Rotational motion

Chapter 9

Rigid objectsA rigid object has a perfectly definite

and unchanging shape and size. In this class, we will approximate

everything as a rigid object

Radians In describing rotational motion, we will

use angles in radians, not degrees.

in radians

r

s

rad 180 rad 2

90

in radiansAn angle in radians is the ratio of two

lengths, so it has no units.We will often write “rad” as the units on

such an angle to make it clear that it’s not in degrees

But in calculations, “rad” doesn’t factor into unit analysis.

Angular velocityRate of change of (omega) is the symbol for angular

velocity

tav

tt

0

lim

Angular velocityAt any instant, all points on a rigid

object have the same angular velocity.The units of angular velocity are rad/s.Sometimes angular velocity is given in

rev/s or rpm.1 rev is 2 radiansAngular speed is the magnitude of

angular velocity

Angular accelerationRate of change of angular velocity (alpha) is the symbol for angular

acceleration

tav

tt

0

lim

Angular accelerationThe units for angular acceleration are

rad/s2.

Comparison x is linear position v is linear velocity a is linear

acceleration

is angular position is angular velocity is angular

acceleration

Rotation with constant angular acceleration

200 2

1attvxx

020

2 2 xxavv

tvv

xx

2

00

atvv 0

200 2

1tt

020

2 2

t

2

00

t 0

ExampleA CD rotates from rest to 500 rev/min in

5.5 s. What is its angular acceleration, assuming

it is constant? How many revolutions does the disk make

in 5.5 s? 9.52 rad/s22.9 rev

Relating linear and angular kinematicsWe might want to know the linear speed

and acceleration of a point on a rotating rigid object.

So we need relationships betweenv and a and

Speed relationship

Note: these are speeds, not velocities

rv

Acceleration relationship

rr

varad

22

Change in direction

ExampleFind the required angular speed, in

rev/min, of an ultracentrifuge for the radial acceleration of a point 2.50 cm from the axis to equal 400,000 times the acceleration due to gravity.

1.25 x 104 rad/s = 1.19 x 105 rev/min

Moment of InertiaRotating objects have inertia, but is

more than just their mass. It depends on how that mass is

distributed.

Moment of inertiaThe moment of inertia, I, of an object is

found by taking the sum of the mass of each particle in the object times the square of it’s perpendicular distance from the axis of rotation.

2222

211 ...

iiirmrmrmI

Moment of InertiaFor continuous distributions of particles,

i.e. large objects, the sum becomes an integral.

The moments of inertia for several familiar shapes with uniform densities are given on page 215 of your book.

Moments of inertia are given in terms of masses and dimensions.

Kinetic energy of rotating objects

2

2

1 IK

Gravitational potential energy of rotating objectsSame as for other objects, but use total

mass and position of the center of mass.

MgYU

ExampleA uniform thin rod of length L and mass

M, pivoted at one end, is held horizontal and then released from rest. Assuming the pivot is frictionless, find The angular velocity of the rod when it

reaches its vertical position Sqrt(3g/L)

On your own An airplane propeller (I=(1/12)ML2) is 2.08 m

in length (from tip to tip) with mass 117 kg. The propeller is rotating at 2400 rev/min about an axis through it’s center. What is its rotational kinetic energy? If it were not rotating, how far would it have to drop

in free fall to acquire the same kinetic energy? 1.33 x 106 J 1.16 km

TorqueThe measure of the tendency of a force

to change the rotational motion of a object.

Torque depends on the perpendicular distance between the force and the axis of rotation

Torque magnitude

Where (tau) [your book uses (gamma)] is the magnitude of the torque Also called moment

F is the magnitude of the force l is the perpendicular distance between

the force and the axis of rotation. Also called lever arm or moment arm

Fl

Torque magnitude

Torque magnitude

Torque Magnitude

Torque signCounterclockwise rotation is caused by

positive torques and clockwise rotation is caused by negative torques.

We can use this symbol to indicate which direction is positive torque.

+

Torque UnitsThe SI-unit of torque is the Newton-

meter.Torque is not work or energy, so it

should not be expressed as Joules.

Torque Vector Direction

Fl

sinrF

Visual aid for torque direction

Think of a normal, right-handed screw.

The torque vector points in the direction the screw moves.

Torque is perpendicular to both r and F.

Discussion QuestionWhy are doorknobs located far from the

hinges?

Example

Forces F1 = 8.60 N and F2 = 2.40 N are applied tangentially to a wheel with a radius of 1.50 m, as shown on the next slide. What is the net torque on the wheel if it rotates on an axis perpendicular to the wheel and passing through its center?

F2

F1

You tryCalculate the torque (magnitude and

direction) about point O due to the force shown below. The bar has a length of 4.00 m and the force is 30.0 N.

O 2 m

F

= 60°

Torque and angular acceleration

Only valid for rigid objects must be in rad/s2 for units to work

I

ExampleA torque of 32.0 N-m on a certain wheel

causes an angular acceleration of 25.0 rad/s2. What is the wheel’s moment of inertia?

On your ownA solid sphere has a radius of 1.90 m.

An applied torque of 960 N-m gives the sphere an angular acceleration of 6.20 rad/s2 about an axis through its center. Find The moment of inertia of the sphere The mass of the sphere

ExampleAn object of mass m is tied to a light

string wound around a wheel that has a moment of inertia I and radius R. The wheel is frictionless, and the string does not slip on the rim. Find the tension in the string and the acceleration of the object.

T=(I/(I+mR2)*mg a=(mR2/(I+mR2))g

On your own

a

On your ownTwo blocks are connected by a string

that passes over a pulley of radius R and moment of inertia I. The block of mass m1 slides on a frictionless, horizontal surface; the block of mass m2 is suspended from the string. Find the acceleration a of the blocks and the tensions T1 and T2 assuming that the string does not slip on the pulley.

a=(m2/(m1+m2+I/R2))m2gT1=(m1/(m1+m2+I/R2))m2gT2=((m1+I/R2)/(m1+m2+I/R2))m2g

Rigid object rotation about a moving axisCombined translation and rotation.

Translation of center of mass Rotation about the center of mass

There is friction, but only static friction to keep the object from slipping

Kinetic EnergyThe kinetic energy is the sum of

translational and rotational kinetic energies.

22

2

1

2

1 comImVK

Rolling without slippingWhen something is rolling without

slipping,

RV

RA

On your ownA hollow cylindrical shell with mass M

and radius R rolls without slipping with speed V on a flat surface. What is its kinetic energy?

MV2

ExampleA solid disk and a hoop with the same

mass and radius roll down an incline of height h without slipping.

Which one reaches the bottom first? The disk

What if they had different masses?Different radii?

Dynamics of translating and rotating objectsWe can use both Newton’s 2nd law and

its rotational counterpart

mAF

I

ExampleA uniform solid ball of mass m and

radius R rolls without slipping down a plane inclined at an angle . A frictional force f is exerted on the ball by the incline. Find the acceleration of the center of mass.

(5/7) gsin

Work and Power

Work done by a constant torque

W

Work and Kinetic Energy

Total work done equals change in K

PowerPower is the rate of doing work

dt

d

dt

dW

P

FvP

Example A uniform disk with a mass of 120 kg and a

radius of 1.4 m rotates initially with an angular speed of 1100 rev/min. A constant tangential force is applied at a radial distance of 0.6 m. What work must this force do to stop the wheel?

780 kJ

If the wheel is brought to rest in 2.5 min, what torque does the force produce?

90.4 N-m

What is the magnitude of the force? 151 N

On your own A playground merry-go-round has a radius of

2.40 m and a moment of inertia 2100 kg-m2 about a vertical axle through its center, and turns with negligible friction. A child applies an 18.0-N Force tangentially to the edge of the merry-go-round for 15.0 s. If the merry-go-round is initially at rest, what is its

angular speed after this 15.0-s interval? How much work did the child do on the merry-go-

round? What is the average power supplied by the child?

0.309 rad/s 100 J 6.67 W

Angular momentumRelationship between angular

momentum and linear momentum is the same as between torque and force.

mvrrpL

rF

UnitsThe units for angular momentum are

kg-m2/s

Angular momentum of rigid objects

Look at L for one particle of the object

2iiiiiii rmrrmvrmL

Sum over all particles for total angular momentum

iii

ii rmLL 2

IL

ExampleA woman with mass 50 kg is standing

on the rim of a large disk that is rotating at 0.50 rev/s about an axis through its center. The disk has mass 110 kg and a radius of 4.0 m. Calculate the magnitude of the total angular momentum of the woman-plus-disk system. You can treat the woman as a point.

5275 kg-m2/s

On your ownFind the magnitude of the angular

momentum of the sweeping second hand on a clock about an axis through the center of the clock face. The clock hand has a length of 15.0 cm and a mass of 6.00 g. Take the second hand to be a slender rod rotating with constant angular velocity about one end.

4.71 x 10-6 kg-m2/s

Conservation of angular momentum If there is no net external torque acting

on a system, then the total angular momentum of the system is conserved.

fi LL

ExampleA uniform circular disk is rotating with an

initial angular speed 1 around a frictionless shaft through its center. Its moment of inertia is I1. It drops onto another disk of moment of inertia I2 that is initially at rest on the same shaft. Because of surface friction between the disks, they eventually attain a common angular speed f. Find f.