Lecture 16

Rotational Dynamics

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Angular Momentum

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For circular motion:

define: angular momentum

p vF m mr

tt t

mr I LrF mr

tt ttL mr mrv

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.

Thus, 0 constant

I I or Ii i ff ff

Lt

L

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) 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 greater energy ?

Dumbbell IIDumbbell II

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 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.

Optional materialSection 11.9

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

Optional materialSection 11.9

Applied tangential force related to the torque vector

Optional materialSection 11.9

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

Optional materialSection 11.9

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?

Gravity

Newton’s Law of Universal Gravitation

Newton’s insight: The force accelerating an apple downward is the same force that keeps the Moon in its orbit.

Universal Gravitation

The gravitational force is always attractive, and points along the line connecting the two masses:

The two forces shown are an action-reaction pair.

If an object is being acted upon by several different gravitational forces, the net force on it is the vector sum of the individual forces.

This is called the principle of superposition.

G is a very small number; this means that the force of gravity is negligible unless there is a very large mass involved (such as the Earth).

Gravitational Attraction of Spherical Bodies

Gravitational force between a point mass and a sphere*: the force is the same as if all the mass of the sphere were concentrated at its center.

a consequence of 1/r2

(inverse square law) *Sphere must be radial symmetric

Gravitational Force at the Earth’s SurfaceThe center of the Earth is one Earth radius away, so this is the distance we use:

The acceleration of gravity decreases slowly with altitude...

...until altitude becomes comparable to the radius of the Earth. Then the decrease in the acceleration of gravity is much larger:

g

In the Space Shuttle

Astronauts in Astronauts in

the space the space

shuttle float shuttle float

because:because:

a) they are so far from Earth that Earth’s gravity doesn’t act any more

b) gravity’s force pulling them inward is cancelled by the centripetal force pushing them outward

c) while gravity is trying to pull them inward, they are trying to continue on a straight-line path

d) their weight is reduced in space so the force of gravity is much weaker

In the Space Shuttle

Astronauts in Astronauts in

the space the space

shuttle float shuttle float

because:because:

a) they are so far from Earth that Earth’s gravity doesn’t act any more

b) gravity’s force pulling them inward is cancelled by the centripetal force pushing them outward

c) while gravity is trying to pull them inward, they are trying to continue on a straight-line path

d) their weight is reduced in space so the force of gravity is much weaker

Astronauts in the space shuttle float because

they are in “free fall” around Earth, just like a

satellite or the Moon. Again, it is gravity that

provides the centripetal force that keeps them

in circular motion.

Follow-upFollow-up: How weak is the value of : How weak is the value of gg at an altitude of at an altitude of 300 km300 km??

Satellite Motion: FG and acp

Consider a satellite in circular motion*:

* not all satellite orbits are circular!

Gravitational Attraction:

Necessary centripetal acceleration:

• Does not depend on mass of the satellite!• larger radius = smaller velocity smaller radius = larger velocity

Relationship between FG and acp will be important for many gravitational orbit problems

A geosynchronous satellite is one whose orbital period is equal to one day. If such a satellite is orbiting above the equator, it will be in a fixed position with respect to the ground.

These satellites are used for communications and weather forecasting.

How high are they?

RE = 6378 kmME = 5.87 x 1024 kg

Averting Disaster

a) it’s in Earth’s gravitational fielda) it’s in Earth’s gravitational field

b) the net force on it is zerob) the net force on it is zero

c) it is beyond the main pull of Earth’s c) it is beyond the main pull of Earth’s gravitygravity

d) it’s being pulled by the Sun as well as by d) it’s being pulled by the Sun as well as by EarthEarth

e) none of the abovee) none of the above

The Moon does not The Moon does not

crash into Earth crash into Earth

because:because:

The Moon does not crash into Earth because of its

high speed. If it stopped moving, it would, of course,

fall directly into Earth. With its high speed, the Moon

would fly off into space if it weren’t for gravity

providing the centripetal force.

Averting Disaster

The Moon does not The Moon does not

crash into Earth crash into Earth

because:because:

Follow-upFollow-up: What happens to a satellite orbiting Earth as it slows?: What happens to a satellite orbiting Earth as it slows?

a) it’s in Earth’s gravitational fielda) it’s in Earth’s gravitational field

b) the net force on it is zerob) the net force on it is zero

c) it is beyond the main pull of Earth’s c) it is beyond the main pull of Earth’s gravitygravity

d) it’s being pulled by the Sun as well as by d) it’s being pulled by the Sun as well as by EarthEarth

e) none of the abovee) none of the above

Two Satellites

a) a) 11//88

b) ¼b) ¼

c) ½c) ½

d) it’s the samed) it’s the same

e) 2e) 2

Two satellites A and B of the same mass Two satellites A and B of the same mass

are going around Earth in concentric are going around Earth in concentric orbits. The distance of satellite B from orbits. The distance of satellite B from Earth’s center is twice that of satellite A. Earth’s center is twice that of satellite A. What is theWhat is the ratio ratio of the centripetal force of the centripetal force acting on B compared to that acting on acting on B compared to that acting on A?A?

Using the Law of Gravitation:

we find that the ratio is .we find that the ratio is .

Two Satellites

a) a) 11//88

b) ¼b) ¼

c) ½c) ½

d) it’s the samed) it’s the same

e) 2e) 2

Two satellites A and B of the same mass Two satellites A and B of the same mass

are going around Earth in concentric are going around Earth in concentric orbits. The distance of satellite B from orbits. The distance of satellite B from Earth’s center is twice that of satellite A. Earth’s center is twice that of satellite A. What is theWhat is the ratio ratio of the centripetal force of the centripetal force acting on B compared to that acting on acting on B compared to that acting on A?A?

Note the 1/R2 factor

Gravitational Potential Energy

Gravitational potential energy, just like all other forms of energy, is a scalar. It therefore has no components; just a sign.

Gravitational potential energy of an object of mass m a distance r from the Earth’s center:

(U =0 at r -> infinity)

Very close to the Earth’s surface, the gravitational potential increases linearly with altitude:

Energy Conservation

Total mechanical energy of an object of mass m a distance r from the center of the Earth:

This confirms what we already know – as an object approaches the Earth, it moves faster and faster.

Escape Speed

Escape speed: the initial upward speed a projectile must have in order to escape from the Earth’s gravity

from total energy:

If initial velocity = ve, then velocity at large distance goes to zero. If

initial velocity is larger than ve, then there is non-zero total energy, and the kinetic energy is non-zero when the body has left the potential well

Maximum height vs. Launch speedSpeed of a projectile as it leaves the Earth, for various launch speeds

Black holesIf an object is sufficiently massive and sufficiently small, the escape speed will equal or exceed the speed of light – light itself will not be able to escape the surface.

This is a black hole.

The light itself has mass (in the mass/energy relationship of Einstein), or spacetime itself is curved

Gravity and lightLight will be bent by any gravitational field; this can be seen when we view a distant galaxy beyond a closer galaxy cluster. This is called gravitational lensing, and many examples have been found.

Kepler’s Laws of Orbital MotionJohannes Kepler made detailed studies of the apparent motions of the

planets over many years, and was able to formulate three empirical laws

You already know about circular motion... circular motion is just a special case of elliptical motion

1. Planets follow elliptical orbits, with the Sun at one focus of the ellipse.

Only force is central gravitational attraction - but for elliptical orbits this has both radial and tangential components

Elliptical orbits are stable under inverse-square force law.

Kepler’s Laws of Orbital Motion

2. As a planet moves in its orbit, it sweeps out an equal amount of area in an equal amount of time.

This is equivalent to conservation of angular momentum

v Δt

r

Kepler’s Laws of Orbital Motion

3. The period, T, of a planet increases as its mean distance from the Sun, r, raised to the 3/2 power.

This can be shown to be a consequence of the inverse square form of the gravitational force.

Orbital Maneuvers

Which stable circular orbit has the higher speed?

How does one move from the larger orbit to the smaller orbit?

Binary systemsIf neither body is “infinite” mass, one should consider

the center of mass of the orbital motion

If you weigh yourself at the equator If you weigh yourself at the equator

of Earth, would you get a bigger, of Earth, would you get a bigger,

smaller, or similar value than if you smaller, or similar value than if you

weigh yourself at one of the poles?weigh yourself at one of the poles?

a) bigger value

b) smaller value

c) same value

Guess My Weight

If you weigh yourself at the equator If you weigh yourself at the equator

of Earth, would you get a bigger, of Earth, would you get a bigger,

smaller, or similar value than if you smaller, or similar value than if you

weigh yourself at one of the poles?weigh yourself at one of the poles?

a) bigger value

b) smaller value

c) same value

The weight that a scale reads is the normal forcenormal force exerted by

the floor (or the scale). At the equator, you are in circular you are in circular

motionmotion, so there must be a net inward forcenet inward force toward Earth’s

center. This means that the normal force must be slightly less normal force must be slightly less

than than mgmg. So the scale would register something less than your

actual weight.

Guess My Weight

Earth and Moon I

a) the Earth pulls harder on the Moona) the Earth pulls harder on the Moon

b) the Moon pulls harder on the Earthb) the Moon pulls harder on the Earth

c) they pull on each other equallyc) they pull on each other equally

d) there is no force between the Earth d) there is no force between the Earth and the Moonand the Moon

e) e) it depends upon where the Moon is in it depends upon where the Moon is in its orbit at that timeits orbit at that time

Which is stronger,

Earth’s pull on the

Moon, or the

Moon’s pull on

Earth?

By Newton’s Third Law, the forces

are equal and opposite.

Earth and Moon I

a) the Earth pulls harder on the Moona) the Earth pulls harder on the Moon

b) the Moon pulls harder on the Earthb) the Moon pulls harder on the Earth

c) they pull on each other equallyc) they pull on each other equally

d) there is no force between the Earth d) there is no force between the Earth and the Moonand the Moon

e) e) it depends upon where the Moon is in it depends upon where the Moon is in its orbit at that timeits orbit at that time

Which is stronger,

Earth’s pull on the

Moon, or the

Moon’s pull on

Earth?