1053C EXAM 3 Fall 2010 NAME_____________________________

A bullet of mass m and velocity vo is fired toward a block of thickness Lo and mass M. The block is initially

at rest on a frictionless surface. The bullet emerges from the block with velocity vo/3.

a. Determine the final speed of block M.

b. If, instead, the block is held fixed and not allowed to slide, the bullet emerges from the block with a

speed vo/2. Determine the loss of kinetic energy of the bullet

A uniform solid cylinder of mass m1 and radius R is mounted on frictionless bearings about a fixed axis

through O. The moment of inertia of the cylinder about the axis is I = ½m1R2. A block of mass m2,

suspended by a cord wrapped around the cylinder as shown above, is released at time t = 0.

a. On the diagram below draw and identify all of the forces acting on the cylinder and on the block.

b. In terms of ml, m2, R. and g, determine each of the following:

i. The linear acceleration of the block

ii. The angular acceleration of the cylinder.

iii. The tension in the cord

iv. The linear velocity of m2 after it has fallen 1 meter

v. calculate the time passed when it has fallen 1 meter

vi. The angular velocity of m1 at this time

An inclined plane makes an angle of with the horizontal, as shown above. A solid sphere of radius R and

mass M is initially at rest in the position shown, such that the lowest point of the sphere is a vertical height h

above the base of the plane. The sphere is released and rolls down the plane without slipping. The moment

of inertia of the sphere about an axis through its center is 2MR2/5. Express your answers in terms of M, R.

h, g, and B.

a. What is the total kinetic energy of the solid sphere at the bottom of the plane?

Determine the following for the solid sphere when it is at the bottom of the plane:

i. Its linear velocity

a. Determine the Potential energy of the satellite on the surface of the earth.

b. Determine the escape velocity (PE=0 at r= infinity)

c. Determine the potential energy of the satellite when it is at A.

c. Determine the velocity of the satellite as it passes point A in its orbit.

d. Determine the Kinetic energy needed to place the satellite in orbit at A.

An explorer plans a mission to place a satellite into a circular orbit around the planet Jupiter, which has mass

MJ = 1.90 x 1027 kg and radius RJ = 7.14 x 107 m.

a. If the radius of the planned orbit is R, use Newton's laws to show each of the following.

Show at least 3 steps in your work for each derivation:

The orbital speed of the planned satellite is given by

i. The period of the orbit is given by

The explorer wants the satellite's orbit to be synchronized with Jupiter's rotation. This requires an

equatorial orbit whose period equals Jupiter's rotation period of 9 hr 51 min = 3.55 x 104 s.

b. Determine the required orbital radius in meters.

c. Determine the orbital velocity at that radius.