Download Sample exam 2

Physics 221
Sample exam 2 (Chapters 5, 6, 7, 13)
Open book, notes, homework assignments, lab notebook, reference sheets; no
collaboration or Internet usage. This exam is worth fifty points and is scheduled for
eighty minutes. Answer all questions.
1. a. With a sufficient speed, a marble can spin in a horizontal
circle inside an inverted cup, as illustrated in the figure. The
side of the cup makes an angle of θ with the vertical, and the
coefficient of static friction between the cup and the marble is
µs. Find an expression for the velocity of the marble in terms of
g, µs, θ, and r, the radius of the circle. Hint: start with drawing
the free-body diagram and setting up the force equations.
b. As θ approaches 90°, what does your expression simplify to? Explain, in a couple of
sentences, how this makes sense physically.
2. a. A point mass m moves in a circular orbit under the influence of the gravitational
force of a large mass M. You may assume M >> m, and that M is fixed in space. Show
that, if v is the velocity of the mass m, and R is the radius of its orbit that:
R v2 = M G
b. Show that the expression above is algebraically equivalent to Kepler’s Third Law of
Planetary Motion.
3. The potential energy curve is shown on the graph below; the potential energy function
is U(x) = A x2, where A is a constant.
a. Write an expression for F(x), the force exerted
by this potential.
b. Sketch a graph of F vs. x; you need not overly
quantify it.
c. If a particle has a total energy of some value
greater than zero, write its range of x (in other
words, in what range of x is the particle allowed
to move?).
4. a. The springs of a car of mass 1,000 kg give it a period of 1.0 s when there are no
passengers. Four people, each with a mass of 60.0 kg get into the car. How far down does
the car move?
b. The car and the four passengers are travelling along a horizontal road when it runs onto
new pavement, which is raised 10.00 cm above the old road. This suddenly raises the
wheels and the bottom ends of the springs 10.00 cm before the car begins to move
upwards. In the ensuing rebound, what is the maximum weight (force exerted by the
passenger on the seat) of a 60.0 kg passenger? Assume there are no shock absorbers.
c. What is the maximum upward velocity attained by the car and passengers?
d. Assuming the springs eventually stop the oscillations, how much energy did they
absorb?
5. The typical result in Lab 6 was that the tension force was measured quite accurately by
the force sensor (often to an error of less than 0.5%) but the acceleration of the cart was
measured as having huge errors bars (perhaps as large as ± 1 m/s2, which is much larger
than the average acceleration itself). Suggest a reason why the tension force could be
measured so well and the acceleration so poorly, and then suggest a “fix” for this problem,
if it can be fixed. You may assume that there were no “human” errors (e.g., calculation
errors, sensor positioning errors, zeroing and calibration errors).