PH6246, Section 3916, Fall 2015, Homework 5
Due at the start of class on Friday, October 9.
Answer all questions. Please write neatly and include your name on the front page of your
answers. To gain maximum credit you should explain your reasoning and show all working.
1. Consider the rotation matrix R in two dimensions, given in cartesian coordinates (x, y),
by:
Rab
R=
cos φ sin φ
=
.
− sin φ cos φ
a) By considering your knowledge of partial derivatives (∂x/∂r, ∂θ/∂y, etc), obtain
the rotation matrix transformed into plane polar coordinates (r, θ), using:
0
R0
=
0
Rab0
Note: Both:
∂xa a ∂xb
=
.
R
∂xa b ∂xb0
0
b
∂xa
b = ∂x
,
and
Λ
=
0
b0
∂xa
∂xb
behave as matrices under multiplication here.
0
Λaa
b) Similarly, obtain the transformed metric ga0 b0 in polar coordinates:
∂xa
∂xb
1 0
0
g = ga0 b0 =
where gab =
0 gab
0,
0 1
∂xa
∂xb
in cartesian coordinates. Because gab is not a matrix, you will have to be careful
about the contractions here.
0 0
c) Find the inverse of the metric g0 = g a b which satisfies:
0 0
0
g a c gc0 b0 = δ ab0 ,
0 0
0
and ga0 c0 g c b = δab0 ,
0
0
where δ ab0 is the unit matrix I, and δab0 is its transpose.
d) Show that:
0
0
Rca0 gc0 d0 Rdb0 = ga0 b0 .
Caution: R0T is NOT the matrix inverse of R0 .
e) Compute:
S0
=
0
S ab0
=
0 0
0
g a c Rdc0 gd0 b0
≡
0 0
ga c
RT
d0
c0
gd0 b0 .
0
0
0
0
0
f) Show that S ac0 Rcb0 = Rac0 S cb0 = δ ab0 . Thus, S0 is THE matrix inverse of R0 .
2. A thin, flat, uniform metal triangle lies in the xy plane with its corners at (1, 0, 0), (0, 1, 0)
and (0, 0, 0).
a) Find the triangle’s inertia tensor I.
b) What are its principle moments?
c) Find unit vectors in the three directions of the corresponding axes.
d) From scratch, recalculate its inertia tensor with the coordinate axes aligned along
the principle axes.
3. Consider a cube, of side a and mass M , rotating about an edge.
a) Write down and evaluate an integral for the moment of inertia of the cube about
that edge.
b) Suppose the cube starts from rest, balancing on its edge, resting on a horizontal
surface, and topples without slipping, onto the surface. What is its kinetic energy
at the instant before impact?
c) During its fall, the center of mass moves. Indicate why all the kinetic energy can
be considered as rotational energy about the edge which has been at rest on the
surface.
d) Find the angular velocity of the cube at the instant of impact.
4. Consider a symmetric top spinning with principle moments (λ1 , λ1 , λ3 ) and its tip fixed
on a horizontal surface.
a) Using Euler angles (φ, θ, ψ), write down, the angular velocity vector ω, the angular
momentum L, and the complete Lagrangian for the top.
b) Find the canonical momenta which are conserved, and write down the equation of
motion for the degree of freedom whose canonical momentum is not conserved.
c) By substituting for all first order derivatives in terms of the conserved momenta,
show that the equation of motion from b) can be integrated to give an expression
for a conserved energy. Identify the effective potential for the problem.
d) Explain what motion occurs if the angle θ0 at which φ̇ vanishes lies strictly between
the turning points (in θ) of the motion.
e) Compare your conserved quantity in part c) with the total energy of the system,
and explain why it is acceptable that these two conserved quantities are not equal.
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5. A uniform rod slides with its ends inside a smooth vertical circle. If the rod subtends an
angle of 2π/3 at the center of the circle, show that the equivalent simple pendulum has a
length equal to the radius of the circle.
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