Physics 21 Final - 3 hours
3 pages - 10 problems
Harry Nelson
Friday, March 18
Write your answers in a blue book. Calculators and two pages of notes allowed. No textbooks allowed.
Please make your work neat, clear, and easy to follow. It is hard to grade sloppy work accurately. Generally,
make a clear diagram, and label quantities. Make it clear what you think is known, and what is unknown and
to be solved for. Except for extremely simple problems, derive symbolic answers, and then plug in numbers
(if numbers provided) after a symbolic answer is available. You can take the acceleration of gravity near the
earth as g= 10 m/s2, to simplify calculations. Put a box around your final answer... otherwise we
may be confused about which answer you really mean, and you could lose credit.
1. You release a mass m1from rest at the top of a ramp shaped like a quarter-circle of radius R on to a
horizontal table, where m1eventually has an elastic collision with mass m2, as shown in Figure 1. You
may choose any value for the mass m1, but everything else in the situation stays the same, like R,m2,
and the acceleration of gravity g; everything is frictionless.
(a) What is the velocity, v1, of the mass m1just before it collides with m2, in terms of the masses, R,
and g?
(b) What choice of m1will result in the largest velocity u2of m2after the collision, and what is the
value of that velocity?
(c) What choice of m1will result in m1to losing the largest fraction of its energy in the collision?
2. A line of rain drops, each with mass mand falling at terminal velocity v, falls on the pan of a weight
scale. Each drop in the line is a distance `from the next drop. When each rain drop falls on the pan,
its water immediately comes to rest, accumulating in a puddle on the pan. At t= 0, the first rain drop
hits the pan. What does weight is read on the scale, as a function of time t? Neglect any ‘jerkiness’
that arises because the drops reach the pan one at a time.
3. A particle of mass m= 1 kg moves in one dimension from the origin to ∞and is subject to the potential
energy:
U(x) = α(1 −x
a) + 1
2βx2(1)
where α= 4 Joules, a= 1 meter, and β= 4 Joules/meter2.
(a) Is there a stable equilibrium point for the particle, and if so, at what value of xdoes it occur
(symbolically and numerically)?
(b) Determine the (circular) frequency ωof small oscillations about any stable equilibrium point (both
symbolically and numerically).
4. A bug of mass m1sits on a bar of mass m2and length L. The midpoint of the bar is connected by a
massless strut of length dto a pivot point, as shown in Figure 2. The bar and strut sit atop a frictionless
table. The bug starts at t= 0 from the point where the strut attaches to the bar, and walks along the
bar with velocity v with respect to the bar. No net external force or torque acts on the system.
