CAIE M2 (Mechanics 2) 2010 November

1
\(A B C D\) is a uniform lamina with \(A B = 1.8 \mathrm {~m} , A D = D C = 0.9 \mathrm {~m}\), and \(A D\) perpendicular to \(A B\) and \(D C\) (see diagram).

1. Find the distance of the centre of mass of the lamina from \(A B\) and the distance from \(A D\). The lamina is freely suspended at \(A\) and hangs in equilibrium.
2. Calculate the angle between \(A B\) and the vertical.

2 A particle \(P\) is projected with speed \(26 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(30 ^ { \circ }\) above the horizontal from a point \(O\) on a horizontal plane.

1. For the instant when the vertical component of the velocity of \(P\) is \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) downwards, find the direction of motion of \(P\) and the height of \(P\) above the plane.
2. \(P\) strikes the plane at the point \(A\). Calculate the time taken by \(P\) to travel from \(O\) to \(A\) and the distance \(O A\).
   \includegraphics[max width=\textwidth, alt={}, center]{022776e2-80bb-4e73-a309-8cd972ae7377-2_679_455_1544_845} Particles \(P\) and \(Q\) have masses 0.8 kg and 0.4 kg respectively. \(P\) is attached to a fixed point \(A\) by a light inextensible string which is inclined at an angle \(\alpha ^ { \circ }\) to the vertical. \(Q\) is attached to a fixed point \(B\), which is vertically below \(A\), by a light inextensible string of length 0.3 m . The string \(B Q\) is horizontal. \(P\) and \(Q\) are joined to each other by a light inextensible string which is vertical. The particles rotate in horizontal circles of radius 0.3 m about the axis through \(A\) and \(B\) with constant angular speed \(5 \mathrm { rad } \mathrm { s } ^ { - 1 }\) (see diagram).
3. By considering the motion of \(Q\), find the tensions in the strings \(P Q\) and \(B Q\).
4. Find the tension in the string \(A P\) and the value of \(\alpha\).

4
\includegraphics[max width=\textwidth, alt={}, center]{022776e2-80bb-4e73-a309-8cd972ae7377-3_371_570_258_790} A uniform \(\operatorname { rod } A B\) has weight 15 N and length 1.2 m . The end \(A\) of the rod is in contact with a rough plane inclined at \(30 ^ { \circ }\) to the horizontal, and the rod is perpendicular to the plane. The rod is held in equilibrium in this position by means of a horizontal force applied at \(B\), acting in the vertical plane containing the rod (see diagram).

1. Show that the magnitude of the force applied at \(B\) is 4.33 N , correct to 3 significant figures.
2. Find the magnitude of the frictional force exerted by the plane on the rod.
3. Given that the rod is in limiting equilibrium, calculate the coefficient of friction between the rod and the plane.

5
\includegraphics[max width=\textwidth, alt={}, center]{022776e2-80bb-4e73-a309-8cd972ae7377-3_287_1068_1306_536} A light elastic string has natural length 2 m and modulus of elasticity \(\lambda \mathrm { N }\). The ends of the string are attached to fixed points \(A\) and \(B\) which are at the same horizontal level and 2.4 m apart. A particle \(P\) of mass 0.6 kg is attached to the mid-point of the string and hangs in equilibrium at a point 0.5 m below \(A B\) (see diagram).

1. Show that \(\lambda = 26\).
   \(P\) is projected vertically downwards from the equilibrium position, and comes to instantaneous rest at a point 0.9 m below \(A B\).
2. Calculate the speed of projection of \(P\).

6
\includegraphics[max width=\textwidth, alt={}, center]{022776e2-80bb-4e73-a309-8cd972ae7377-4_341_572_258_790} A particle \(P\) of mass 0.2 kg is projected with velocity \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) upwards along a line of greatest slope on a plane inclined at \(30 ^ { \circ }\) to the horizontal (see diagram). Air resistance of magnitude \(0.5 v \mathrm {~N}\) opposes the motion of \(P\), where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the velocity of \(P\) at time \(t \mathrm {~s}\) after projection. The coefficient of friction between \(P\) and the plane is \(\frac { 1 } { 2 \sqrt { } 3 }\). The particle \(P\) reaches a position of instantaneous rest when \(t = T\).

1. Show that, while \(P\) is moving up the plane, \(\frac { \mathrm { d } v } { \mathrm {~d} t } = - 2.5 ( 3 + v )\).
2. Calculate \(T\).
3. Calculate the speed of \(P\) when \(t = 2 T\). \footnotetext{Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable effort has been made by the publisher (UCLES) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will be pleased to make amends at the earliest possible opportunity. University of Cambridge International Examinations is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge. }