Rotational Dynamics

Rotational Dynamics

Lecture 15

Reading and Review

Moment of Inertia

The moment of inertia I:

The total kinetic energy of a rolling object is the sum of its linear and rotational kinetic energies:

Rolling DownRolling Down

Two spheres start rolling down a ramp from the same height at the same time. One is made of solid gold, and the other of solid aluminum.

Which one reaches the bottom first?

a) a) solid aluminum

b) solid goldb) solid gold

c) samec) same

d) can’t tell without more information

Rolling DownRolling Down

Two spheres start rolling down a ramp from the same height at the same time. One is made of solid gold, and the other of solid aluminum.



b) solid goldb) solid gold

c) samec) same


Moment of inertia depends on mass and distance from axis squared. For a sphere:

I = 2/5 MR2

But you don’t need to know that! All you need to know is that it depends on MR2

initial PE: mgh

final KE:

MR2 cancels out!Mass and radius don’t matter, only the distribution of mass

(shape)!

Moment of InertiaMoment of Inertia

Two spheres start rolling down a ramp at the same time. One is made of solid aluminum, and the other is made from a hollow shell of gold.



b) hollow goldb) hollow gold

c) samec) same


Moment of InertiaMoment of Inertia


b) hollow goldb) hollow gold

c) samec) same


A solid sphere has more of its mass close to the center. A shell has all of its mass at a large radius.

initial PE: mgh

final KE:

A shell has a larger moment of inertia than a solid object of the same mass, radius and shape

Larger moment of inertia -> lower velocity for the same energy.

Two spheres start rolling down a ramp at the same time. One is made of solid aluminum, and the other is made from a hollow shell of gold.


•Power output of the Crab pulsar, in radio and X-rays, is about 6 x 1031 W (which is about 150,000 times the power output of our sun). Since the pulsar is out of nuclear fuel, where does all this energy come from ?

KE i 1

2I 2

KE f 1

2I 2

1

2I 2

1

2I 2 1

2I 2

KE I I 2

Power output of the Crab pulsar

•calculate the rotational kinetic energy at the beginning and at the end of a second, by taking the moment of inertia to be 1.2x1038 kg-m2 and the initial angular speed to be 190 s-1. Δω over one second is given by the angular acceleration.

•The angular speed of the pulsar, and so the rotational kinetic energy, is going down over time. This kinetic energy is converted into the energy coming out of that star.

•Power output of the Crab pulsar, in radio and X-rays, is about 6 x 1031 W (which is about 150,000 times the power output of our sun). Since the pulsar is out of nuclear fuel, where does all this energy come from ?

Power output of the Crab pulsar

•calculate the rotational kinetic energy at the beginning and at the end of a second, by taking the moment of inertia to be 1.2x1038 kg-m2 and the initial angular speed to be 190 s-1. Δω over one second is given by the angular acceleration.

•The angular speed of the pulsar, and so the rotational kinetic energy, is going down over time. This kinetic energy is converted into the energy coming out of that star.

Torque

We know that the same force will be much more effective at rotating an object such as a nut or a door if our hand is not too close to the axis.

This is why we have long-handled wrenches, and why doorknobs are not next to hinges.

The torque increases as the force increases, and also as the distance increases.

Only the tangential component of force causes a torque

A more general definition of torque:

Fsinθ

Fcosθ

You can think of this as either:

- the projection of force on to the tangential directionOR

- the perpendicular distance from the axis of rotation to line of the force

r F

Right Hand Rule

Torque

If the torque causes a counterclockwise angular acceleration, it is positive; if it causes a clockwise angular acceleration, it is negative.

Using a WrenchUsing a Wrench

You are using a wrench to

loosen a rusty nut. Which

arrangement will be the

most effective in

tightening the nut?

a

cd

b

e) all are equally effective

You are using a wrench to

loosen a rusty nut. Which

arrangement will be the

most effective in

tightening the nut?

a

cd

b

Because the forces are all the same, the only difference is the lever arm. The arrangement with the largest largest lever armlever arm (case #2case #2) will

provide the largest torquelargest torque. e) all are equally effective

Using a WrenchUsing a Wrench

The gardening tool shown is used to pull weeds. If a 1.23 N-m torque is required to pull a given weed, what force did the weed exert on the tool?

What force was used on the tool?

Force and Angular Acceleration

Consider a mass m rotating around an axis a distance r away.

Or equivalently,

Newton’s second law:

a = r α

Torque and Angular Acceleration

Once again, we have analogies between linear and angular motion:

The L-shaped object shown below consists of three masses connected by light rods. What torque must be applied to this object to give it an angular acceleration of 1.2 rad/s2 if it is rotated about (a) the x axis,(b) the y axis(c) the z axis (through the origin and perpendicular to the page) (a)

(b)

(c)

Torque

Project the force onto the tangential direction

Fsinθ

Fcosθ

Only the tangential component of force causes a torque

sin

r F

rF rF

Torque and Angular Acceleration

Angular motion is analogous to linear motion

The L-shaped object shown below consists of three masses connected by light rods. What torque must be applied to this object to give it an angular acceleration of 1.2 rad/s2 if it is rotated about an axis parallel to the y axis, and through the 2.5kg mass?

A 2.85-kg bucket is attached to a disk-shaped pulley of radius 0.121 m and mass 0.742 kg. If the bucket is allowed to fall, (a) what is its linear acceleration? (b) What is the angular acceleration of the pulley? (c) How far does the bucket drop in 1.50 s?

A 2.85-kg bucket is attached to a disk-shaped pulley of radius 0.121 m and mass 0.742 kg. If the bucket is allowed to fall, (a) what is its linear acceleration? (b) What is the angular acceleration of the pulley? (c) How far does the bucket drop in 1.50 s?

Pulley spins as bucket falls

(c)

(b)

(a)

Static Equilibrium

X

Static equilibrium describes an object at rest – neither rotating nor translating.

If the net torque is zero, it doesn’t matter which axis we consider rotation to be around; you choose the axis of rotationThis can greatly simplify a problem

Center of Mass and Gravitational Force on an Extended Object

center of massmjm1

xj

...

xj

mjm1

Fj = mj g

...X

axis of rotation

xcm

F = Mg

X

axis of rotation

Balance

If an extended object is to be balanced, it must be supported through its center of mass.

Center of Mass and BalanceThis fact can be used to find the center of mass of an object – suspend it from different axes and trace a vertical line. The center of mass is where the lines meet.

Balancing RodBalancing Rod

1kg

1m

A 1-kg ball is hung at the end of a rod

1-m long. If the system balances at a

point on the rod 0.25 m from the end

holding the mass, what is the mass of

the rod?

a) ¼ kg

b) ½ kg

c) 1 kg

d) 2 kg

e) 4 kg

1 kg

X

CM of rod

same distance mROD = 1 kg

A 1-kg ball is hung at the end of a rod

1-m long. If the system balances at a

point on the rod 0.25 m from the end

holding the mass, what is the mass of

the rod?

The total torque about the The total torque about the

pivot must be zero !!pivot must be zero !! The CM

of the rod is at its center, 0.25 0.25

m to the right of the pivotm to the right of the pivot.

Because this must balance the

ball, which is the same same

distance to the left of the pivotdistance to the left of the pivot,

the masses must be the

same !!

a) ¼ kg

b) ½ kg

c) 1 kg

d) 2 kg

e) 4 kg

Balancing RodBalancing Rod

When you arrive at Duke’s Dude Ranch, you are greeted by the large wooden sign shown below. The left end of the sign is held in place by a bolt, the right end is tied to a rope that makes an angle of 20.0° with the horizontal. If the sign is uniform, 3.20 m long, and has a mass of 16.0 kg, what is (a) the tension in the rope, and (b) the horizontal and vertical components of the force, exerted by the bolt?

When you arrive at Duke’s Dude Ranch, you are greeted by the large wooden sign shown below. The left end of the sign is held in place by a bolt, the right end is tied to a rope that makes an angle of 20.0° with the horizontal. If the sign is uniform, 3.20 m long, and has a mass of 16.0 kg, what is (a) the tension in the rope, and (b) the horizontal and vertical components of the force exerted by the bolt? Torque, vertical force, and horizontal force are all zero

But I don’t know two of the forces!

I can get rid of one of them, by choosing my axis of rotation where the force is applied.

Choose the bolt as the axis of rotation, then:

(b)

Dumbbell IDumbbell I

a) case (a)a) case (a)

b) case (b)b) case (b)

c) no differencec) no difference

d) it depends on the rotational d) it depends on the rotational inertia of the dumbbellinertia of the dumbbell

A force is applied to a dumbbell for a certain period of time, first as in (a) and then as in (b). In which case does the dumbbell acquire the greatercenter-of-mass speed ?

Dumbbell IDumbbell I

a) case (a)a) case (a)

b) case (b)b) case (b)

c) no differencec) no difference

d) it depends on the rotational d) it depends on the rotational inertia of the dumbbellinertia of the dumbbell

A force is applied to a dumbbell for a certain period of time, first as in (a) and then as in (b). In which case does the dumbbell acquire the greatercenter-of-mass speed ?

Because the same force acts for the same time interval in both cases, the change in momentum must be the same, thus the CM velocity must be the same.

Linear momentum was the concept that tied together Newton’s Laws, is there something similar for rotational motion?

F = ma implies Newton’s first law: without a force, there is no acceleration

Now we have

Angular Momentum

Consider a particle moving in a circle of radius r,

I = mr2

L = Iω = mr2ω = rm(rω) = rmvt = rpt

Angular Momentum

For more general motion (not necessarily circular),

The tangential component of the momentum, times the distance

Angular Momentum

For an object of constant moment of inertia, consider the rate of change of angular momentum

analogous to 2nd Law for Linear Motion

Conservation of Angular Momentum

If the net external torque on a system is zero, the angular momentum is conserved.

As the moment of inertia decreases, the angular speed increases, so the

angular momentum does not change.

Conservation of Angular Momentum

Angular momentum is also conserved in rotational collisions

Figure SkaterFigure Skater

a)a) the samethe same

b)b) larger because she’s rotating larger because she’s rotating fasterfaster

c) smaller because her rotational c) smaller because her rotational inertia is smallerinertia is smaller

A figure skater spins with her arms extended. When she pulls in her arms, she reduces her rotational inertiaand spins faster so that her angular momentum is conserved. Comparedto her initial rotational kinetic energy, her rotational kinetic energy after she pulls in her arms must be:

Figure SkaterFigure Skater

a)a) the samethe same

b)b) larger because she’s rotating larger because she’s rotating fasterfaster

c) smaller because her rotational c) smaller because her rotational inertia is smallerinertia is smaller

A figure skater spins with her arms extended. When she pulls in her arms, she reduces her rotational inertiaand spins faster so that her angular momentum is conserved. Comparedto her initial rotational kinetic energy, her rotational kinetic energy after she pulls in her arms must be:

KErot = I 2 = L2 I (used L =

I). Because L is conserved,

smaller I means larger KErot. The

“extra” energy comes from the work she does on her arms.

Rotational WorkA torque acting through an angular displacement does work, just as a force acting through a distance does.

The work-energy theorem applies as usual.

Consider a tangential force on a mass in circular motion: τ = r F

s = r ΔθW = s F

Work is force times the distance on the arc:

W = (r Δθ) F = rF Δθ = τ Δθ

Rotational Work and Power

Power is the rate at which work is done, for rotational motion as well as for translational motion.

Again, note the analogy to the linear form (for constant force along motion):

The Vector Nature of Rotational Motion

The direction of the angular velocity vector is along the axis of rotation. A right-hand rule gives the sign. Right-hand Rule:

your fingers should follow the velocity vector around the circle

The Torque VectorSimilarly, the right-hand rule gives the direction of the torque vector, which also lies along the assumed axis or rotation

Right-hand Rule: your fingers should follow the force vector around the circle

The linear momentum of components related to the vector angular momentum of the

system

Applied tangential force related to the torque vector

Applied torque over time related to change in the vector angular momentum.

When a tape is played on a cassette deck, there is a tension in the tape that applies a torque to the supply reel. Assuming the tension remains constant during playback, how does this applied torque vary as the supply reel becomes empty?

a) torque increases

b) torque decreases

c) torque remains constant

Cassette PlayerCassette Player

When a tape is played on a cassette deck, there is a tension in the tape that applies a torque to the supply reel. Assuming the tension remains constant during playback, how does this applied torque vary as the supply reel becomes empty?

a) torque increases

b) torque decreases


Cassette PlayerCassette Player

As the supply reel empties, the lever arm decreases because the radius of the reel (with tape on it) is decreasing. Thus, as the playback continues, the applied torque diminishes.

a) torque increases

b) torque decreases


Rotational Dynamics

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