PHYS 419: Classical Mechanics, Assignment 9
Due 11/14/08
1. A double pendulum is attached to a cart of mass 2mwhich moves without friction on a
horizontal surface. Each pendulum has length b, the mass of the bob m, and the rod is
massless. Find the equations of motion for the system. Do not solve these equations.
2. A particle of mass mis constrained to move on a circle of radius R. The circle rotates in
its plane about one point on the circle with a constant angular speed ω. The position of the
particle relative to the circle can be described by an angle φthat the particle’s radius (from
the origin of the circle) forms with the circle’s diameter passing through the pivot point. In
the absence of a gravitational force, show that the particle’s motion in coordinate φis the
same as that of a simple pendulum in a uniform gravitational field. Explain why this result
could have been expected.
3. Taylor: Problem 7.27.
4. Taylor: Problem 7.29.
5. Taylor: Problem 7.33.
As formulated in the text, if the angular velocity ˙
θ(t) changed from zero to ωin a very short
time ∆t, the object would jump above the surface. To avoid this complication, assume that
the plane moves with the constant ˙
θ(t) = ωall the time but at time t= 0 the object is at
rest and the plane is horizontal.
Additional question: Find an expression for the normal force acting on the object as a
function of time.
6. A particle of mass mcan slide freely along a straight wire placed in the x−yplane whose
perpendicular distance to the origin Ois h. Denote the projection of Oon the wire by C.
The line OC rotates around the origin (in x−yplane) at a constant angular velocity ω.
The particle is subject to a gravitational force acting down the yaxis. Find the equations
of motion under the initial conditions θ(0) = 0, q(0) = 0, and ˙q(0) = 0, where θis the polar
angle of OC and qis the distance of the particle from C. Sketch the solution.
7. Taylor: Problem 7.37.
8. Taylor: Problem 7.41.
1
