Physics 321 – Spring 2016
Homework #3, Due at beginning of class Wednesday Feb 3.
1. [6 pts] A particle of mass M is moving in a plane, with its Cartesian coordinates (x, y)
given by
x = A [Bt − sin(Bt)]
y = A [1 − cos(Bt)]
where A and B are positive constants.
(a) Find the times at which the speed is a maximum.
(b) Find the tangential component of acceleration, i.e., the magnitude of the component
of acceleration in the direction of motion, as a function of the time t.
(c) Find the “radial” component of acceleration, i.e., the magnitude of the component
of acceleration that is perpendicular to the direction of motion.
2. [6 pts] A particle with electric charge Q and mass M is initially traveling with velocity v0
in the x direction at time t = 0. There is a constant electric field of strength E in the
y direction, and a constant magnetic field B in the z direction. To save a lot of ink, set
Q = M = 1 while you work this problem.
(a) Find the velocity (vx , vy , vz ) as a function of the time t.
(b) Find the position (x, y, z) as a function of time, assuming that it started at (0, 0, 0)
at t = 0.
(c) Rewrite your answer to part (b) with the Q and M factors properly included.
3. [8 pts] Ted and his iceboat have a combined mass M1 . At time t = 0 the boat is at rest on
a frozen lake. While standing on the boat, Ted holds one end of a long rope with mass M2
and length B, which is initially laid out in a straight line on the ice. Ted causes the boat
to move by pulling the rope onto the boat at a steady rate. The last of the rope comes on
board at time T . The rope is not attached to anything, and there is no friction.
(a) The center of mass of the boat+rope+Ted system does not move. Why?
(b) Use the result of part (a) to calculate where the boat is as a function of time. (Treat
the boat plus Ted plus the part of the rope that he has pulled on board as a single
point.)
(c) Use your result from part (b) to find the velocity of the boat as a function of time t.
(d) Use your result from part (c) to calculate the total momentum as a function of time.
(e) Find the kinetic energy of the boat+Ted+rope sytem at the final time T .
(f) Find the total amount of work done by Ted’s muscles during the time interval from
t = 0 to t = T .
(Last updated 1/28/2016.)