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PHYSICS 221 Summer 2013

Homework #4 Due Friday July 12, 2013

1. Two blocks A and B of masses mA = 1.00 kg and mB = 2.00 kg on a horizontal, frictionless table head toward each other with initial speeds vA = 3.00 m/s and vB = 2.00 m/s. A light spring is attached to block A, so when the two block collide, they compress the spring and bounce off with velocities v′A and v′B.

(a) Find the final velocities v′A and v ′ B.

(b) What are the velocities of the blocks when the spring reaches its maximum compression.

(c) How much potential energy was stored in the spring at the instant of maximum compression?

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PHYSICS 221 Summer 2013

Homework #4 Due Friday July 12, 2013

2. Find the center of mass of the following systems:

(a) Three point masses as shown figure (a) below.

(b) Three identical uniform rectangles of sides a and b forming a U-shape as shown in figure (b) below.

figure (a) figure (b)

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PHYSICS 221 Summer 2013

Homework #4 Due Friday July 12, 2013

3. A uniform metal plate of edge 25 cm has a square section of edge 5.0 cm cut out of it as in the figure.

25 cm

5.0 cm

P

(a) Where is the center of gravity of the plate?

(b) What is the moment of Inertia of the plate about an axis per- pendicular to it’s plane through the low left corner P.

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PHYSICS 221 Summer 2013

Homework #4 Due Friday July 12, 2013

4. Assume a disk of uniform mass density σ and radius R. A circular hole of radius a is cut off the disk at distance b from the center as shown in the figure.

R

a

P

b

(a) Where is the center of gravity of the new shape?

(b) What is the moment of Inertia of the new shape about an axis per- pendicular to it’s plane through the center of the uncut cylinder, P?

(c) Substitute in answers in (a) and (b) for R = 10.0 cm, a = 2.00 cm and b = 7.00 cm.

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PHYSICS 221 Summer 2013

Homework #4 Due Friday July 12, 2013

5. A solid disk of radius R has a mass density as a function of radial distance:

σ (r) = σ0

( 1 −

r

R

) (a) Locate the Center of Mass of this object.

(b) Calculate the moment of inertia of the disk about it’s axle. Express the answer it terms of the disk’s total mass M.

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PHYSICS 221 Summer 2013

Homework #4 Due Friday July 12, 2013

6. A cylinder of uniform mass M and radius r is released from point A of height h on a track as shown in the figure. The end of the track is a circular path of radius R. The cylinder rolls without slipping. Determine the radial and tangential acceleration of the cylinder at points B, C (90◦ from B on the circular arc) and D as shown. Express your answers in terms of given variables, M, h, r, R and/or g. Assume h is high enough that cylinder will make it to point D without falling off.

h

r

A

B

C

D

R

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PHYSICS 221 Summer 2013

Homework #4 Due Friday July 12, 2013

7. Assume a block is placed on the top of an upside down semi-spherical bowl as shown in the figure. The block starts to slide without frictions. At what angle θ from the vartical will the block fly off the bowl surface? Now consider a circular ring instead of the block and the fact it will roll down the surface without slipping. What will be the value of the angle θ at which the ring will fly off the bowl surface?

R

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PHYSICS 221 Summer 2013

Homework #4 Due Friday July 12, 2013

8. A flywheel in the shape of a solid cylinder of radius R = 0.60 m and mass M = 15 kg can be brought to an angular speed of 12 rad/s in 0.60 s by a motor exerting a constant torque. After the motor is turned off, the flywheel makes 20 revolutions before coming to rest because of frictional forces (assumed to be constant during rotation). What percentage of the power generated by the motor is used to overcome the frictional losses?

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