A slender, uniform metal rod of mass M and length l is pivoted without friction about an axisthrough its midpoint and perpendicular to the rod. A horizontalspring, assumed massless and with force constant k, is attached to the lower end of the rod, withthe other end of the spring attached to a rigid support.(a) We start by analyzing the torques acting on the rod when it isdeflected by a small angle theta from the vertical. Considerfirst the torque due to gravity. Which of the following statementsmost accurately describes the effect of gravity on the rod?a. Under theaction of gravity alone the rod would move to a horizontalposition. But for small deflections from the vertical the torquedue to gravity is sufficiently small to be ignored. b. Under theaction of gravity alone the rod would move to a vertical position. But for small deflections from the vertical the restoring force dueto gravity is sufficiently small to be ignored. c. There is notorque due to gravity on the rod.(b) Find the torque Ï due to the spring. Assume that θ is small enough that the spring remains effectively horizontal and you can approximate sin(θ)âθ (and cos(θ)â1 ).Express the torque as a function oftheta and other parameters ofthe problem.(c) What is the angular frequency omega of oscillations of the rod?Express the angular frequency interms of parameters given in the introduction.

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