Physics II	Name: ________________________________________

Rotational Motion

1 radian = an angle at the center of a circle subtended by an arc whose length is equal to the circle's radius

1 radian = 180°/ π = 360°/2π ≈ 57.3°

90º = ½ π radians. 180º = π radians 360º = 2π radians

ω (omega) = rotational velocity; tells how fast something is rotating, in radians/sec.

v = linear velocity

α (alpha) = rotational acceleration; tells how fast rotation is speeding up, in radians/sec/sec.

a = linear acceleration

T = torque

I = moment of inertia; tells how easy or difficult it is to get an object rotating. Units are kg m2. Moment of inertia is analagous to mass (from linear calculations).

T = Fr (Torque acting on a body equals a tangential force, F, multiplied the distance, r, from the point of application of the force to the body's fulcrum.)

ΣT = Iα (Net torque on a rotating body equals its moment of inertia multiplied by its rotational acceleration; similar to F=ma)

KErotational = ½ Iω2 (similar to ½ mv2, for translational kinetic energy)

Total KE of a rolling object = ½ Iω2 + ½ mv2

The linear velocity of a revolving object revolving a radius r away from a central point is given by

v = ωr.

The linear acceleration of a revolving object revolving a radius r away from a central point is given by

a = αr

Torque Practice Problems:

1.	A bicyclist applies a force of 250N to the bicycle’s pedal, which is 0.25m from the center of the bike’s chain ring. The bicycle’s chain ring has a radius of 0.1m, and the sprocket has a radius of 0.04m.

What torque is applied to the pedal/chain ring?

What force does the chain ring apply to the chain?

What torque is applied to the sprocket?

What "backward" (to the right) force does the tire apply to the road?

How far does a link in the chain move for each revolution of the pedal?

How many rotations will the sprocket make for each rotation of the pedal?

How far will the bicycle travel for each revolution of the pedal/chain ring?

If the pedal is making one revolution per second, how fast is the bicycle moving, in m/s?

2.	For this question, imagine a situation similar to the one above... Another bicyclist applies a downward force of 400N to the bicycle’s pedal, which is 0.2m from the center of the bike’s chain ring. The bicycle’s chain ring has a radius of 0.15m, and the sprocket has a radius of 0.08m.

What torque is applied to the pedal/chain ring?

What force does the chain ring apply to the chain?

What torque is applied to the sprocket?

What "backward" (to the right) force does the tire apply to the road?

How far does a link in the chain move for each revolution of the pedal?

How many rotations will the sprocket make for each rotation of the pedal?

How far will the bicycle travel for each revolution of the pedal/chain ring?

If the pedal is making one revolution per second, how fast is the bicycle moving, in m/s?

Torque Challenge: weighing by balancing torques. Using only a string and one weight of known mass, determine the mass of a metal rod.

Moment of Inertia Activity

Practice Questions:

Suppose you have a frictionless pulley like the one on the right. The hanging object has a mass of 50g. The mass of the string is negligible. If you wind up the string and then let go of the mass, it begins to rotate the pulley as it unwinds the string. As it unwinds, this mass falls 0.8meters.

1.	What is the linear acceleration (m/s2) of the mass as it drops?

2.	What is the tension in the string as the mass is dropping (in Newtons)?

3.	How much torque does the string apply to the pulley?

4.	What is the rotational acceleration (α) of the pulley?

5.	What is the pulley’s moment of inertia (I)?

6.	Now suppose you have a ring with a moment of inertia of 0.002kg m2 and a mass of 1kg. You also have a ramp that slopes from a height of 2m on one end to a height of 0m on the other. The surface of the board is 4m long. If you allow the ring to roll down the board, what will its translational (not rotational) speed be when it reaches the end of the ramp?

Actual Lab

Set up a rotational motion apparatus (with nothing added) so that a mass on a string unwinds from the apparatus’ pulley and thereby gets the apparatus spinning.

Wind up the string. Then simultaneously let go of the mass and start a timer. Time how long it takes the mass to drop a certain distance [It’s convenient to let it drop to the floor.]. Measure the distance. Repeat this a few times and get a representative time. Then…

1.	Determine the linear acceleration (m/s2) of the mass as it drops.

2.	Figure out the tension in the string as the mass is dropping (in Newtons). [Assume for all of these calculations that the mass of the string is negligible.]

3.	Determine the amount of torque that is applied to the apparatus by the string.

4.	Use your mass’ linear acceleration to determine the rotational acceleration of the apparatus.

5.	Calculate the apparatus’ moment of inertia.

6.	Now add a ring to the apparatus. Calculate the new moment of inertia of the apparatus with the ring.

7.	Determine the moment of inertia of the ring by subtracting the moment of inertia of the apparatus only.

8.	Set up a big ramp. Measure the ring’s mass. If you allow the ring to roll down the ramp, what should its final translational speed be at the end? Calculate it.

9.	Now actually roll the ring and check your answer.

Physics II

2009 Test: Rotational Motion

Suppose you have a disc with a smaller disc attached at its center. A string is wrapped around the outside edge of the smaller disc, and a 200g mass is attached to the end of the string. The mass of the string is negligible. When the mass is released, the mass drops, the string to unwinds, and the disc rotates. Starting from rest, the mass falls 0.7m in 10 seconds. The radius of the smaller disc is 0.05m.

1.	What is the linear acceleration (m/s2) of the mass as it drops?

2.	What is the tension (approximate or actual) in the string as the mass is dropping (in Newtons)?

3.	How much torque does the string apply to the disc?

4.	What is the angular acceleration (α) of the disc?

5.	What is the disc’s moment of inertia (I)?

A second disc is allowed to roll down a ramp. The ramp’s vertical drop is 1.5m, and the disc’s velocity at the bottom of the ramp is 5m/s. The disc’s radius is 0.8m.

6.	What is this second disc’s moment of inertia?

7.	What is this second disc’s rotational kinetic energy when it reaches the bottom of the ramp?

The diagram below shows a penny smearing machine. An operator turns the big crank while feeding a penny into the space where the smearing wheel meets the smearing plate. The big crank turns sprocket A, which turns sprocket B via the chain. Radii are listed in the table.

Big Crank length	0.3m
Sprocket A radius	0.02m
Sprocket B radius	0.8m
Smearing Wheel Radius	0.83m

8.	The smearing wheel applies a force to the penny that pushes it to the right. What is the magnitude of this rightward force?

9.	How far will a point on the outside edge of the smearing wheel move during each rotation of the big crank?

10.	If an operator turns the big crank at a rate of one revolution every 1.8 seconds, how fast is the outside edge of the smearing wheel moving (in m/s)?

String is wrapped around a pulley. A 150g mass is suspended from the string, held at rest, and then allowed to fall 0.4m. The mass falls the 0.4m in 2 seconds.

11.	What is the mass’ linear acceleration during its fall?

12.	What is the mass' kinetic energy at the bottom of its 0.4m fall?

13.	What is the pulley’s kinetic energy when the mass reaches the bottom of its 0.4m fall?

Diagram and Table Support #14-22.

Length of string wrapped around axle (m)	0.1
Force of rubber band at moment when string is released from drive axle (N)	4
Force of rubber band when string is fully wound (N)	12
Total car mass (kg)	0.2
Drive Wheel Radius (m)	0.03
Front Wheel Radius (m)	0.02
Drive wheel and axle Moment of Inertia (kgm^2)	0.00006
Front wheel and axle Moment of Inertia (kgm^2)	0.00004
Drive Axle Radius (m)	0.003

14.	What is the average force applied by the rubber band as it pulls string from the car’s drive axle?

15.	What is the distance over which that force is applied?

16.	How potential energy is stored in the rubber band when the string is fully wound (assuming 100% efficiency).

17.	What initial backward force do the car’s drive wheels apply to the road?

18.	Suppose the car is released at the top of a ramp. The string is not wound, so the car is powered only by gravity. The car travels down the ramp and rolls out onto a level surface. During its descent, the car's center of gravity drops 0.6m vertically. After rolling a total linear distance of 25m, the car comes to a stop. What is the net force of friction that is opposing the car’s motion? [Assume that this force of friction is the same on the ramp and the level floor.]

19.	The car is fully wound and released from rest on a level surface. How far will the car travel before its string unwinds?

20.	If the car is fully wound in the beginning, what is the car’s approximate total kinetic energy at the moment the string unwinds?

21.	What is the car’s velocity at the moment that all of the string is unwound?

22.	What is the total distance that this car will travel before coming to a stop?