Question

A slender, uniform metal rod of mass and length is pivoted without friction about an axisthrough its midpoint and perpendicularto the rod. A horizontalspring, assumed massless and with force constant , 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 from the vertical. Considerfirst the torque due to gravity. Which of the following statementsmost accurately describes the effectof gravity on the rod?

Under theaction of gravity alone the rod would move to a horizontalposition. But for smalldeflections from the vertical the torquedue to gravity is sufficiently small to be ignored. Under theaction of gravity alone the rod would move to a vertical position.But for small deflectionsfrom the vertical the restoring force dueto gravity is sufficiently small to be ignored. There is notorque due to gravity on the rod.

(b) Find the torque due to the spring. Assume that is small enough that the springremains effectively horizontal andyou can approximate (and).Express the torque as a function of and other parametersofthe problem.

(c) What is the angular frequency of oscillations ofthe rod?Express the angular frequency interms of parameters given in the introduction.

Thanks

Answer 1

Diagram:Given thatThe metal rod of mass is = MThe length of the rod is = LThe x-componenet of force on the rod is where x =L/2 sinθso.The torque =we also know that The angular frequencyisThe time period isthis may helps you

Answer 2

A. There is no torque due to gravity on the rod.

B. -k*θ* (L/2)<sup>2

C.</sup> √(3k/M)=sqar(3k/m)