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Boat \(A\) is travelling with constant speed \(7.9 \mathrm {~ms} ^ { - 1 }\) on a course with bearing \(035 ^ { \circ }\). Boat \(B\) is travelling with constant speed \(10.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) on a course with bearing \(330 ^ { \circ }\). At one instant, the boats are 1500 m apart with \(B\) on a bearing of \(125 ^ { \circ }\) from \(A\) (see diagram).
- Find the magnitude and the bearing of the velocity of \(B\) relative to \(A\).
- Find the shortest distance between \(A\) and \(B\) in the subsequent motion.
- Find the time taken from the instant when \(A\) and \(B\) are 1500 m apart to the instant when \(A\) and \(B\) are at the point of closest approach.
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Two uniform rods \(A B\) and \(B C\), each of length \(a\) and mass \(m\), are rigidly joined together so that \(A B\) is perpendicular to \(B C\). The rod \(A B\) is freely hinged to a fixed point at \(A\). The rods can rotate in a vertical plane about a smooth fixed horizontal axis through \(A\). One end of a light elastic string of natural length \(a\) and modulus of elasticity \(\lambda m g\) is attached to \(B\). The other end of the string is attached to a fixed point \(D\) vertically above \(A\), where \(A D = a\). The string \(B D\) makes an angle \(\theta\) radians with the downward vertical (see diagram). - Taking \(D\) as the reference level for gravitational potential energy, show that the total potential energy \(V\) of the system is given by
$$\mathrm { V } = \frac { 1 } { 2 } \mathrm { mga } ( \sin 2 \theta - 3 \cos 2 \theta ) + \frac { 1 } { 2 } \lambda \mathrm { mga } ( 2 \cos \theta - 1 ) ^ { 2 } - 2 \mathrm { mga } .$$
- Given that \(\theta = \frac { 1 } { 4 } \pi\) is a position of equilibrium, find the exact value of \(\lambda\).
- Find \(\frac { d ^ { 2 } V } { d \theta ^ { 2 } }\) and hence determine whether the position of equilibrium at \(\theta = \frac { 1 } { 4 } \pi\) is stable or unstable.