1. (15 points) A mass particle moves in a constant vertical gravitational field along the curve
defined by y = ax4
, where y is the vertical direction. Find the equation of motion for small
oscillations around the equilibrium position.
2. (25 points) Find the normal modes of vibration for a system described by the following
kinetic and potential energy:
T =
1
2
mR2
(
˙θ
2
1 + ˙θ
2
2 + ˙θ
2
3
), V =
1
2
kR2
[(θ1 − θ2)
2 + (θ2 − θ3)
2 + (θ3 − θ1)
2
]. (1)
3. (30 points) Consider a system under small oscillations. Express its kinetic and potential
energy in terms of normal coordinates. Try to show that the time average of the kinetic energy
is equal to that of the potential energy.
4. (30 points) A simple pendulum of length l and mass m is attached to a block of mass 2m,
which can slide on a frictionless surface. Assume the motion is in the vertical plane, solve the
small oscillation problem for this system.
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