Physics 225, Fall 2002 Final Exam (Section 2)

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 = 0.1s, the position of the block (relative to the equilibrium position) is 0.0995m and the velocity of the block is –0.3883m/s. (a) What is the angular frequency of the motion?  (b) What is the phase angle?  (c) What is the amplitude of the motion?

 

 

 

 

a)

b) From  and  we have

 so , which yields

c)

 

Problem #2) The uniform solid cylinder shown in the figure below has a mass of 2kg and a radius of 0.05m.  It rolls without slipping for a distance of 1.0m along the 30deg incline plane shown below, and it continues rolling without slipping along the horizontal surface.  (a) When the cylinder reaches the horizontal surface, what is its total kinetic energy?  (b) What is its rotational kinetic energy?  (c) What is its translational kinetic energy?  (d) What is the translational velocity of the CM of the cylinder?

 

 

 

 

 

 

 

a) The cylinder drops a vertical distance of 1.0m·sin(30deg) = 0.5m. So KE_tot = mgh= 2kg·9.8m/s²·0.5m = 9.8J.

 

b) KE_rot = KE_tot-KE_CM = 9.8J-6.53J = 3.27J  (see part c).

c) From , we see that the translational KE (or KE_CM) = KE_tot/1.5 = 6.53J.

 

d)

 

 

Problem #3) The bar shown in the figure below has a mass of 20 kg.  It is supported by two ropes.  The rope on the left makes an angle of 45 deg with the bar, while the rope on the right is horizontal.  Find the tensions in each of the ropes.

 

 

 

 

 

 

 

 

 

 

Considering first the forces in the vertical direction, , so .  Next, considering the horizontal forces .

 

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) A thin rod 1m in length is held vertically with one end on the floor.  It is then allowed to fall.  As it falls the end originally on the floor does not slip.  What is the speed of the other end of the rod as it hits the floor?  (Hint: Use conservation of energy.)

 

As the rod falls gravitational potential energy is converted into rotational kinetic energy, since the rod is rotating about its end.  Note that the CM of the rod has dropped 0.5m when the rod hits the ground.  So , where I is the rotational inertia about the end of the rod, and h = 0.5m.  So,  or , which implies that the other end of the rod is moving with a velocity of 5.42m/s when it hits.

 

Problem #6) A baseball is thrown horizontally from the top of a building that is 10 m high.  The ball lands 5 m 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) From the equations of motion in the y-direction, , where t is the time it takes for the ball to hit the ground.  This is 1.429s.  So , which implies that v_0x = 3.5m/s.

 

b)

 

Problem #7) A wooden block with a mass of 1.5 kg is at rest on a horizontal surface.  The coefficient of kinetic friction between the block and the surface is 0.15.  A lump of clay with a mass of 0.4 kg is thrown horizontally at the block with a velocity of 2.0 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 = 0.421m/s.

 

b) From , so

 

c)

 

 

Problem #8) A 0.75kg 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 exactly half the speed it hit with.  The ball is in contact with the floor for 0.012s.  (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 3.13m/s, so the change in momentum for the ball is 0.75kg(3.13-(-6.26))m/s = 7.04kg·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.