Physical Dynamics (SPA5304) – Homework 8
Due Friday 17-Mar-2017 at 16:00. Attempt to answer all questions.
Problem 1 (Quick part A-type questions) [9 marks]
i) State what is meant by a principal axis system for a rigid body.
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ii) State what is meant by the intermediate axis theorem for rigid bodies, also known as the
tennis racket theorem.
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iii) What are the conserved quantities for a free asymmetric top?
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iv) If the principal moments of inertia of a rigid body are all equal, I1 = I2 = I3 , does that
imply that the mass-distribution ρ(x) in the rigid body is spherically symmetric? State
yes or no, and then give a short explanation.
[3]
Problem 2 [25 marks]
A conservative mechanical system consists of two identical uniform rigid rods of mass m and
length l. Two endpoints of the rods are joined together at C with a weightless hinge, and
this common endpoint C of the rods is constrained to move without friction along the vertical
ŷ axis. The remaining two endpoints A and B of the rods can slide without friction along
the x̂ axis as the common endpoint C slides along the ŷ axis. Gravity acts along the vertical
direction. See the figure below for a representation of the system.
(i) How many degrees of freedom does the system have?
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(ii) Calculate the moment of inertia of a single rod with respect to an axis orthogonal to
the rod and passing through its centre of mass. Assume that the width of the rod is
negligible compared to its length.
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(iii) Write down the Lagrangian of the system and the corresponding Euler-Lagrange equations.
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1
(iv) Write down the energy E of the system, and check explicitly that the time derivative Ė
vanishes on the solution to the equations of motion.
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(v) The system is initially at rest in a configuration such that the angle θ indicated in the
figure is equal to π/6. The point C is then set free to slide along the ŷ axis until it
hits the x̂ axis. Use energy conservation to determine the value θ̇hit of the angular
velocity at the time when the common endpoint C of the two rods hits the x̂ axis.
Finally, determine the time elapsed between the moment when the point C starts falling
and the moment when it hits the x̂ axis. You may find the following integral useful:
R π/6
√ 1dθ
' 1.52 .
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0
2
−sin θ
Problem 3 [16 marks]
Consider a frisbee moving through air with angular velocity ω. The friebee’s principal moments of inertia Ii (i = 1, 2, 3) satisfy I1 = I2 , where we have taken x̂II
3 as the symmetry axis
of the frisbee.
(i) Assuming that air is exerting on the frisbee a frictional torque K = −Kω where K is
a positive constant, write down the three Euler equations describing the time evolution
of ω in the frisbee’s body-fixed frame (which we called SII in the lectures).
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(ii) Show that ω3 exponentially decays in time with the decay constant K/I3 .
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Hint: When a quantity A(t) decays in time exponentially as A(t) ∝ e−λt where λ > 0 is
a constant, λ is called the decay constant.
(iii) Show that ω12 + ω22 exponentially decays in time with the decay constant 2K/I1 .
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Hint: Start by eliminating ω3 from the equations for ω̇1 and ω̇2 .
(iv) In the lectures, we learned that, if a symmetric top is free, the angular velocity ω
II
precesses about the x̂II
3 -axis with the angle θ between ω and the x̂3 -axis staying constant
in time. Now, in the present case with a frictional torque, derive how the angle θ evolves
in time, and state whether the angle grows or decays in time.
Hint: This part can be solved
even if you have difficulty in solving (i)–(iii). Find the
p
2
time evolution of tan θ = ω1 + ω22 /ω3 (there was a typo here in the original version of
this homework text, which I have corrected).
[5]
2