Part a we start by analyzing the torques acting on

Part A
We start by analyzing the torques acting on the rod when it is deflected by a small angle from the vertical. Consider first
the torque due to gravity. Which of the following statements most accurately describes the effect of gravity on the rod?
Choose the best answer.
ANSWER:
Assume that the spring is relaxed (exerts no torque on the rod) when the rod is vertical. The rod is displaced by a small angle
from the vertical.
Under the action of gravity alone the rod would move to a horizontal position. But for small deflections from the
vertical the torque due to gravity is sufficiently small to be ignored.
Under the action of gravity alone the rod would move to a vertical position. But for small deflections from the
vertical the restoring force due to gravity is sufficiently small to be ignored.
There is no torque due to gravity on the rod.
θ
τ θ
sin ( θ ) ≈ θ cos( θ ) ≈ 1
Correct
Part B
Find the torque due to the spring. Assume that is small enough that the spring remains effectively horizontal and you
can approximate (and ).
θ
Express the torque as a function of and other parameters of the problem.
Hint 1. Find the change in spring length
δ
Deflecting the rod will stretch or compress the spring by a length . The spring will react with a restoring force
θ
δ

5/26/16, 12 : 58 PM homework #1
given by Hooke's law:
.
What is ?
Express your answer in terms of and
Hint 1. Relation between displacement and rotation angle
Consider an object at a distance from the pivot it is rotating about. If it rotates through an angle , then
its displacement is given by
.
F = − k δ
δ
l θ
r ϕ
∆ x
∆ x = r ϕ
.
2
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Correct
Hint 2. Find the moment arm
The torque about a point is defined as the product of the force acting on a body times the moment arm
(perpendicular distance from the line of action of the force to the center point):
What is for the given situation?
Express your answer in terms of quantities given in the problem introduction.
2
l
4

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Correct
Since the torque is opposed to the deflection and increases linearly with it, the system will undergo angular simple
harmonic motion.
Part C
What is the angular frequency of oscillations of the rod?
Express the angular frequency in terms of parameters given in the introduction.
Hint 1. How to find the oscillation frequency
can be found from the equation of motion by comparing it to the standard form
.
This equation describes any angular simple harmonic motion. Bring your equation of motion into this standard
form, and hence extract the expression for in terms of the parameters of this specific problem.