Description
1. Taken from the 2017 Classical and Statistical Mechanics Qualifier:
A wedge of mass M and angle θ moves frictionlessly along the x axis. A small mass,
m a distance ` from the top of the wedge moves frictionlessly along the wedge. Other
than gravity and the normal force on the wedge from the ground, there are no external
forces on the system.
x
θ
(a) (2 pts) Find the kinetic energy of the system in terms of the generalized coordinates x and `.
(b) (1 pts) Find the Lagrangian.
(c) (2 pts) Find the equations of motion for the generalized coordinates and the ratio
µ = m/(m + M).
(d) (1 pt) Is there a speed of the wedge in which the acceleration of the mass is up
the wedge? If so, find it. If not, prove mathematically or explain why.
(e) (2 pts) Integrate the equations of motion and find how long it takes for the particle
to slide off if it starts at a height, h above the ground with both wedge and particle
at rest.
(f) (1 pt) Show that in the limit of M → ∞ this agrees with the expected result.
(g) (1 pt) How far did the wedge move during this time?
2. Taken from the Spring 2020 Classical and Statistical Mechanics Qualifier:
A heavy rope of mass m, and length L has uniform density and is attached at one end
to the edge of a (fixed) balcony. It is initially at rest with its free end at a height equal
to the point of support so that y(t = 0) = 0.
The end of the rope is released and it
falls downward due to the force of gravity. You should treat the rope as ideal, so that
the portion at the left hand side is at rest, and all motion is elastic.
y(t)
(a) Determine the position of the center of mass of the rope as a function of y. (2
points)
(b) Using the above expression for the center of mass, write down the Lagrangian for
the rope. (2 points)
(c) Determine the speed at which the end of the rope falls as a function of y. (2
points)
(d) Determine the acceleration of the end of the rope as a function of height. (2
points)
(e) How does your answer compare to g? Explain your result. (2 points)
3. Taken from the 2018 Classical and Statistical Mechanics Qualifier
Two blocks each of mass M are connected by a rope of length ` and mass m = λ`,
where λ is the linear density of the rope. One block slides frictionlessly on a horizontal
surface attached to the wall by spring with spring constant, k. The rope, attached to
the other side, goes over an ideal, massless pulley and is attached to the second block
which hangs freely.
The figure on the left is the initial condition, where D is the distance between the
sliding mass and the edge of the surface when the spring exerts no force (it is neither
compressed nor stretched) and the entire system is at rest. In this initial arrangement
the length of the rope supporting the suspended block is a = ` − D. (Note that this
is not an equilibrium position.)
The figure on the right shows the system a short time
after the blocks are released from rest, so that the spring is now stretched and the
blocks are moving. We define x as the displacement of the block on the horizontal
surface from the neutral position of the spring and s = x + a is length of the rope
supporting the suspended block . (Initially, s = a.)
This problem should be worked using the generalized coordinate s and the variables
M, m, k, λ, a and g, the acceleration due to gravity. You should assume that the
radius of the pulley is so small that it can be neglected.
(a) Find the potential energy. (2 Points)
a
D
s
D
x
Initial position Displaced position
(b) Find the kinetic energy and Lagrangian. (2 Points)
(c) Find the equation of motion for s using the Lagrangian. (2 Points)
(d) Find the equilibrium position for the system. (2 Points)
(e) Find the frequency of oscillation for the system about the equilibrium position.
(2 Points)
4. Taken from the 1996 CSM Qualifier:
A point particle of mass m moves under the influence of gravity (V (x, y, z) = mgz).
The particle itself is constrained to stay on a frictionless surface given by
z = α
x
2 + y
2
where α > 0.
(a) Derive the equations of motion for x, y, and z by minimizing the appropriately
constrained action, using the method of Lagrange multipliers. 2pt
(b) Consider the class of trajectories (i.e. solutions) for which z = constant ≡ z0. 2pt
Calculate the force required to maintain the constraint for these trajectories.
(c) Consider the class of trajectories for which y = 0. (These are different trajectories
than those in part (b).) Assume that the particle reaches a maximum height of
z1
i. Using conservation of energy derive an expression for ˙z ≡
dz
dt as a function of 2pt
z, and z1. (That is, remove the dependence on any other spatial coordinate.)
ii. From the expression derived in (i) and the equations of motion you calcu- 2pt
lated above, determine the force in the z-direction required to maintain the
constraint.
iii. Calculate the total force required to maintain the constraint for this class of 2pt
trajectory.
5. Many solid state systems are crystals, and the problems involving them have a rotational symmetry where the system is rotated by θ = π/2 (four-fold symmetry) or
θ = π/3 (six-fold rotational symmetry). Is there a conserved quantity associated with
this symmetry by Noether’s Theorem? If so, what is it? If not, why not?
6. Consider a two particle interacting system in 3D with the Lagrangian:
L =
1
2
m1
˙~r1
2
+ m2
˙~r2
2
− V (~r1 − ~r2)
(a) Show that the transformation:
~ri → ~ri + ~v0t
where ~v0 is a constant vector, leaves the Lagrangian unchanged to order up to
a total derivative.
(b) Calculate the conserved quantity associated with with this symmetry of the action.
(c) What is this new conserved quantity? Is it useful?
While the above problem is posed in three dimensions, you can work in one dimension
and generalize.
