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Unformatted text preview: Physics 7A- Section 1, Fall 2004 (Lanzara) FINAL EXAM
Monday December 15, 2004 6:00-8.00 pm CLOSED BOOK GOOD LUCK!
Print your name, student ID number, GSI name, and your discussion section number on
the front of your blue book. This exam contains 6 questions, and will be graded out of a
total of 130 points. You should answer all the questions to the best of your ability. You
are allowed both sides of three sheets of handwritten notes, and the use of a calculator,
but no QWERTY keyboards are allowed. Express all numerical results to 3 significant
figures. The following are useful constants: The acceleration due to Earth’s gravity in the
Bay Area is g = 9.80 m/s2. The Earths’s radius is rE = 6.38 ×103 km. The mass of the Earth is
ME = 5 .97×1024 kg. Newton’s constant for universal gravitation is G = 6.67 × 10-11Nm kg .
Please show all your work in your blue book. Explain the steps in your reasoning in
coherent, English sentences. Define all symbols that you use. If you do not show relevant
work for any part of the problem, you will not be awarded any credit, even if the answer
is correct. If you recognize that an answer does not make physical sense, and you do not
have time to find your error, write that you know that the answer cannot be correct, and
explain how you know this to be true. (We will award some credit for recognizing there
is an error.) For full credit, explain your reasoning carefully, show all steps neatly, and
box your answers. Cross out any work you decide is incorrect, with an explanation in the
Do the easiest problems first. You may answer the questions in any order you wish, but
please clearly label each problem by number to ensure that it is properly graded.
DO NOT OPEN THIS EXAM UNTIL YOU ARE TOLD TO DO SO. STOP ALL
WORK WHEN TOLD TO SO AT THE END OF THE EXAM. GOOD LUCK!
Problem 1- [15 points]
A man of mass m=60kg clings to a rope ladder suspended below a hot-air balloon of
mass M=400kg. The balloon is stationary with respect to the ground.
a) [10 points] If the man begins to climb the ladder at speed v=2km/h (with respect
to the ladder), in what direction and with what speed (with respect to the ground)
will the balloon move?
b) [5 points] What is the state of motion of the balloon after the
man stops climbing? v m Problem 2- [20 points]
A large spool of rope of mass M stands on the ground with the end of the rope lying on
the top edge of the spool. The radius of the spool is R. A person grabs the end of the rope
and walks a distance L (see panel a). The spool rolls without slipping.
1) (5 points) What length of the rope unwinds from the spool?
2) (5 points) How far does the spool’s center of mass move?
3) (10 points) At one point the spool will hit a step of height h, against which the
spool rests (see panel b). What force should the person apply in order for the
spool to climb the step? (You can ignore inertia in this final part, i.e. the spool
doesn't jump the curb simply because it's moving fast). R h
(a) (b) Problem 3- [25 points]
A spaceship is traveling in a circular path about the Earth. This trajectory is due to the
force of gravity and a force produced by the spaceship’s engines. The engines produce a
force with a constant magnitude F0 and which is always directed at the center of the
Earth. Assume the mass of the spaceship is a constant m. The mass of the Earth is ME.
The radius of the circular path is R.
1) (5 points) Find the time that it takes to go around once. Your answer should be
expressed in terms of G, ME, m, R and F0.
2) (5 points) How much work do the engines do on the spaceship each time it goes
around the Earth? Explain.
3) (10 points) A large cloud of dust appears and creates a drag force with a
magnitude FD. Because the spaceship is slowing down, the astronauts must
reduce the force due to the engines (which continues to point at the center of the
Earth) as a function of time, in order to maintain the radius of the circular path.
Find F(t), the magnitude of the engines’ force as a function of time, with F(0)=F0.
Assume that FD is independent of the speed of the spaceship. Your answer for
F(t) should go to zero for a certain finite value of t. You do not have to calculate
F(t) for times after this.
4) (5 points) Suppose that at one point the spaceship comes out of its circular orbit
and strikes the Earth at the equator with a velocity v. Assume that the crash
velocity is nearly parallel to the Earth’s surface; i.e. ignore any vertical velocity
component. By what factor would this change the Earth’s rotational frequency? Problem 4 [30 points]
A pendulum consisting of a loaded spring gun and a gun
support is attached to a pivot point by a massless, rigid rod
of length L. The gun is loaded with a single bullet of mass
m2. The mass of the support and the gun (without the
bullet) is m1, where m2<<m1. The gun can be tilted at an
angle α with respect to the support, as shown in the figure
(0< α <180o).
Initially, the pendulum swings freely without friction,
reaching a maximum height (with respect to the bottom of
its trajectory) of H0. At some point, the gun is fired and the
bullet is shot away from the pendulum. The bullet leaves
the gun with a velocity u relative to the gun, at an angle α
with respect to the pendulum’s direction of motion, as
shown in the diagram.
You may assume that the small oscillation approximation is
valid both before and after the gun is fired. Ignore friction
and air resistance.
1) (10points) Find the velocity vf of the pendulum after the gun is shot as a function
of m1, m2, u, α and the velocity v0 of the pendulum right before the gunshot.
1) (5 points) What is the maximum height that the pendulum can reach after the gun
is triggered if the ball is shot along the attachment rod (α=90) and why?
2) (10 points) In general shooting of the gun will result in a change of the oscillation
amplitude of the pendulum. Find the angle α, and the position on the pendulum’s
trajectory where the gunshot will result in the largest increase of the amplitude.
3) (5 points) What is the maximum height that can be reached by the pendulum after
the gun is shot?
Problem 5 [20 points]
A bucket filled up with a fluid of density ρL is sitting on a scale as shown in figure 1.
The total reading of the balance is A (panel a).
First a steel block of mass Ms is lowered into the water, suspended by a bar of negligible
mass and length L (panel b). The water does not overflow.
1) (5 points) Does the weight registered by the scale go up, go down or stay the same?
Explain your answer.
Next, the block is removed and a cork of mass Mc (Mc=1/10 Ms) is floated in the water
(panel c). The water does not overflow.
2) (5 points) Does the weight registered by the scale go up, go down or stay the same
compared to the original case (just the bucket of water)? Explain your answer.
(Problem 5 continues on the next page) Suppose that we drill a hole into the side of the bucket at a height h from the bottom of
the bucket (panel d). The level of the fluid inside the tank is d. (NOTE: The diagram is
not to scale. You should assume d>H0 in solving part 3.)
3) (10 points) Determine h such that the water stream exiting the hole lands on the ground
as far away from the base of the balance as possible, i.e. calculate the h that maximizes
the horizontal range R of the water stream. Neglect viscosity of the water. Assume the
water is uniformly distributed in the bucket. L d h H0 (a) (b) (c) (d) R Problem 6 [20 points]
A uniform rod of mass M and length L is supported
by a hinge attached to the ceiling on one end and by
a rope of very small mass on the other end (i.e. you
can neglect the mass of the rope for this part). The
rope makes an angle of 90 degrees with the rod and
θ with the ceiling. The system doesn’t move.
1) (5 points) Draw a free body diagram for the rod.
2) (5 points) Find the horizontal and vertical components of the force exerted by the
hinge on the rod. Give both magnitude and direction (left/right; up/down) of
3) (10 points) Now assume that the mass of the rod M=2.00 Kg, the length L of the
rod (not the rope!) is 1.0 m, the angle θ=30.0 degrees and the mass of the rope is 5
grams. This rope mass is so small that you may use any relevant results from part
A to calculate the tension. I put a standing wave on the rope with a frequency of
47.6 Hz. How many nodes does this standing wave have (not counting the ends)? ...
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This note was uploaded on 10/17/2011 for the course PHYSICS 7A taught by Professor Lanzara during the Spring '08 term at University of California, Berkeley.
- Spring '08