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Precalculus II
University Physics (Phy 231, Phy 241) Final Exam
Completely document your work and your reasoning.
You will be graded on your documentation, your reasoning, and the
correctness of your conclusions.
Test should be printed using Internet Explorer. If
printed from different browser check to be sure test items have not been cut off. If
items are cut off then print in Landscape Mode (choose File, Print, click on Properties
and check the box next to Landscape, etc.).
Name and Signature of Student
_____________________________
Signed by Attendant, with Current Date and Time:
______________________
If picture ID has been matched with student and name as
given above, Attendant please sign here: _________
Instructions:
- Test is to be taken without reference to text or
outside notes.
- Graphing Calculator is allowed, as is blank paper or
testing center paper.
- No time limit but test is to be taken in one
sitting.
- Please place completed test in Dave Smith's folder,
OR mail to Dave Smith, Science and Engineering, Va. Highlands CC, Abingdon, Va.,
24212-0828 OR email copy of document to dsmith@vhcc.edu,
OR fax to 276-739-2590. Test must be returned by individual or agency supervising test. Test is not to be returned to student after it has been taken. Student may, if proctor deems it feasible, make and retain a copy of the test..
Directions for Student:
- Completely document your work.
- Numerical answers should be correct to 3 significant
figures. You may round off given numerical information to a precision consistent
with this standard.
- Undocumented and unjustified answers may be counted
wrong, and in the case of two-choice or limited-choice answers (e.g., true-false or
yes-no) will be counted wrong. Undocumented and unjustified answers, if wrong, never get
partial credit. So show your work and explain your reasoning.
- Due to a scanner malfunction and other errors some
test items may be hard to read, incomplete or even illegible. If this is judged by
the instructor to be the case you will not be penalized for these items, but if you
complete them and if they help your grade they will be counted. Therefore it is to
your advantage to attempt to complete them, if necessary sensibly filling in any
questionable parts.
- Please write on one side of paper only, and staple
test pages together.
Test Problems:
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Problem Number 1
If a simple harmonic oscillator of mass .3 kg is subjected to a restoring force
of 8.5 Newtons when displaced .1071 meters from equilibrium, what will be its KE and its PE at
equilibrium and halfway to equilibrium if it is released from rest at a displacement of .18
meters from equilibrium?
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Problem Number 2
Prove that if the gravitational field strength at distance r from the center of
a planet of mass M is G M / r^2, the work required to move a mass m from a point at
distance r1 to a point at distance r2 from the planet with no net change in velocity is G
M m ( 1/r1 - 1/r2). Derive the expression for the velocity of an object in a
circular orbit at distance r from the center of the planet. Use this result to show
that the KE change between circular orbits has half the magnitude of the PE change between
those orbits.
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Problem Number 3
A simple harmonic oscillator of mass 8 kg has a period of 3 seconds.
What is its restoring force constant?
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Problem Number 4
A white dwarf star might have about the mass of our Sun, around 2 * 10^30 kg,
packed into a very nearly perfect sphere of radius roughly 1600 km (the radius of the
Moon). If you suddenly appeared at the surface of a dwarf star you would
vaporize-they're hot, even if they are small.
- Suppose you decided to orbit the white dwarf in a heat-resistant craft at a
radius of 2160 km from the center. At what rate would you orbit?
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Problem Number 5
An elevator of mass 1370 kg drops freely through a distance of 44 meters when its
cable breaks, landing on a metal spring with force constant 112 * 10^4 N/m. How much
work does gravity do on the elevator, and how much does the spring compress? What is
the maximum speed of the elevator? code `t
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Problem Number 6
A car moving at 17 m/s drives over the top of a hill. The top of the hill
forms an arc of a vertical circle 148 meters in diameter.
- What is the centripetal force holding the car in the circle?
- What therefore is the normal force between the car's tires and the road?
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Problem Number 7
Explain why the work required to pull a dynamics cart up an incline, in the absence of
friction, should be the same as the work required to lift the cart vertically through the
vertical displacement it experiences in the process.
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Problem Number 8
A ball slides down a frictionless ramp of length L to the end of the ramp, which
protrudes over the edge of a table, and falls freely the remaining distance `dy to the
floor. The vertical change in elevation on the ramp is h.
- What is the horizontal range range(h) of the ball?
- What is d / dh (range(h))?
- For a ramp of length 55 cm, vertical rise 10 cm and a vertical fall `dy = 100 cm,
according to a differential estimate by how much would the range be expected to change if
h changed by .3 cm?
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Problem Number 9
A pendulum is released from rest at a displacement of .45 meters from its
equilibrium position. It is stopped abruptly and uniformly at its equilibrium
position and it is observed that a loose bit of metal slides without resistance off the
top of the pendulum and falls to the floor 2.09 meters below.
- If the projectile started off with a velocity in just its horizontal direction,
and if travels .29 meters in the horizontal direction during its fall, what was the
velocity of the pendulum at equilibrium?
- What would be the velocity of the pendulum at a point .3105 meters from its
equilibrium position?
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Problem Number 10
The force exerted by a rubber band at stretch x is given by the function F(x) =
k x^ .71, with k = 150 N / m^ .71.
- If an object of mass .8 kg is accelerated from rest by the rubber band, which is
initially stretched by .113 meters, and as a result rolls up a frictionless incline at some
small angle with horizontal, then how high will the object rise vertically as it rolls up
the incline (assume that none of the potential energy in the rubber band is dissipated and
that the mass of the rubber band is negligible)?
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Problem Number 11
A uniform rod of mass 4.9 kg
and length 84 cm is constrained to rotate on an axis about its center. An unknown
uniform torque is applied to the rod as it rotates through .27 radians from rest, which
requires .4 seconds. The applied torque is then removed and, coasting only under the
influence of friction, the rod comes to rest after rotating through 3 radians, which
requires 10 seconds.
- Determine the angular acceleration of the moving rod before and after the torque is
applied.
- Determine the net torque for each phase of the motion.
- What is the applied torque?
- How much work is done by the net torque, by the applied torque and by friction during
the first phase of motion?
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Problem Number 12
A block of mass 1.5 Kg is held stationary on a level
frictionless tabletop. A mass of .54 Kg is attached to the block by a string over
which runs horizontally fromt the block to a pulley located at the edge of the table; the
mass hangs freely from the string over the pulley.
- As the hanging mass descends 1.7 meters from rest the
block is pulled 1.7 meters along the tabletop (it's a good-sized table--and pretty high
too). How much work does gravity do on each mass during the process?
- Using the definition of KE, determine the velocity
of the system after having traveled the 1.7 meters.
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Problem Number 13
Show that if a net force Fnet = - k x acts on an object of mass m, the equation
of motion of the object must be x(t) = A sin(`omega * t) + B cos(`omega * t).
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Problem Number 14
What is the effective force constant for a simple pendulum of mass 1.6 kg and
length .92 meters? What therefore is its period of oscillation?
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Problem Number 15
A ball rolls down a ramp from rest, starting at different
positions on the ramp.
- If it coasts distances of 2.315286, .4393912, 2.453806 and 1.092554 cm, starting
from rest each time, and requires respective times of 9.5 sec, 2 sec, 10.25 sec and
8.75 sec, is the hypothesis that acceleration is independent of position or velocity
supported or not?