CAIE M2 (Mechanics 2) 2012 November

\(ABC\) is a uniform semicircular arc with diameter \(AC = 0.5\) m. The arc rotates about a fixed axis through \(A\) and \(C\) with angular speed \(2.4\) rad s\(^{-1}\). Calculate the speed of the centre of mass of the arc. [3]

\includegraphics{figure_2} A uniform rod \(AB\) has weight \(6\) N and length \(0.8\) m. The rod rests in limiting equilibrium with \(B\) in contact with a rough horizontal surface and \(AB\) inclined at \(60°\) to the horizontal. Equilibrium is maintained by a force, in the vertical plane containing \(AB\), acting at \(A\) at an angle of \(45°\) to \(AB\) (see diagram). Calculate

1. the magnitude of the force applied at \(A\), [3]
2. the least possible value of the coefficient of friction at \(B\). [4]

A particle \(P\) of mass \(0.2\) kg is released from rest and falls vertically. At time \(t\) s after release \(P\) has speed \(v\) m s\(^{-1}\). A resisting force of magnitude \(0.8v\) N acts on \(P\).

1. Show that the acceleration of \(P\) is \((10 - 4v)\) m s\(^{-2}\). [2]
2. Find the value of \(v\) when \(t = 0.6\). [5]

\includegraphics{figure_4} A particle \(P\) is moving inside a smooth hollow cone which has its vertex downwards and its axis vertical, and whose semi-vertical angle is \(45°\). A light inextensible string parallel to the surface of the cone connects \(P\) to the vertex. \(P\) moves with constant angular speed in a horizontal circle of radius \(0.67\) m (see diagram). The tension in the string is equal to the weight of \(P\). Calculate the angular speed of \(P\). [6]

A particle \(P\) is projected with speed \(30\) m s\(^{-1}\) at an angle of \(60°\) above the horizontal from a point \(O\) on horizontal ground. For the instant when the speed of \(P\) is \(17\) m s\(^{-1}\) and increasing,

1. show that the vertical component of the velocity of \(P\) is \(8\) m s\(^{-1}\) downwards, [2]
2. calculate the distance of \(P\) from \(O\). [5]

\includegraphics{figure_6} A uniform lamina \(OABCD\) consists of a semicircle \(BCD\) with centre \(O\) and radius \(0.6\) m and an isosceles triangle \(OAB\), joined along \(OB\) (see diagram). The triangle has area \(0.36\) m\(^2\) and \(AB = AO\).

1. Show that the centre of mass of the lamina lies on \(OB\). [4]
2. Calculate the distance of the centre of mass of the lamina from \(O\). [4]

A light elastic string has natural length \(3\) m and modulus of elasticity \(45\) N. A particle \(P\) of weight \(6\) N is attached to the mid-point of the string. The ends of the string are attached to fixed points \(A\) and \(B\) which lie in the same vertical line with \(A\) above \(B\) and \(AB = 4\) m. The particle \(P\) is released from rest at the point \(1.5\) m vertically below \(A\).

1. Calculate the distance \(P\) moves after its release before first coming to instantaneous rest at a point vertically above \(B\). (You may assume that at this point the part of the string joining \(P\) to \(B\) is slack.) [4]
2. Show that the greatest speed of \(P\) occurs when it is \(2.1\) m below \(A\), and calculate this greatest speed. [5]
3. Calculate the greatest magnitude of the acceleration of \(P\). [3]