CAIE M2 (Mechanics 2) 2011 June

1
\includegraphics[max width=\textwidth, alt={}, center]{1d2e8f3a-dab6-4306-bc4a-d47805947cd2-2_518_609_255_769} A uniform \(\operatorname { rod } A B\) of weight 16 N is freely hinged at \(A\) to a fixed point. A force of magnitude 4 N acting perpendicular to the rod is applied at \(B\) (see diagram). Given that the rod is in equilibrium,

1. calculate the angle the rod makes with the horizontal,
2. find the magnitude and direction of the force exerted on the rod at \(A\).

2 A uniform lamina \(A B C D\) consists of a semicircle \(B C D\) with centre \(O\) and diameter 0.4 m , and an isosceles triangle \(A B D\) with base \(B D = 0.4 \mathrm {~m}\) and perpendicular height \(h \mathrm {~m}\). The centre of mass of the lamina is at \(O\).

1. Find the value of \(h\).
2. 
   \includegraphics[max width=\textwidth, alt={}, center]{1d2e8f3a-dab6-4306-bc4a-d47805947cd2-2_680_627_1466_797} The lamina is suspended from a vertical string attached to a point \(X\) on the side \(A D\) of the triangle (see diagram). Given the lamina is in equilibrium with \(A D\) horizontal, calculate \(X D\).

3
\includegraphics[max width=\textwidth, alt={}, center]{1d2e8f3a-dab6-4306-bc4a-d47805947cd2-3_385_1154_253_497} A particle \(P\) of mass 0.5 kg is attached to the vertex \(V\) of a fixed solid cone by a light inextensible string. \(P\) lies on the smooth curved surface of the cone and moves in a horizontal circle of radius 0.1 m with centre on the axis of the cone. The cone has semi-vertical angle \(60 ^ { \circ }\) (see diagram).

1. Calculate the speed of \(P\), given that the tension in the string and the contact force between the cone and \(P\) have the same magnitude.
2. Calculate the greatest angular speed at which \(P\) can move on the surface of the cone.

4 One end of a light elastic string of natural length 0.5 m and modulus of elasticity 12 N is attached to a fixed point \(O\). The other end of the string is attached to a particle \(P\) of mass \(0.24 \mathrm {~kg} . P\) is projected vertically upwards with speed \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a position 0.8 m vertically below \(O\).

1. Calculate the speed of the particle when it is moving upwards with zero acceleration.
2. Show that the particle moves 0.6 m while it is moving upwards with constant acceleration.

5 A particle \(P\) of mass 0.4 kg moves in a straight line on a horizontal surface and has velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at time \(t \mathrm {~s}\). A horizontal force of magnitude \(k \sqrt { } v \mathrm {~N}\) opposes the motion of \(P\). When \(t = 0 , v = 9\) and when \(t = 2 , v = 4\).

1. Express \(\frac { \mathrm { d } v } { \mathrm {~d} t }\) in terms of \(k\) and \(v\), and hence show that \(v = \frac { 1 } { 4 } ( t - 6 ) ^ { 2 }\).
2. Find the distance travelled by \(P\) in the first 3 seconds of its motion.

6 A particle \(P\) is projected with speed \(26 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(30 ^ { \circ }\) below the horizontal, from a point \(O\) which is 80 m above horizontal ground.

1. Calculate the distance from \(O\) of the particle 2.3 s after projection.
2. Find the horizontal distance travelled by \(P\) before it reaches the ground.
3. Calculate the speed and direction of motion of \(P\) immediately before it reaches the ground.