#### Problem 1:

A mechanical system known as an Atwood machine consists of three weights of
mass m_{1}, m_{2}, and m_{3}, 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 m_{1}.
Let m_{1} = m, m_{2} = m, m_{3} = 0.25 m, M = 0.5 m and
I = 0.25 mR^{2}.

#### 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.

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 = (dx^{2} + dy^{2})^{½}
= ((dx/dθ)^{2} + (dy/dθ)^{2})^{½}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.

#### Problem 3:

A particle of mass m is described by the Lagrangian function

L = ½m[(dx/dt)^{2} + dy/dt)^{2} + (dz/dt)^{2}] +
½ωl_{3},

where l_{3} 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.

#### 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?

####
**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
= R_{0} + R't.

At t = 0 the speed of the particle is v_{0}.

(a) Are there any constants of motion?

(b) Let E(t_{1}) be the energy of the particle at the time the
radius of the ring is 2R_{0} and E_{0} its energy at t = 0.

Find the ration E(t_{1})/E_{0}.