PHYSICS 218 Final Exam

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PHYSICS 218 Final Exam Fall 2007 Belyanin
Name______________________ Signature___________
_________ Student ID__________________ E-mail___
___________________ Section Number _____________

• __________________________________________________
  ________________
• No calculators are allowed in the test.
• Be sure to put a box around your final answers
  and clearly indicate your work to your grader.
• All work must be shown to get credit for the
  answer marked. If the answer marked does not
  obviously follow from the shown work, even if the
  answer is correct, you will not get credit for
  the answer.
• Clearly erase any unwanted marks. No credit will
  be given if we cant figure out which answer you
  are choosing, or which answer you want us to
  consider.
• Partial credit can be given only if your work is
  clearly explained and labeled. Partial credit
  will be given if you explain which law you use
  for solving the problem.
• Put your initials here after reading the above
  instructions

For grader use only Problem 1 (15)
___________ Problem 2 (20) ___________ Problem 3
(20) ___________ Problem 4 (15)___________ Problem
5 (20) ___________ Problem 6 (15)___________ Tot
al (105) ___________

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Some formulas
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Problem 1 (15 points)

• This is a one-dimensional problem. A particle
  with mass m is acted on by a single force with
  potential energy function equal to
  where ? is a known positive constant.
• The particle is initially placed at x x0 and is
  given an initial velocity
  where ? is a known positive constant.
• Calculate the coordinate of the turning point
  where the direction of motion of a particle
  reverses.
• Calculate the acceleration of the particle at the
  turning point.
• Find the velocity Vx of the particle when, after
  turning around, it passes the initial point x0.
• Eventually the particle goes to a very large,
  effectively infinite distance from the origin.
  Find the velocity and acceleration of a particle
  there.

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Problem 2 (20 points)
A skier with mass m starts from rest at the top
of the frictionless ski slope H meters high. He
immediately loses control over his skis and goes
straight downhill. Fortunately, at the bottom of
the slope he enters an upward ramp of constant
slope angle ?. The ramp has a soft snow surface
with a coefficient of friction ?. (a) What
is the distance that the skier moves up the ramp
before coming to a halt? (b) Find this distance
if the coefficient of friction is given by
where c is a known constant and x
is counted from the bottom of the ramp along its
surface. Stop when you have one equation for one
unknown. Dont solve it.
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Problem 3 (20 points)

• Two blocks of masses m1 and m2 are connected by a
  massless string over a pulley in the shape of a
  solid disk of radius R and moment of inertia I
  around its axis. At t 0 the blocks start moving
  from rest on a fixed wedge of angle ?, with block
  m2 sliding down from initial height H above the
  bottom of the wedge. Assume that the pulley
  rotates without friction and there is no slip of
  the string. The coefficient of friction between
  the blocks and the surface is ? for both blocks.
• Find the acceleration of the two blocks,
• Find the velocity of the block m2 at the time
  when it has reached the bottom of the plane.

m1
I, R
m2
H
?
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Problem 4 (15 points)
A bullet of mass m moving with a velocity V1 hits
a wooden block of mass M through its center and
continues with velocity V2 in the same direction.
(a) Find the velocity of the box after the
collision. (b) Find the fraction of the kinetic
energy of the bullet that was transformed into
heat in the collision. (c) After the collision
the block continues moving for some time and then
stops due to friction. Find the distance traveled
by the block if the coefficient of friction is ?.

V1
V2
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Problem 5 (15 points)
An asteroid of mass m strikes the Earth at the
equator with a velocity V, as shown below. Assume
that the asteroid remains stuck where it hit the
ground. Assume that the Earth is a sphere of
radius R, mass M, and moment of inertia I. (a)
By what factor will this collision change the
angular velocity of the Earth, which was equal to
?0 (one revolution in 24 hrs) before the
collision? (b) Bonus question (extra 5 points)
By what factor would this collision affect your
weight on the equator?
View from the North pole. Note that the Earth
rotates counter-clockwise as viewed from the
North pole.
m
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Problem 6 (15 points)

• A spring with constant k has a block of mass M
  attached to it on a frictionless table. The
  spring is initially unstretched. At t 0 the
  bullet of mass m moving with velocity Vb hits the
  block and gets stuck in it. The collision is
  nearly instantaneous.
• Find the position and the velocity of the block
  as a function of time.
• How long will it take the block to return to its
  equilibrium position?
• What is the maximum displacement of the block?
• Find the total mechanical energy of the block as
  a function of time.

k
Vb
M