Physics 225, Fall 2002 Final Exam (Section 1)

Dr. Shapiro

 

Closed book – two 3"×5" note cards and the handouts may be used. Please do not write on either the test or the handout. Both must be turned in with your test answers.  Partial Credit will be given. Please write clearly and show all your work (answers without supporting work will not receive credit).

 

Problem #1) The oscillator shown below consists of a block attached to a spring.  The spring constant k = 400N/m, the mass of the block is 0.25kg, and the surface is frictionless.  At t = 0 s, the block is displaced 0.1m and released (a) What is the angular frequency of the motion?  (b) At t = 0.1s what is the position of the block?  (c) At t = 0.1s what is velocity of the block?  (d) At t = 0.1s what is the acceleration of the block?

 

 

 

 

a)

 

b)

 

c)

 

d)

 

Problem #2) The uniform solid cylinder shown in the figure below has a mass of 1.0kg and a radius of 0.05m.  It is rolling without slipping on the horizontal surface.  The center of mass of the cylinder is moving with a velocity of 0.6m/s.  The cylinder then encounters an incline plane that makes an angle of 20 degrees with the horizontal.  It continues to roll up the incline plane without slipping.  (a) What is the total kinetic energy of the rolling cylinder before it encounters the incline plane?  (b) How far does it roll along the incline before it stops and begins to roll back down?

 

 

 

 

 

 

a) , where , so

 

b) The cylinder will roll up the incline plane until all the kinetic energy has been converted to potential energy, so   So

 

 

Problem #3) The uniform bar shown in the figure below has a mass of 20kg and is 2m long.  The bar rests on two supports as shown.  One support is located at the left end of the bar.  The other support is located 1.5m to the right of the first one.  (a) What force does the left support exert on the bar?  (b) What force does the right support exert on the bar?

 

 

 

 

 

 

 

 

The net force in the y-direction is zero, so , and the sum of the torques must be zero, so .  From the second equation F₂ = 130.7N, and from the first equation F₁ = mg – F₂ = 65.3N.

 

a) 65.3N

b) 130.7N

 

 

Problem #4) A spaceship is on a straight-line path between Earth and the Moon.  At what distance from Earth is the net gravitational force on the spaceship zero?  (Note that the mean distance from the Earth to the Moon is 3.82·10⁸m, the mass of the Earth is 5.98·10²⁴kg, and the mass of the moon is 7.36·10²²kg)

 

The net gravitational force on the spaceship will be zero when the forces on it from the Moon and Earth are equal, so , where r is the distance from the Earth to the ship, and R is the Earth-Moon distance.  So

 

 

Problem #5) An ice skater is rotating with an angular speed of 5 rad/s.  Her arms are outstretched.  Her total rotational inertia at this point is 8.0kg·m².  She pulls in her arms and her angular speed increases to 6.5 rad/s.  (a) What is her new rotational inertia?  (b) What was her original kinetic energy?  (c) What is her final kinetic energy?

 

a) From conservation of angular momentum , so

b)

c)  

 

Problem #6) A baseball is thrown horizontally from the top of a building that is 10m high.  The ball just clears a 2m high fence located 5m away from the building.  (a) What was the initial horizontal velocity of the baseball?  (b) What is the velocity of the baseball as it hits the ground?

 

a) The baseball drops 8m vertically in the time it takes to travel 5m horizontally.  So from  we have  for the time it takes the ball to drop 8m.  Then from the equation for its horizontal motion

 

b) The time the ball takes to hit the ground is given by , so the vertical velocity is , so the velocity that the ball has when it hits is

 

Problem #7) A wooden block with a mass of 2.0 kg is at rest on a horizontal surface.  The coefficient of kinetic friction between the block and the surface is 0.25.  A lump of clay with a mass of 0.3 kg is thrown horizontally at the block with a velocity of 2.5 m/s.  The lump hits the block and sticks to it.  (a) What is the velocity of the lump plus block immediately after the collision?  (b) Assuming that the lump hits the block in such a way that the block always remains in contact with the surface, how far does the block move after the collision?  (c) How much energy was lost in the collision of the lump with the block?

 

 

 

 

 

 

a) From conservation of momentum , where m is the mass of the lump, v is its initial velocity, M is the mass of the block, and V is the velocity of the combination immediately after the collision.  So V = 0.326m/s.

 

b) From , so

 

c)

 

 

 

Problem #8) A 0.5kg hard rubber ball is dropped from a height of 2.0m.  It hits a concrete floor and rebounds with a speed (just after the collision) that is 75% of the speed it hit with.  The ball is in contact with the floor for 0.015s.  (a) With what velocity does the ball hit the ground?  (b) What impulse acts on the ball during its collision with the ground?  (c) What average force does the ball exert on the floor during the collision?

 

a) From the equations of motion , which implies that it takes 0.639s for the ball to hit the floor.  So it hits with a velocity of –6.26m/s.

 

b) The ball rebounds with a speed of 4.70m/s, so the change in momentum for the ball is 0.5kg(4.70-(-6.26))m/s = 5.48kg·m/s.  This is the impulse on the ball.

 

c) , so .  This is the upward average force on the ball.  The force on the floor is equal in magnitude and opposite in direction.