M5 January 2006 Q4

4. A uniform rod \(AB\), of mass \(m\) and length \(2a\), is free to rotate in a vertical plane about a fixed smooth horizontal axis through \(A\) and perpendicular to the plane. The rod hangs in equilibrium with \(B\) below \(A\). The rod is rotated through a small angle and released from rest at time \(t = 0\).

(a) Show that the motion of the rod is approximately simple harmonic. (4)

(b) Using this approximation, find the time \(t\) when the rod is first vertical after being released. (2)

Sketch: rod hanging from A at angle theta to the vertical, weight mg — Mark scheme (a)

Scheme	Marks
M(\(A\)): \(\tfrac{4}{3}ma^2\ddot{\theta} = -mga\sin\theta\)	M1 A1
Small \(\theta \Rightarrow \sin\theta \simeq \theta \;\rightarrow\; \ddot{\theta} = -\dfrac{3g}{4a}\theta\)	M1
\(\therefore\) Approx. SHM	A1
	(4)

Scheme	Marks
Time \(= \tfrac{1}{4}\) period \(= \tfrac{1}{4}\cdot 2\pi\sqrt{\dfrac{4a}{3g}} = \pi\sqrt{\dfrac{a}{3g}}\)	M1 A1
	(2)
	(6 marks)