Physics 1D03

Rotation (III)Rotation (III)

• Torque and angular acceleration• “Moment of inertia”

Text Sections : 10.7, and part of 10.4

Physics 1D03

Angular velocity vector: parallel to the axis of rotation, following a similar right-hand rule:

Angular acceleration vector: parallel to the angular velocity, if is increasing.

rotation direction

Force causes linear acceleration: Fnet = ma

Torque causes angular acceleration: net = I?

Physics 1D03

Q: How much torque does it take to rotate a particular object?

Better: How much torque does it take to change the rate ofrotation?

What property of an object determines the response (angular acceleration) to an unbalanced external torque?

Force causes linear acceleration: Fnet = ma

Torque causes angular acceleration: net = I?

Physics 1D03

The angular acceleration of a particle is proportional to the net torque applied to it.

Example: A particle accelerates in a circle. Break the net force on it into radial and tangential components. Only Ft causes tangential acceleration:

Ft

Fr

r

Ft = mat = m(r ), since at = r

Multiply by r : rFt = mr 2

or torque = (mr 2)

Physics 1D03

Fext

f-f

ri

mi

For a rigid body, is the same for all particles. The net force on each particle is composed of internal forces f, and external forces Fext.

The total torque is the sum of the torques on the individual particles:

i

iii

iii

i rmrm )( 22

Torques due to the internal forces f and –f cancel when we sum over all particles. The quantity in brackets is the “moment of inertia”:

i

iirm 2I

and Newton’s 2nd law for rotation is I external

Physics 1D03

The Moment of Inertia: i

iirm 2I

“I” measures the rotational inertia of an object. It depends on:

(r is the distance from a particle to the axis of rotation.)

Units: kg m2

1) the total mass. I is proportional to mass.

2) how the mass is distributed (distance from the rotation axis). I is proportional to (linear size)2, and is larger if the mass is concentrated farther from the rotaion axis.

3) which axis the object rotates about.

Physics 1D03

Moments of inertia for uniform objects can be calculated using integral calculus. A few results (see table 10.2 in the text):

I=MR2 I = ½ MR2

L

I = 1/12 ML2

L

I = 1/3 ML2

2

5

2MRI

R

2

3

2MRI

R

Physics 1D03

Quiz

Three point particles, each of mass m, are arranged in the x-y plane in an equilateral triangle of side 2b as shown. What is the moment of inertia:

1) about the x axis? - clicker2) about the y axis?3) about the z-axis?

a) mb2

b) 2mb2

c) 3mb2

d) 4mb2

e) 5mb2

x

y

b b

2b2b

Physics 1D03

Quiz

Two spheres are made of the same material. However, all the dimensions of sphere B are twice as large as those of sphere A. If the spheres are launched with the same torques, applied for equal times, sphere A accelerates faster than B by a factor of :

a) 2

b) 4

c) 8

d) 16

e) 32

Physics 1D03

Quiz

Two spinning tops have equal radius and mass, but whereas top B is a thin disc, top A has light spokes connecting the hub to an outer ring. The tops are spun by applying equal torques to the spindle. Which spins the fastest?

Top View

a) A

b) B

c) spin at same rate

d) not enough info.A B

Physics 1D03

Quiz

Two spinning tops are made from circular disks with a pointed spindle inserted through the center. The tops have the same radii and total mass, but since they are made of different material, top A is thicker. The tops are launched by equal torques to the spindles. Which spin is the fastest?

a) A

b) B

c) spin at the same rate

d) not enough info

A B

Physics 1D03

Quiz

A uniform board is attached to the edge of a table at one end by a hinge. A coin is placed on the other end, and the free end of the board is held so that the board is horizontal.

When the board is released:

a) the board falls faster than the coin

b) the board and coin fall freely together

c) the coin would fall faster, but presses on the board

d) it depends on the length of the board

coin

Physics 1D03

Example

Two (thin, uniform) sticks of different lengths are held nearly upright (at a slight angle to the vertical) on a table and released

simultaneously. Which hits the ground first? (Try this at home.)

Physics 1D03

Summary

The moment of inertia measures the rotational inertia of a body.

Particle: I = mr2

i

iirm 2IExtended Body:

I externalNewton’s 2nd law for rotation about a fixed axis: