10. Unequal masses M and m are suspended from a pulley as shown in the
(a) Analyze the forces in which mass m participates, using a table the
format shown in section 5.3. [The forces in which the other masses
participate will of course be similar, but not numerically the same.]
(b) Find the magnitude of the accelerations of the two masses. [Hints:
(1) Pick a coordinate system, and use positive and negative signs consis-
tently to indicate the directions of the forces and accelerations. (2) The
two accelerations of the two masses have to be equal in magnitude but of
opposite signs, since one side eats up rope at the same rate at which the
other side pays it out. (3) You need to apply Newton’s second law twice,
once to each mass, and then solve the two equations for the unknowns:
the acceleration, a, and the tension in the rope, T.]
(c) Many people expect that in the special case of M=m, the two masses
will naturally settle down to an equilibrium position side by side. Based on
your answer from part (b), is this correct.
11. A tugboat of mass m pulls a ship of mass M, accelerating it. The speeds
are low enough that you can ignore fluid friction acting on their hulls,
although there will of course need to be fluid friction acting on the tug’s
(a) Analyze the forces in which the tugboat participates, using a table in
the format shown in section 5.3. Don’t worry about vertical forces.
(b) Do the same for the ship.
(c) Assume now that water friction on the two vessels’ hulls is negligible.
If the force acting on the tug’s propeller is F, what is the tension, T, in the
cable connecting the two ships. [Hint: Write down two equations, one for
Newton’s second law applied to each object. Solve these for the two
unknowns T and a.]
(d) Interpret your answer in the special cases of M=0 and M=
12. Explain why it wouldn't make sense to have kinetic friction be stron-
ger than static friction.
13. In the system shown in the figure, the pulleys on the left and right are
fixed, but the pulley in the center can move to the left or right. The two
masses are identical. Show that the mass on the left will have an upward
acceleration equal to g/5.
14 S. The figure shows two different ways of combining a pair of identical
springs, each with spring constant k. We refer to the top setup as parallel,
and the bottom one as a series arrangement. (a) For the parallel arrange-
ment, analyze the forces acting on the connector piece on the left, and
then use this analysis to determine the equivalent spring constant of the
whole setup. Explain whether the combined spring constant should be
interpreted as being stiffer or less stiff. (b) For the series arrangement,
analyze the forces acting on each spring and figure out the same things.
15. Generalize the results of problem 14 to the case where the two spring
constants are unequal.