CAIE M2 (Mechanics 2) 2012 November

2
\includegraphics[max width=\textwidth, alt={}, center]{2c6b2e42-09cb-4653-9378-6c6add7771cc-2_463_567_479_790} A uniform rod \(A B\) 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 \(A B\) inclined at \(60 ^ { \circ }\) to the horizontal. Equilibrium is maintained by a force, in the vertical plane containing \(A B\), acting at \(A\) at an angle of \(45 ^ { \circ }\) to \(A B\) (see diagram). Calculate

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

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

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

4
\includegraphics[max width=\textwidth, alt={}, center]{2c6b2e42-09cb-4653-9378-6c6add7771cc-2_538_885_1809_628} 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 \(45 ^ { \circ }\). 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 \(P\).

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

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

6
\includegraphics[max width=\textwidth, alt={}, center]{2c6b2e42-09cb-4653-9378-6c6add7771cc-3_582_862_577_644} A uniform lamina \(O A B C D\) consists of a semicircle \(B C D\) with centre \(O\) and radius 0.6 m and an isosceles triangle \(O A B\), joined along \(O B\) (see diagram). The triangle has area \(0.36 \mathrm {~m} ^ { 2 }\) and \(A B = A O\).

1. Show that the centre of mass of the lamina lies on \(O B\).
2. Calculate the distance of the centre of mass of the lamina from \(O\).

7 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 \(A B = 4 \mathrm {~m}\). 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.)
2. Show that the greatest speed of \(P\) occurs when it is 2.1 m below \(A\), and calculate this greatest speed.
3. Calculate the greatest magnitude of the acceleration of \(P\).