M5 June 2014 (R) Q3

3. A uniform rectangular lamina \(ABCD\), where \(AB = a\) and \(BC = 2a\), has mass \(2m\). The lamina is free to rotate about its edge \(AB\), which is fixed and vertical. The lamina is at rest when it is struck at \(C\) by a particle \(P\) of mass \(m\). The particle \(P\) is moving horizontally with speed \(U\) in a direction which is perpendicular to the lamina. The coefficient of restitution between \(P\) and the lamina is 0.5

Find the angular speed of the lamina immediately after the impact. (8)

Mark scheme

Scheme	Marks
\(I_{AB} = \tfrac{4}{3}(2m)a^2 = \tfrac{8}{3}ma^2\)	B1
CAM: \(\quad mU2a = \tfrac{8}{3}ma^2\omega + mv2a\)	M1 A2
NIL: \(\quad \tfrac{1}{2}U = 2a\omega - v\)	M1 A1
\(\omega = \dfrac{9U}{20a}\)	M1 A1
	(8 marks)