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Chapter 8

- Torque and Angular Momentum

Torque and Angular Momentum

- Rotational Kinetic Energy
- Rotational Inertia
- Torque
- Work Done by a Torque
- Equilibrium (revisited)
- Rotational Form of Newtons 2nd Law
- Rolling Objects
- Angular Momentum

Rotational KE and Inertia

For a rotating solid body

For a rotating body vi ?ri where ri is the

distance from the rotation axis to the mass mi.

Moment of Inertia

The quantity

is called rotational inertia or moment of inertia.

Use the above expression when the number of

masses that make up a body is small. Use the

moments of inertia in the table in the textbook

for extended bodies.

Moments of Inertia

(No Transcript)

Moment of Inertia

Example The masses are m1 and m2 and they are

separated by a distance r. Assume the rod

connecting the masses is massless. Q (a) Find

the moment of inertia of the system below.

r1 and r2 are the distances between mass 1 and

the rotation axis and mass 2 and the rotation

axis (the dashed, vertical line) respectively.

Moment of Inertia

Take m1 2.00 kg, m2 1.00 kg, r1 0.33 m , and

r2 0.67 m.

(b) What is the moment of inertia if the axis is

moved so that is passes through m1?

Moment of Inertia

What is the rotational inertia of a solid iron

disk of mass 49.0 kg with a thickness of 5.00 cm

and a radius of 20.0 cm, about an axis through

its center and perpendicular to it?

From the table

Torque

A torque is caused by the application of a force,

on an object, at a point other than its center of

mass or its pivot point.

Q Where on a door do you normally push to open

it? A Away from the hinge.

A rotating (spinning) body will continue to

rotate unless it is acted upon by a torque.

Torque

Torque method 1

Top view of door

r the distance from the rotation axis (hinge)

to the point where the force F is applied.

F? is the component of the force F that is

perpendicular to the door (here it is Fsin?).

Torque

The units of torque are Newton-meters (Nm) (not

joules!).

By convention

- When the applied force causes the object to

rotate counterclockwise (CCW) then ? is positive. - When the applied force causes the object to

rotate clockwise (CW) then ? is negative.

Torque

Torque method 2

r? is called the lever arm and F is the magnitude

of the applied force.

Lever arm is the perpendicular distance to the

line of action of the force.

Torque

Top view of door

Torque Problem

The pull cord of a lawnmower engine is wound

around a drum of radius 6.00 cm, while the cord

is pulled with a force of 75.0 N to start the

engine. What magnitude torque does the cord

apply to the drum?

F75 N

R6.00 cm

Torque Problem

Calculate the torque due to the three forces

shown about the left end of the bar (the red X).

The length of the bar is 4m and F2 acts in the

middle of the bar.

Torque Problem

The lever arms are

Torque Problem

The torques are

The net torque is 65.8 Nm and is the sum of the

above results.

Work done by the Torque

The work done by a torque ? is

where ?? is the angle (in radians) that the

object turns through.

Following the analogy between linear and

rotational motion Linear Work is Force x

displacement. In the rotational picture force

becomes torque and displacement becomes the angle

Work done by the Torque

A flywheel of mass 182 kg has a radius of 0.62 m

(assume the flywheel is a hoop).

(a) What is the torque required to bring the

flywheel from rest to a speed of 120 rpm in an

interval of 30 sec?

Work done by the Torque

(b) How much work is done in this 30 sec period?

Equilibrium

The conditions for equilibrium are

Linear motion

Rotational motion

For motion in a plane we now have three equations

to satisfy.

Using Torque

A sign is supported by a uniform horizontal boom

of length 3.00 m and weight 80.0 N. A cable,

inclined at a 35? angle with the boom, is

attached at a distance of 2.38 m from the hinge

at the wall. The weight of the sign is 120.0 N.

What is the tension in the cable and what are the

horizontal and vertical forces exerted on the

boom by the hinge?

Using Torque

This is important! You need two components for

F, not just the expected perpendicualr normal

force.

FBD for the bar

Apply the conditions for equilibrium to the bar

Using Torque

Equation (3) can be solved for T

Equation (1) can be solved for Fx

Equation (2) can be solved for Fy

Equilibrium in the Human Body

Find the force exerted by the biceps muscle in

holding a one liter milk carton with the forearm

parallel to the floor. Assume that the hand is

35.0 cm from the elbow and that the upper arm is

30.0 cm long. The elbow is bent at a right

angle and one tendon of the biceps is attached at

a position 5.00 cm from the elbow and the other

is attached 30.0 cm from the elbow. The weight

of the forearm and empty hand is 18.0 N and the

center of gravity is at a distance of 16.5 cm

from the elbow.

MCAT type problem

Newtons 2nd Law in Rotational Form

Compare to

Rolling Object

A bicycle wheel (a hoop) of radius 0.3 m and mass

2 kg is rotating at 4.00 rev/sec. After 50 sec

the wheel comes to a stop because of friction.

What is the magnitude of the average torque due

to frictional forces?

Rolling Objects

An object that is rolling combines translational

motion (its center of mass moves) and rotational

motion (points in the body rotate around the

center of mass).

For a rolling object

If the object rolls without slipping then vcm

R?.

Note the similarity in the form of the two

kinetic energies.

Rolling Example

Two objects (a solid disk and a solid sphere) are

rolling down a ramp. Both objects start from

rest and from the same height. Which object

reaches the bottom of the ramp first?

This we know - The object with the largest linear

velocity (v) at the bottom of the ramp will win

the race.

Rolling Example

Apply conservation of mechanical energy

Solving for v

Rolling Example

Example continued

Note that the mass and radius are the same.

The moments of inertia are

For the disk

Since Vspheregt Vdisk the sphere wins the race.

For the sphere

Compare these to a box sliding down the ramp.

The Disk or the Ring?

How do objects in the previous example roll?

FBD

Both the normal force and the weight act through

the center of mass so ?? 0. This means that

the object cannot rotate when only these two

forces are applied.

x

The round object wont rotate, but most students

have difficulty imagining a sphere that doesnt

rotate when moving down hill.

Add Friction

FBD

Also need acm ?R and

The above system of equations can be solved for v

at the bottom of the ramp. The result is the

same as when using energy methods. (See text

example 8.13.)

It is the addition of static friction that makes

an object roll.

Angular Momentum

Units of p are kg m/s

Units of L are kg m2/s

When no net external forces act, the momentum of

a system remains constant (pi pf)

When no net external torques act, the angular

momentum of a system remains constant (Li Lf).

Angular Momentum

Units of p are kg m/s

Units of L are kg m2/s

When no net external forces act, the momentum of

a system remains constant (pi pf)

When no net external torques act, the angular

momentum of a system remains constant (Li Lf).

Angular Momentum Example

A turntable of mass 5.00 kg has a radius of 0.100

m and spins with a frequency of 0.500 rev/sec.

Assume a uniform disk. What is the angular

momentum?

Angular Momentum Example

A skater is initially spinning at a rate of 10.0

rad/sec with I2.50 kg m2 when her arms are

extended. What is her angular velocity after

she pulls her arms in and reduces I to 1.60 kg m2?

The skater is on ice, so we can ignore external

torques.

The Vector Nature of Angular Momentum

Angular momentum is a vector. Its direction is

defined with a right-hand rule.

The Right-Hand Rule

Curl the fingers of your right hand so that they

curl in the direction a point on the object

moves, and your thumb will point in the direction

of the angular momentum.

Angular Momentum is also an example of a vector

cross product

The Vector Cross Product

The magnitude of C C ABsin(F) The direction of

C is perpendicular to the plane of A and B.

Physically it means the product of A and the

portion of B that is perpendicular to A.

The Cross Product by Components

Since A and B are in the x-y plane A x B is along

the z-axis.

Memorizing the Cross Product

The Gyroscope Demo

Angular Momentum Demo

Consider a person holding a spinning wheel. When

viewed from the front, the wheel spins CCW.

Holding the wheel horizontal, they step on to a

platform that is free to rotate about a vertical

axis.

Angular Momentum Demo

Initially, nothing happens. They then move the

wheel so that it is over their head. As a

result, the platform turns CW (when viewed from

above).

This is a result of conserving angular momentum.

Angular Momentum Demo

Initially there is no angular momentum about the

vertical axis. When the wheel is moved so that

it has angular momentum about this axis, the

platform must spin in the opposite direction so

that the net angular momentum stays zero.

Is angular momentum conserved about the direction

of the wheels initial, horizontal axis?

It is not. The floor exerts a torque on the

system (platform person), thus angular momentum

is not conserved here.

Summary

- Rotational Kinetic Energy
- Moment of Inertia
- Torque (two methods)
- Conditions for Equilibrium
- Newtons 2nd Law in Rotational Form
- Angular Momentum
- Conservation of Angular Momentum

X

Z

Y

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