A slender, uniform metal rod of mass MMM and length lll is pivoted without friction about an axis through its midpoint and perpendicular to the rod. A horizontal spring, assumed massless and with force constant kkk, is attached to the lower end of the rod, with the other end of the spring attached to a rigid support. (Figure 1)
We start by analyzing the torques acting on the rod when it is deflected by a small angle θθtheta 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.
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.
Find the torque ττtau due to the spring. Assume that θθtheta is small enough that the spring remains effectively horizontal and you can approximate sin(θ)≈θsin⁡(θ)≈θ (and cos(θ)≈1cos⁡(θ)≈1).
Express the torque as a function of θθtheta and other parameters of the problem.
What is the angular frequency ωωomega of oscillations of the rod?
Express the angular frequency in terms of parameters given in the introduction.