lecture 14 rotational kinematics. office hours: several requests to meet today i’ll be available...

Lecture 14

Rotational Kinematics

Office Hours:

Several requests to meet today

I’ll be available 1:30 – 2:30

Rolling MotionIf a round object rolls without slipping, there is a fixed relationship between the translational and rotational speeds:

Rolling MotionWe may also consider rolling motion to be a combination of pure rotational and pure translational motion:

+ =

Rolling Wheel

a) A wheel rolls without slipping. Which vector best represents the velocity of point A?

A

b) c)

d) e) the velocity at

point A is zero

Rolling Wheel

a) A wheel rolls without slipping. Which vector best represents the velocity of point A?

A

b) c)

d) e) the velocity at

point A is zero

+vtrans

vrotv

Rotational Kinetic Energy

For this mass,

Rotational Kinetic Energy

For these two masses,

Ktotal = K1 + K2 = mr2ω2

Moment of InertiaWe can also write the kinetic energy as

Where I, the moment of inertia, is given by

What is the moment of inertia for two equal masses on the ends of a (massless) rod, spinning about the center of the rod?

Moment of InertiaWe can also write the kinetic energy as

Where I, the moment of inertia, is given by

What is the moment of inertia for a uniform ring of mass M and radius R, rolling around the center of the ring?

Moment of Inertia for various shapesMoments of inertia of various objects can be calculated:

and so one can calculate the kinetic energy for rotational motion:

This is a concept and will not be on the formula sheet!

A block of mass 1.5 kg is attached to a string that is wrapped around the circumference of a wheel of radius 30 cm and mass 5.0 kg, with uniform mass density. Initially the mass and wheel are at rest, but then the mass is allowed to fall.

What is the velocity of the mass after it falls 1 meter?

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

Since velocity and angular velocity are related for rolling objects, the kinetic energy of a rolling object is a multiple of the kinetic energy of translation.

Conservation of Energy

An object rolls down a ramp - what is its translation and rotational kinetic energy at the bottom?

Conservation of Energy

An object rolls down a ramp - what is its translation and rotational kinetic energy at the bottom?

From conservation of energy:

Velocity at any height:

Note: no dependence on mass, only on distribution of mass

Conservation of EnergyIf these two objects, of the same radius, are released simultaneously, which will reach the bottom first?

The disk will reach the bottom first – it has a smaller moment of inertia. More of its gravitational potential energy becomes translational kinetic energy, and less rotational.

Conservation of EnergyIf these two objects, of the same radius, are released simultaneously, which will reach the bottom first?

What if they have different radii?

mR2 cancels, leaving only the geometric factor (in this case, 1/2 vs. 1)

Rolling 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 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 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 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. Total mass and radius cancel in expression for fraction of K tied up in angular (rolling) motion.

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.


ConcepTest ConcepTest

A) at a greater height as when it was released.B) at a lesser height as when it was released. • at the same height as when it was released. • impossible to tell without knowing the mass of the ball. • impossible to tell without knowing the radius of the ball.

A ball is released from rest on a no-slip surface, as shown. After reaching its lowest point, the ball begins to rise again, this time on a frictionless surface as shown in the figure. When the ball reaches its maximum height on the frictionless surface, it is:

ConcepTest ConcepTest

A) at a greater height as when it was released.B) at a lesser height as when it was released. • at the same height as when it was released. • impossible to tell without knowing the mass of the ball. • impossible to tell without knowing the radius of the ball.

A ball is released from rest on a no-slip surface, as shown. After reaching its lowest point, the ball begins to rise again, this time on a frictionless surface as shown in the figure. When the ball reaches its maximum height on the frictionless surface, it is:

Q: What if both sides of the half-pipe were no-slip?

The pulsar in the Crab nebula was created by a supernova explosion that was observed on Earth in a.d. 1054. Its current period of rotation (33.0 ms) is observed to be increasing by 1.26 x 10-5 s per year.

•What is the angular acceleration of the pulsar in rad/s2 ?•Assuming the angular acceleration of the pulsar to be constant, how many years will it take for the pulsar to slow to a stop? •Under the same assumption, what was the period of the pulsar when it was created?



(a)



(b)

(c)

•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 change in rotational kinetic energy from the beginning to 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

•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.

•calculate the change in rotational kinetic energy from the beginning to 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.

Lecture 14

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 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 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 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 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.

lecture 14 rotational kinematics. office hours: several requests to meet today i’ll be available...

Documents

rotational kinetic energy

distribution of mass

conservation of energy

translational kinetic

kinetic energy of translation

total kinetic energy

rotational kinematics

rolling wheel