Figure 1: Diagram for Problem 1
1) A car, mass m, begins at the right end of a barge, mass 4m and length `. The center of mass of the barge
is at its midpoint. The barge begins in contact with the shore, and the top of the barge is a height h above
the shore. (See figure 1.) Assume that the barge moves through the water without friction.
(i) Taking the shore to be the origin x = 0, what is the x-coordinate for the center of mass of the car and
barge?
(ii) The car drives to the left towards the shore with speed vcar as measured by an observer on the shore.
What is the speed of the barge relative to the shore?
(iii) When the car reaches the left side of the barge, how far is it from the shore in the x direction? (I.e.,
how far has the barge moved from the shore?)
(iv) If the car then drives off the left side of the barge (only horizontal initial velocity), what minimum speed,
relative to the shore, does the car need to reach the shore?
Figure 2: Diagram for Problem 2
2) A rock of mass m is attached to a massless spring with spring constant k. At the spring’s equilibrium
position (x = 0), the rock sits on the boundary of two surfaces. To the left, the surface is smooth, and the
rock slides without friction. To the right, the surface is rough, and the rock slides (never rolls) with kinetic
friction coefficient µk . The rock begins at x = −x0 (spring compressed).
(i) The rock is released. What is the speed of the rock when it reaches x = 0?
(ii) What is the maximum displacement x1 of the rock beyond its equilibrium point (on the rough surface)?
Remember, it’s still attached to the spring.
(iii) If the rock gets stuck at its maximum displacement on the rough surface, what is the minimum value of
the coefficient of static friction µs between the rock and rough surface in terms of x1 , m, g, and k? (You do
not need to have solved part (ii) to answer either this or the next part.)
(iv) If the rock does not get stuck but is pulled back and stops exactly on its equilibrium point, compute the
value of the coefficient of kinetic friction µk in terms of x1 , m, g, and k?
Figure 3: Diagram for Problem 3
3) A block (mass m) slides to the right on a frictionless surface with speed v0 in the lab frame. A frictionless
curved ramp (mass 5m), also free to slide on the frictionless surface, begins at rest.
(i) What is the center-of-mass velocity of the block and ramp in the lab frame?
(ii) What is the total kinetic energy of the block and ramp (1) in the lab frame and (2) in the center-of-mass
frame?
(iii) What is the maximum (vertical) height on the ramp reached by the block? (Assume the block is not
going so fast that it flies off the top of the ramp.)
(iv) After the block and ramp separate, what is the velocity of the ramp in the lab frame?