All Problems

Mechanical Oscillations

Problem 4.47

A uniform rod of mass \(m\) and length \(l\) performs small oscillations about the horizontal axis passing through its upper end. Find the mean kinetic energy of the rod averaged over one oscillation period if at the initial moment it was deflected from the vertical by an angle \(\theta_{0}\) and then imparted an angular velocity \(\dot{\theta}_{0}\).

Reveal Answer
⟨T⟩=1/8mglθ02+1/12ml2θ˙02\langle T\rangle=1 / 8 m g l \theta_{0}^{2}+1 /{ }_{12} m l^{2} \dot{\theta}_{0}^{2}
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