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PHY W3003: Practice Final Examination
All problems have equal weight and each subquestion contributes equally to the total for
that problem. The examination continues on the other side of the page!
Please answer all questions, and show your work as partial credit will be given.
You may bring to the exam on sheet (both sides) of paper. Use of calculators or other
electronic devices is not allowed during the examination. Where numerical answers are
requested, answers to one significant figure are acceptable.
1 A pendulum is constructed by putting a bob of mass m at the end
of a massless spring which can stretch along its length but does not
bend (see Fig). The spring constant is k and the relaxed length of
the spring is 0. In addition to the spring the mass is subject to the
force of gravity which points vertically down. The mass can move in
x
θ (shown on fig) and radially along the length of the spring but not in
the azimuthal direction, so motion is confined to a plane.
(a) Please write the Lagrangian of the system, using as generalized
coordinates θ and the length x of the spring.
(b) Please write (but do not solve!) the Euler-Lagrange equations
(c) Please find the equilibrium values of the generalized coordinates.
(d) Please find the frequencies of the normal modes describing small oscillations about
the equilibrium position
2 A particle of mass m = 1 moves in one dimension subject to a harmonic potential and
an additional force linearly proportional to time such that its equation of motion is
x
¨ + 4x˙ + 4x = t
If the particle is released from rest at position x = 0 at time t = 0 please find its position
x(t) for all t > 0.
3
Three
k
x=0
particles
equal
k
k
x1
of
x2
k
x3
mass m are coupled by springs as shown
in the figure. At time t = 0 particles 1 and 3 are at
rest in their equilibrium positions while particle 2
is at rest but with a small displacement q2 in the
+x direction. Please find the position of particle
3 for all t > 0.
x=4a
1
4 Consider the rectangular parallelepiped of dimensions a, b and c shown in the Figure,
with mass M at each of the 8 corners. Take the origin of coordinates as shown to be the
front lower left corner of the object, and choose x to be along b, y to be along a and z to
be along c
(a) Please find the inertia tensor for
rotations about the origin.
(b) If the object rotates with angular
velocity Ω = 1radian/sec about the axis
shown, the masses M are each 0.5kG and
a = 1, b = 2 and c = 0.5 with distances
measured in meters, please find the three
components of the angular momentum
Origin
vector (numerical values, to one significant figure).
5 Consider a particle of mass m moving in two dimensions and subject to the central
potential
k
V (r) = − 2
r + b2
Please
(a) find (in terms of m, k and b) the values of the angular momentum L (measured about
an axis perpendicular to the plane of motion and passing through the origin) for which
stable circular motion is possible, and for these values, the radius of the circular motion
(terms of m, k and b and L)
(b) for values of L such that stable circular motion is possible please sketch the effective
potential and the phase space orbits (in the half plane r > 0, r˙ arbitrary) indicating the
bound unbound and separatrix orbits (if any) and give the energy of the separatrix orbit
(if any)
(c) Find the value of L for which the stable circular orbit has radius b.
Consider a particle moving on the orbit found in (c). Suppose that at time t = 0 the
particle receives a tangential impulse such that its angular momentum is instantaneously
doubled (L → 2L) but the radial velocity is unchanged.
(d) After the impulse the orbit is no longer circular. Please determine the smallest and
largest values of r reached by the particle.