Lecture 14

Rotational Kinematics

Reading and Review

Arc length s, measured in radians: s = rθ

Analogies between linear and rotational kinematics:

Rotational Motion

Linear and Angular Velocity

Bonnie and Klyde Bonnie and Klyde

BonnieBonnieKlydeKlyde

a) Klyde

b) Bonnie

c) both the same

d) the net kinetic energy for both of them is zero since there is no average motion of center of mass

Bonnie sits on the outer rim of amerry-go-round, and Klyde sits midway between the center and the rim. The merry-go-round makes one revolution every 2 seconds. Klyde and Bonnie have about the same mass. Who has the larger kinetic energy?

Bonnie and Klyde Bonnie and Klyde

a) Klyde

b) Bonnie

c) both the same

d) the net kinetic energy for both of them is zero since there is no average motion of center of mass

Bonnie sits on the outer rim of amerry-go-round, and Klyde sits midway between the center and the rim. The merry-go-round makes one revolution every 2 seconds. Klyde and Bonnie have about the same mass. Who has the larger kinetic energy?

BonnieBonnie

KlydeKlyde

Their linear speeds linear speeds vv will be

different because v = r v = r

and Bonnie is located farther Bonnie is located farther

outout (larger radius r) than

Klyde.

Therefore KEBonnie > KEKlyde

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

+ =

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

Suppose that the speedometer of a truck is set to read the linear speed of the truck but uses a device that actually measures the angular speed of the tires. If larger diameter tires are mounted on the truck instead, how will that affect the speedometer reading as compared to the true linear speed of the truck?

a) speedometer reads a higher speed than the true linear

speed

b) speedometer reads a lower speed than the true linear speed

c) speedometer still reads the true linear speed

Truck SpeedometerTruck Speedometer

Suppose that the speedometer of a truck is set to read the linear speed of the truck but uses a device that actually measures the angular speed of the tires. If larger diameter tires are mounted on the truck instead, how will that affect the speedometer reading as compared to the true linear speed of the truck?

a) speedometer reads a higher speed than the true linear

speed

b) speedometer reads a lower speed than the true linear speed

c) speedometer still reads the true linear speed

The linear speed is v = R. So when the speedometer measures the same angular speed as before, the linear speed v is actually higher, because the tire radius is larger than before.

Truck SpeedometerTruck Speedometer

Jeff of the Jungle swings on a vine that is 7.20 m long. At the bottom of the swing, just before hitting the tree, Jeff’s linear speed is 8.50 m/s. (a) Find Jeff’s angular speed at this time.(b) What centripetal acceleration does Jeff experience at the bottom of his swing?(c) What exerts the force that is responsible for Jeff’s centripetal acceleration?

Jeff of the Jungle swings on a vine that is 7.20 m long. At the bottom of the swing, just before hitting the tree, Jeff’s linear speed is 8.50 m/s. (a) Find Jeff’s angular speed at this time.(b) What centripetal acceleration does Jeff experience at the bottom of his swing?(c) What exerts the force that is responsible for Jeff’s centripetal acceleration?

a)

b)

c) This is the force that is responsible for keeping Jeff in circular motion: the vine.

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:

2I MR

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?

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.

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?

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 decreasing 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?

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 decreasing 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)

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 decreasing 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?

(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 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.

Rotational Dynamics

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)

PHYS2010 midterm 2, Fall 2008

4) Two uniform solid spheres have the same mass, but one has twice the radius of the

other. The ratio of the larger sphere's moment of inertia to that of the smaller sphere is

A) 8/5

B) 4

C) 2

D) 1/2

E) 4/5

suggested time: 1-2 minutes

Please do not ask questions about this problem at

discussion sessions before 10/21

Provided in lecture

notes on: 10/19

PHYS2010 midterm 2, Fall 2008

7) A solid disk is released from rest and rolls without slipping down an inclined plane

that makes an angle of 25.0o with the horizontal. What is the speed of the disk after it

has rolled 3.00 m, measured along the plane? (Moment of inertia of a solid disk of

mass M and radius R is 1/2(MR2)).

A) 5.71 m/s

B) 2.04 m/s

C) 3.53 m/s

D) 4.07 m/s

E) 6.29 m/s

suggested time: 4-5 minutes

Please do not ask questions about this problem at

discussion sessions before 10/21

Provided in lecture

notes on: 10/19