Physics 410 F16
HW #7
1)
Lagrangian of a skewed cylindrical pendulum: A pendulum consists of a mass
m on a massless rope that is wrapped around a cylinder of moment I that rotates about its
axis. A mass mb is attached by a massless rigid rod of length L to the cylinder. Assume
that the masses and lengths allow the equilibrium angle θ0 to be less than π/2. Derive the
angular frequency of oscillation ω for small oscillations as a function of the masses, sizes
and I.
R
L
mbg
mg
2)
Lagrangian for wrapped rope: A rope is wrapped around a cylinder of radius R
and mass M on an inclined plane. It is attached by a massless rope over a massless pulley
to a mass m. The cylinder rolls without slipping. Use Lagrange’s undertermined
multipliers to solve for the acceleration of the system. What is the condition for
equilibrium? What happens to the friction force in equilibrium? (Note on Lagrange’s
undetermined multipliers can be found at http://www.works.bepress.com/ddnolte in the
file Notes and Corrections.)
I = MR 2
mg
3)
Lagrangian for sliding wedge: A cylinder of mass M and radius R rolls without
slipping down an inclined plane of equal mass M. The inclined plane can slide without
friction on the surface of the table. Solve for the accelerations of the cylinder and wedge.
I = m R2
M
4)
IMD 2.16