Physical Dynamics (SPA5304) – Homework 1
Due Friday 20-Jan-2017 at 16:00. Attempt to answer all questions.
Problem 1 (Quick part A-type questions) [15 marks]
(i) Explain what is meant by Newton’s principle of determinacy.
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(ii) State under what condition the angular momentum of a particle is conserved.
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(iii) Give the definition of work of a force F~ along a path P(1, 2) going from point ~r1 to
point ~r2 .
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(iv) Prove that the work done by a conservative force on a particle of mass m is independent of the particle’s path.
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(v) Show that, for a conservative mechanical system consisting of a single particle of
~ (r), the time derivative of the energy, d E(~r, ~r˙ ),
mass m subject to a force F~ = −∇V
dt
is equal to zero.
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Problem 2 [15 marks]
A particle of mass m moving on a line is subject to the potential
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V (x) = a(1 − e−x ) ,
a>0.
(i) Draw the potential.
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(ii) Write down the equations of motion.
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(iii) Determine the equilibrium position(s) – these are defined as those positions where
the force is zero.
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(iv) Consider a solution of the equations of motion for which the total energy E is equal
to a/2. Show that the motion is limited and determine the range of x.
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(v) What happens if we choose E < 0?
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Hint: To answer (iv) and (v), use the fact that the kinetic energy must be a positive
quantity – no motion is allowed if the kinetic energy becomes negative.
Problem 3 [20 marks]
Consider two parallel, infinitely long, infinitely thin rods with mass density per unit length
ρ. They both lie in the x-y plane and extend in the x direction. Let one be at y = L, z = 0
and the other at y = −L, z = 0.
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(i) Take the rod at y = L, and consider its infinitesimal piece between x and x + dx.
What is the gravitational force F~+ (x) exerted by this piece on a particle with mass
m at point Q(0, 0, z) on the z-axis? See the figure below for illustration:
Hint: recall that the gravitational force exerted by an object with mass M at point
P on an object with mass m at point Q is
GM m −→
F~grav = − −→ P Q.
|P Q|3
where G is Newton’s constant.
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(ii) Now consider the rod at y = −L. What is the gravitational force F~− (x) by the
infinitesimal piece in [x, x + dx] acting on the same particle?
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(iii) Show that the total gravitational force F~tot exerted by the two entire rods has vanishing x, y components.
Hint: The gravitational force exerted by the piece of the +L rod at x and that by the
piece of the −L rod at −x have cancelling x, y components.
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(iv) By summing (integrating) over the contributions from the infinitesimal pieces you
found in (i), (ii), show that the total gravitational force is
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G
m
ρ
z
F~tot = 0, 0, − 2
.
z + L2
R
dx
√ x
.
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Hint: use the formula (x2 +1)
3/2 =
x2 +1
(v) If we let the mass freely move along the z axis, it gets attracted by the gravitational
force from the rods and oscillate between z > 0 and z < 0. Assuming that z L,
show that the motion of the particle is harmonic and determine the frequency of
small oscillations.
Hint: you can answer this, even if you have difficulties in answering the previous
ones! See also Problem 4 of the Exercise Class 1 problems (the solutions are already
available).
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