ROTATIONAL MOTION

Student is expected to understand the physics of rotating objects.

TORQUE

• Torque is the rotational equivalent of force.

• It measures the “effectiveness” of a force at causing an object to rotate about a pivot.

• A torque causes angular acceleration.

• Problem 9.13, 9.19

FIND THE CENTER OF GRAVITY

• For symmetrical objects, CG lies at its geometrical center.

• For irregular-shaped objects,

...

...

321

332211

mmm

mxmxmxxcg

...

...

321

332211

mmm

mymymyycg

CENTER OF GRAVITY

• A uniform carpenter's square has the shape of an L, as shown in the figure.  Locate the center of mass relative to the origin of the coordinate system.

CAN DO!

• Take a soda can that is full and try and balance it on its edge. What happened?

• This time, have the soda can only about 1/3 of the way full (3.5 oz or 100 ml), and try it again. What happened?

CENTER OF GRAVITY

CHALLENGE

• Balance the nails

ROTATION OF A RIGID BODY

• Rigid body is an extended object whose size and shape do not change as it moves.

• Every point in a rotating rigid body has the same angular velocity.

Linear & Circular Motion

• Position, x

• Velocity, v

• Acceleration, a

• Angular position, θ radian

• Angular velocity, ω rad/s

• Angular acceleration, α

t

xv

t

va

t

t

Spinning up a computer disk

• The disk in a computer disk drive spins up to 5,400 revolutions per minute (rpm) in 2.00 s. What is the angular acceleration of the disk? At the end of 2.00 s, how many revolutions has the disk made?

Useful conversion:

1 revolution = 2π radians

Tangential Acceleration & Angular Acceleration in Uniform Circular Motion

• Tangential acceleration measures the rate at which the particle’s speed around the circle increases.

• If the angular velocity of a bicycle wheel is changing, then the wheel has an angular acceleration.

Relationship between Tangential Acceleration & Angular Acceleration

Linear & Circular Motion

• Displacement at constant speed

• Change in velocity at constant acceleration

• Displacement at constant acceleration

• Angular displacement at constant angular speed

• Change in angular velocity at constant angular acceleration

• Angular displacement at constant angular acceleration

)( tvx )( t

)( tav )( t

2)(2

1)( tatvx o 2)(

2

1)( tto

THINK!

• A ball on the end of a string swings in a horizontal circle once every second. State whether the magnitude of each of the following quantities is ZERO, CONSTANT (BUT NOT ZERO), or CHANGING:

• A) velocity• B) angular velocity• C) centripetal

acceleration• D) angular

acceleration• E) tangential

acceleration

MOMENT OF INERTIA

• It is the rotational equivalent of mass.

• It is a measure of object’s resistance to angular acceleration about an axis.

ROTATIONAL DYNAMICS

Demo: Hammering home inertia!

Inet

Newton’s second law for rotation

where, I is the moment of inertia.

MOMENT OF INERTIA

• Problem 9.39

...233

222

211

2 rmrmrmrmI ii

MOMENT OF INERTIA Table 9.1, page 260

ROTATIONAL KINETIC ENERGY

2

2

1 IKE

Demo: Rolling cylinders, gyroscope

Solve Problem 9.49

ANGULAR MOMENTUM

• Linear dynamics• Force, F• Mass, m• Velocity, v• Momentum, p

• Rotational dynamics• Torque, τ• Moment of Inertia, I• Angular velocity, ω• Angular momentum, L

Conservation of Angular Momentum

• Joey, whose mass is 36 kg, stands at the center of a 200 kg merry-go-round that is rotating once every 2.5 s. While it is rotating, Joey walks out to the edge of the merry-go-round, 2.0 m from its center. What is the rotational period of the merry-go-round when Joey gets to the edge?

Conservation of Angular Momentum

• An ice skater spins around on the tips of his blades while holding a 5.0 kg weight in each hand. He begins with his arms straight out from his body and his hands 140 cm apart. While spinning at 2.0 rev/s, he pulls the weights in and holds them 50 cm apart against his shoulders. If we neglect the mass of the skater, how fast is he spinning after pulling the weights in?