Physics 225, Fall 2003 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 mass of the block is 0.25kg, and the surface is frictionless.  At t = 0 s, the block is displaced 0.1m and released.  The period of the motion is measured and found to be 0.4 s. (a) What is the angular frequency of the motion?  (b) What is the spring constant?  (c) At t = 0.1 s what is velocity of the block?  (d) At t = 0.1 s what is the acceleration of the block?

 

 

 

 

(a) and  so

(b) From  we have

(c) From  we see that

(d) From  we have

 

Problem #2) The horizontal bar in the diagram below is rigid and uniform and has a mass of 2.0 kg.  It is 1.5 m in length, and it is attached to the wall with a hinge.  A cord that is attached to the midpoint of the bar and to the wall makes a 45 deg angle with the bar.  A 5 kg mass is suspended from the far end of the bar.  (a) Find the tension in the cord.  (b) Find the horizontal force from the wall on the bar.  (c) Find the vertical force from the wall on the bar.

 

 

 

 

 

 

 

 

 

 

 

 

 

(a) Balancing torques (using the hinge as the axis of rotation) we have  so T = 166.3N.

 

(b) Balancing forces in the x-direction, we have  so F_hx = 117.6 N.

 

(c) Balancing forces in the y-direction, we have  so F_ky = -49.0 N.

 

 

Problem #3) A uniform solid ball rolls smoothly without slipping along a floor, then up a 15 deg ramp.  The ball stops momentarily when it has rolled 1.25 m along the ramp.  What was the initial velocity of the center of mass of the ball?  (Hint: use energy conservation, and note that for a solid ball .)

 

When the ball has traveled 1.25 m along the ramp, it has risen so it must have had a total energy of .  Equating this to the initial kinetic energy, we have  where we have taken advantage of the fact that  for an object that rolls without slipping.  So

 

 

Problem #4) A 5 g bullet moving directly upward at 500 m/s hits a 5 kg block of wood.  The bullet stops in the block.  (a) What is the kinetic energy of the bullet just before the collision?  (b) What is the momentum of the bullet just before the collision?  (c) What is the momentum of the block (with the bullet embedded in it) just after the collision?  (d) How much energy was lost in the collision?  (e) How high does the block rise above its initial position?

 

 

 

(a) Let m represent the mass of the bullet and M represent the mass of the block.  Then .

 

(b)

 

(c) Since momentum is conserved in the collision this also is the momentum of the block (with bullet imbedded) just after the collision.  Let’s call it P.

 

(d)The energy just after the collision is  so nearly all the energy of the bullet is lost – 624.4 J to be exact.

 

(e) The block will rise until all of its kinetic energy has been converted to gravitational potential energy, so  which implies that h = 0.012 m or 1.2 cm.

 

 

Problem #5) A 20 kg body is moving in the positive x direction with a speed of 200 m/s when, owing to an internal explosion, it breaks into three parts.  One part, with a mass of 10 kg moves away from the point of explosion with a speed of 100 m/s in the positive y direction.  A second fragment, with a mass of 4 kg, moves in the negative x direction with a speed of 500 m/s.  (a) What is the x component of the velocity of the third (6 kg) fragment?  (b) What is the y component of the velocity of the third (6 kg) fragment?  (c) What is the minimum amount of energy that was released in the explosion?  (Hints: momentum must be conserved in both the x and y directions, and you can ignore all frictional and gravitational effects.)

 

(a) Before the explosion all the momentum was in the x-direction and .  Since the momentum in the x-direction is conserved, after the explosion we have  where v_3x is the x-component of the third fragment’s total velocity.  Solving, we find .

 

(b) Before the explosion there was no momentum in the y-direction, so .  Solving, we find .

 

(c) The kinetic energy before the explosion was .  The total kinetic energy after the explosion is so a minimum of had to be released in the explosion.

 

 

Problem #6) In the diagram shown below a cord is looped over a pulley, which can be treated as a solid cylinder.  The pulley has a mass of 0.25 kg and a radius of 0.05 m.  A 1.0 kg mass is attached to the left end of the cord, and a 1.5 kg mass is attached to the right end of the cord.  Assume that the axle of the pulley is frictionless and that the cord does not slip.  Initially the two masses are held in place, then they are released.  (a) After the masses are released, what is the torque on the pulley?  (b) What is the angular acceleration of the pulley?  (c) What is the linear acceleration of each mass?  (Note that  for a solid cylinder.  Hint: the tension on the left is not the same as the tension on the right.)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Let T₁ be the tension on the left, and T₂ the tension on the right.  Then

 and since must be negative (the pulley turns clockwise).

Then where the – sign is needed since the acceleration obviously upward for the mass on the left.  And,  where we have taken advantage of the fact that the linear acceleration of the masses is numerically equal to the tangential acceleration of the pulley.  Solving for the tensions, and inserting in the first equation we have  where M is the mass of the pulley.  Solving for the angular acceleration we have .  Then,

(a) .

(b) .

(c)  in magnitude.  The acceleration is positive for the 1 kg mass and negative for the 1.5 kg mass according to our usual conventions.

 

 

Problem #7) A railroad car moves at a constant speed of 4.0 m/s under a grain elevator.  Grain drops into it at the rate of 500 kg/min.  What is the magnitude of the force needed to keep the car moving at constant speed if friction is negligible?

 

 since the velocity of the car is constant.

 

 

Problem #8) A 0.06 kg rubber ball is thrown vertically downward hitting a cement floor with a velocity of 10 m/s.  The ball is in contact with the floor for 0.015 s, and it rebounds with an initial velocity of 7.5 m/s. (a) What average force did the ball exert on the floor while it was in contact?  (b) How high does the ball rise above the floor after it rebounds?

 

(a) From the impulse-momentum theorem we have , so since the motion is entirely in the y-direction .

(b) Equating the kinetic energy just after rebound with the gravitational potential energy at the maximum height, we have  so .