Download Physics PHYS 352 Mechanics II Problem Set #4

Physics PHYS 352
Mechanics II
Problem Set #4
1.
Bead on a Hoop (Princeton Problems 1.9)
A bead of mass m slides without friction on a circular loop of radius a. The loop lies in a
vertical plane and rotates about a vertical diameter with constant angular velocity .


a
m
a.
b.
For angular velocity  greater than some critical velocity c, the bead can undergo
small oscillations about some stable equilibrium point 0. Find c and 0().
Obtain the equations of motion for the small oscillations about 0 as a function of 
and find the period of the oscillations.
2.
Funny Pendulum (Marion 7-18).
A pendulum is constructed by attaching a mass m to a string of length  . The upper end of the
string is connected to the uppermost part of a vertical disk of radius R ( R    ) as in the figure.
a.
Obtain the pendulum’s equation
of motion, and find the
frequency of small oscillations.
b.
Find the line about which the
angular motion extends equally
in either direction (i.e. 1   2 ).
R
2
1
m
3.
Sliding Wedge (Fowles and Cassiday 10.20)
A particle of mass m1 slides down the smooth circular surface of radius of curvature a of a
wedge of mass m2 that is free to move horizontally along the smooth horizontal surface on
which it rests.
m1
a
m2
a.
b.
Find the equations of motion of each mass.
Find the normal force of constraint exerted by the wedge on the particle. Use the
method of Lagrange multipliers.
2