Assignment 7, solutions

Problem 1: 

A mechanical system known as an Atwood machine consists of three weights of mass m1, m2, and m3, respectively, connected by a light (massless) inextensible cords of length l - πR and l' - πR, respectively, which pass over identical pulleys with radius R, mass M, and moment of inertia I.  Find the acceleration of m1.  Let m1 = m, m2 = m, m3 = 0.25 m, M = 0.5 m and I = 0.25 mR2.

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Solution:

Problem 2:

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.

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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)½ = ((dx/dθ)2 + (dy/dθ)2)½dθ 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.

Solution:

Problem 3:

A particle of mass m is described by the Lagrangian function

L = ½m[(dx/dt)2 + dy/dt)2 + (dz/dt)2] + ½ωl3,

where l3 is the z-component of the angular momentum and ω is a constant angular frequency.
(a)  Find the equations of motion, write them in terms of the variables (x + iy) and z and solve them.
(b)  Construct the Hamiltonian function and find the kinematic and canonical momenta. 
Show that the particle has only kinetic energy and that the latter is conserved.

Solution:

Problem 4:

For a symmetrical prism (one in which the apex angle lies at the top of an isosceles triangle), the total deviation angle ϕ of a light ray is minimized when the ray inside the prism travels parallel to the prism's base.  
Assume that a beam of light passes through a glass equilateral prism with refractive index 1.5.  The prism is in air and is mounted on a rotation stage, as shown in the figure.  When the prism is rotated, the angle by which the beam is deviated changes.  What is the minimum angle ϕ by which the beam is deflected?

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Solution:

Problem 5:

A point particle with mass m is restricted to move on the inside surface of a horizontal ring.  The radius of the ring increases steadily with time as R = R0 + R't.
At t = 0 the speed of the particle is v0.
(a)  Are there any constants of motion?
(b)  Let E(t1) be the energy of the particle at the time the radius of the ring is 2R0 and E0 its energy at t = 0. 
Find the ration E(t1)/E0.

Solution: