Rotational Motion

Chap. 10.4-7

NEW CONCEPT‘Rotational force’: Torque

Torque is the “twisting force” that causes rotational motion. It is equal to the magnitude of the component of an applied force perpendicular to the arm transmitting the force.

F

RA

The torque around point A is T = R x F

= I (compare to F = ma)

Example 10-6Spinning Wheel

Solve using torque.Then, using energy.

Example: torque’s in balance

2r 4f

2mm

Figure 10-7Angular and Linear Speed

Conceptual Checkpoint 10-1How do the angular speeds compare?

V=r

How do the linear speeds compare?

Figure 10-8Centripetal and Tangential Acceleration

IMPORTANT:For uniform circular motion, The centripetal acceleration is:

r

vac

2

For constant angular speed, at = 0. Then, the acceleration is RADIAL, inwards.

Figure 10-9Rolling Without Slipping

Figure 10-11Velocities in Rolling Motion

Figure 10-10Rotational and Translational Motions of a Wheel

Figure 10-12Kinetic Energy of a Rotating Object

2

2

1mvK

But… rv

So…

22

2

2

2

12

12

1

mr

rm

mvK

Define the moment of inertia, I…

2mrI

(it’s different for different shapes!)

2

2

1 IKROT

Figure 10-13Kinetic Energy of a Rotating Object of Arbitrary Shape

i

iirmI 2

2

2

1 IK

Figure 10-15The Moment of Inertia of a Hoop

i

iirmI 2

All of the mass elements are at the same radius, so….

2mRIHOOP

Figure 10-16The Moment of Inertia of a Disk

i

iirmI 2

A disk is like a set of nested hoops.

With some algebra….

2

2

1MRIDISK

Table 10-1aMoments of Inertia for Uniform, Rigid Objects

of Various Shapes and Total Mass M

Conceptual Checkpoint 10-2How does the moment of inertia depend on the axis of rotation?

I = 2 (MR2)

I=M(2R)2=4MR2

Kinetic energy of rotating object

Total kinetic energy = Energy of translation plus energy of rotation.

Energy of translation: one-half m vee squared.

Energy of rotation: one-half I omega squared.

22

2

1

2

1 ImvK

A hoop and a disk roll without slipping with the same speed. They have the same mass and radius. Is the kinetic energy of the disk more, less or the same as the hoop?

1. The disk has more K than the hoop.2. The disk has less K than the hoop.3. The disk and the hoop have the

same K.

V

V

Kinetic energy of disk and hoop.

V

V

The kinetic energy is given by:22

2

1

2

1 IMVK

Disk and hoop have same mass,radius, and velocity. How does K compare?

The only difference is the moment of inertia. The disk has SMALLER moment of inertia than the hoop. The further out the mass from the axle, the higher the I.

Example 10-6Spinning Wheel

Solve using torque.Then, using energy.

Figure 10-24Problem 10-60

0 / 100Place your bets!

1. The disk reaches the bottom first.2. The hoop reaches the bottom first.3. They reach the bottom at the same time.

Ihoop = MR2

Idisk = ½ MR2

Who wins?

Both objects have the same potential energy, U = MGH.

More of the potential energy goes into rotational kinetic energy for the hoop, because it has a larger moment of inertia. That leaves less translational energy for the hoop, so it moves slower.

Kinetic energy of “rolling without slipping” objects.

Rx 2

V

2

t

t1.

2.

3. RR

t

xv

22

22

22

22

2

1

2

1

2

1

2

1

2

1

VR

IM

R

VIMV

IMVK

X-treme roller ball.

1. The ball reaches a greater height on the frictionless side.2. The ball reaches a lower height on the frictionless side.3. The ball goes to the same height.

HINT: Consider conservation of energy. What energy is changed into potential energy on the frictionless side?

X-treme roller ball explained.

Lower!At the bottom of the jump, the ball has both translational and rotational energy. Some of the potential energy U=MGH went into rotational energy. On the frictionless side, the ball continues to rotate, no rotational motion is “lost”. So, there is less kinetic energy to change into potential energy, or height.

Rolling down an inclined plane.

Mgsin()f

Note: Rolling friction is STATIC friction!

I

Forces acting on rolling body:Translation

Rotation

IfR

mafmg

sin

“Torque equals I times alpha”

Ra

Ra

Rv

/

For a rolling object:

2

2

sin

sin

R

Imamg

R

aIf

mafmg

Finally:

2

2

1

sin

sin

mRI

gRI

m

mga

Rolling down an inclined plane, summary.

Mgsin()f

2

2

1

sin

sin

mRI

gRI

m

mga

If moment of inertia is zero, then we get the same result as for sliding without friction.

Check the result:

The moment of inertia for symmetrical objects is in the form:I = (number) X MR2

All objects with same SHAPE roll at same speed, independent of mass!!

Atwood Machine with Massive Pulley

Pulley with moment of inertia I, radius R. Given M1, M2, and H, what is the speed of M1 just before it hits the ground?

Strategy: Use conservation of mechanical energy.

Initial kinetic energy is 0. Initial potential energy is M1gH.Final kinetic energy is translational energy of both blocks plus rotational energy of pulley.Final potential energy is M2gH.Set final energy equal to initial energy.