Lecture 15 Rotational Dynamics. Moment of Inertia The moment of inertia I: The total kinetic energy...

Lecture 15

Rotational Dynamics

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:

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

Torque

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

You are using a wrench to

tighten a rusty nut. Which

arrangement will be the

most effective in

tightening the nut?

a

cd

b

e) all are equally effective

Using a WrenchUsing a Wrench

You are using a wrench to

tighten 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 bcase b) 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)

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?

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 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.

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

So, forget about the weight of all the individual pieces. The net torque will be equivalent to the total weight of the object, pulling from the center of mass of the object

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)

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 and linear acceleration

2

2

2 0 00 0

20 0

2 00

13

2 32 cos

2 3 2Wood:

1 2 2 4: 0 2

2 3 3

Ball:1

: 0 and sin2

10 2

2

fall fall fall fall

fall fall fall

fall fall

I ML

MgL MgL gMg L

I ML L

L Ltt tt

g g

t y t y gt y L L

LL gtt

g Wood !fallt

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.

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 = L (used L = I ).

Because L is conserved, larger

means larger KErot.

Where does the “extra” energy come from?

KErot = I 2 = L (used L = I ).

Because L is conserved, larger

means larger KErot.

Where does the “extra” energy come from?

As her hands come in, the velocity of her arms is not only tangential... but also radial.

So the arms are accelerated inward, and the force required times the Δr does the work to raise the kinetic energy

Conservation of Angular Momentum

Angular momentum is also conserved in rotational collisions

larger I, same total angular momentum, smaller angular

velocity

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):





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 greater energy ?

Dumbbell IIDumbbell II





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 greater energy ?

Dumbbell IIDumbbell II

If the CM velocities are the same, the translational kinetic energies must be the same. Because dumbbell (b) is also rotating, it has rotational kinetic energy in

addition.

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)

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

Optional materialSection 11.9

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: point your RtHand fingers along the force, then follow it “around”. Thumb points in direction of torque.


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.


a) remain stationarya) remain stationary

b) start to spin in the same b) start to spin in the same direction as before flippingdirection as before flipping

c) start to spin in the same c) start to spin in the same direction as after flippingdirection as after flipping

You are holding a spinning bicycle wheel while standing on a stationary turntable. If you suddenly flip the wheel over so that it is spinning in the opposite direction, the turntable will:

Spinning Bicycle WheelSpinning Bicycle Wheel

What is the torque (from gravity) around the supporting point?Which direction does it point?

Without the spinning wheel: does this make sense?

With the spinning wheel: how is L changing?

Why does the wheel not fall? Does this violate Newton’s 2nd?

Lecture 15 Rotational Dynamics. Moment of Inertia The moment of inertia I: The total kinetic energy...

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Transcript of Lecture 15 Rotational Dynamics. Moment of Inertia The moment of inertia I: The total kinetic energy...