Assignment 4

Problem 1:

Two spheres are of the same radius R and mass M, but one is solid and the other is a hollow shell (of negligible thickness).  Both spheres roll (without sliding) down a ramp of incline θ.
(a)  Which sphere will have the greater acceleration down the ramp?
(b)  Determine the Lagrangian for the motion of the sphere and derive the equation of motion for both cases.

Problem 2:

Obtain Lagrange's equations of motion for a spherical pendulum (a mass point suspended by a rigid, weightless rod).

Problem 3:

Consider a bead of mass m sliding freely on a smooth circular wire of radius b which rotates in a horizontal plane about one of its points O, with constant angular velocity Ω.  Let θ be the counterclockwise angle between the diameter that passes through the mass and the diameter that passes through the point O, with θ = 0 the case where the mass is farthest from O.

 

(a)  Find the equation of motion for θ.  Compare this equation with the equation of motion for a simple pendulum (point mass and massless rod).
(b)  For the initial conditions θ = 0, dθ/dt = ω0 at t = 0, describe the θ motion that occurs for |ω0| < 2Ω and for |ω0| > 2Ω.  (Note:  The same equations have the same solutions.)
(c)  Describe the θ motion that occurs for |ω0| << 2Ω.
(d)  Find the force that the wire exerts on the bead as a function of θ and dθ/dt.



Problem 4:

The Lagrangian of a system of N degrees of freedom is

What is the Hamiltonian for a symmetric mass matrix Mij = Mji?

Problem 5:

A bead, of mass m, slides without friction on a wire that is in the shape of a cycloid with equations
x = a(2θ + sin2θ),
y = a(1 - cos2θ),
- π/2 ≤ θ ≤ π/2.


A uniform gravitational field g points in the negative y-direction.
(a)  Find the Lagrangian and the second order differential equation of motion for the coordinate θ.
(b)  The bead moves on a trajectory s with elements of arc length ds.
Integrate ds = (dx2 + dy2) = ((dx/dθ)2 + (dy/dθ)2)dθ with the condition s = 0 at θ = 0 to find s as a function of θ.
(c)  Rewrite the equation of motion, switching from the coordinate θ to the coordinate s and solve it.  Describe the motion.