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Simple Harmonic Motion

A slender, uniform metal rod of mass M and length l is pivoted without friction about an axis through its midpoint and perpendicular to the rod. A horizontal spring, assumed massless and with force constant k, is attached to the lower end of the rod, with the other end of the spring attached to a rigid support.

a). Find the torque due to the spring. Assume that theta is small enough that the spring remains effectively horizontal and you can approximate :

sin(theta) = theta (approximately) and cos(theta) = 1 (approximately)

Express the torque as a function of theta and other parameters of the problem.

b). What is the angular frequency omega of oscillations of the rod?

Express the angular frequency in terms of parameters given in the introduction.


Solution Preview

Please refer to the attachment.

Let the rod be displaced from its mean position (vertical) by a small angle θ. Lateral displacement of the lower end of the rod = (L/2)θ [arc/radius = angle subtended by the arc]

As the angular displacement is very small, the spring can be considered as horizontal even after displacement and lateral ...

Solution Summary

The expert examines simple harmonic motion. A step by step solution provided.