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Rotational Motion (cont.)

TorqueMoment of Inertia

Angular MomentumKinetic Energy/WorkEquilibrium (revisited)

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Definition ofTorque

Torqueis defined as the tendency toproduce a change in rotational motion.

Examples:

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Torque is Determined by Three Factors:

The magnitude of the applied force.

The direction of the applied force.

The location of the applied force.

20 N

Magnitude of force

40 N

The 40-N force

produces twice thetorque as does the

20-N force.

Each ofthe 20-N

forces has a differenttorque due to the

direction offorce. 20 N

Direction of Force

20N

UU

20 N

20 N

Location of forceThe forces nearerthe

endofthe wrenchhave greatertorques.

20 N

20 N

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Estimating Torque

Torque is proportional to the magnitude ofFand to the distancerfrom the axis. Thus,atentativeformula might be:

Units: Nm

X sinFrFr !v!

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Torque

If a force is applied at an angle to the radial line, Only the component of the force perpendicular to the radial

line contributes to the torque.

The parallel component of the force points along the radialline and contributes nothing to the torque.

If is the angle between the radial line and the force, thenthe perpendicular component of the force is, F= F sin.

= r F= r F sin

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Example 1

You spin a bicycle wheel (diameter of 0.85 m, mass of

4.5 kg), applying a force of 24 N tangentially. What is

the torque on the wheel?

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Example 2

An 80-N force acts at the end of a 12-cm wrench as shown.

Find the torque.

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Moment ofInertia

Moment of Inertia: I : the ability of an object to resistchanges in its rotational motion. The rotational equivalentof mass.

Dependent upon the mass of the object

Dependent upon the position of the mass of the object

relative to the axis of rotation. Bigger I - harder to accelerate (rotationally)

If the object is closer to the axis it will be easier to rotate.

If the object is further from the axis it will be harder to rotate.

2

iirmI !

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Common Moments of Inertia

Particle in a circle I = MR2

Solid disk / wheel I = 1/2MR2

Sphere I = 2/5MR2

Spherical Shell I = 2/3MR2

Thin rod rotating around middle I = 1/12ML2

Thin rod rotating around end I = 1/3ML2

Axis of rotation: the axis about which rotation

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Other Moments ofInertiabicycle rim

filled can of coke

baton

baseball bat

basketball

boulder

cylindricalshell : I! MR2

solidcylinder: I!1

2MR

2

rodabout

center: I

!

1

12ML

2

rodaboutend : I!1

3ML

2

sphericalshell : I!2

3MR2

solidsphere : I!2

5MR

2

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Example 3

A uniform wheel of mass m = 2kg and radiusr = 0.25m rotates about its axis. Find itsmoment of inertia.

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Rotational Dynamics / NSL

EX

UX

I

FrFr

aF

!

!v!

!

sin

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Rotational Kinetic Energy

KEof center-of-mass motion

KE due to rotation

Each point of a rigid body rotates with angular

velocity [.

Including the linear motion

KE!1

2mivi

2 !

1

2miri

2[

2

KE!

1

2 I[

2

KE!

1

2 mv2

1

2 I[

2

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Example 4W

hat is the kinetic energy of theEarth due to thedaily rotation?

Given: Mearth=5.98 x1024 kg, Rearth = 6.36 x10

6 m.

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Example 5

A uniform wheel of mass M = 2 kg and radius R = 0.25mrotates about its axis. If the wheel starts at rest, whatangular speed is the wheel rotating at if a torque =150Nm is applied to the edge of the disk for t = 3:00s?

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L ! I[

L ! mvr! m[r2

Angular Momentum

Analogy between L and pAngular Momentum Linear momentum

L = I[ p = mv

X= (L/(t F = (p/(t

Conserved if no netoutside torques

Conserved if no netoutside forces

Rigid body

Point particle

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Example 6

A 145 g one-holed stopper is tied to a string and twirledin a circle with a radius of 75cm. If the stopper makes 10revolutions per 18.5s, what is the angular momentum?

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Workdone by Torque

UX (!W

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Equilibrium

An object undergoing uniform motion is said tobe in equilibrium. Two conditions of equilibrium:

0.2

0.1

!!

!!

EX I

amF

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Definition ofa Vector Product

Themagnitude of the vector (cross)product of two vectorsA andB is defined asfollows:

A x B = l A l l B l SinU

F x r = l Fll rl SinUMagnitude onlyMagnitude only

F

(F SinU ) r or F (r SinU)

In our example, the crossproduct of F and r is:

In effect, this becomes simply:U

r

F Sin U

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Direction of the Vector Product.

TheThedirectiondirectionof aof avector product isvector product isdetermined by thedetermined by the

right hand rule.right hand rule. A

C

B

B

-CA

A x B = C (up)A x B = C (up)

B x A =B x A = --C (Down)C (Down)

Curl fingers of right handCurl fingers of right hand

in direction of cross proin direction of cross pro--duct (duct (AA totoBB) or () or (BB totoAA).).ThumbThumbwill point in thewill point in thedirection of productdirection of productCC..