block is able to slide while the ball has enough friction to roll.

What are the velocities when they reach thebottom of a 0.25m highslope assuming frictionalloses are negligible?