Find amplitude of SHM

1. A particle \(P\) moves in a straight line with simple harmonic motion. The period of the motion is \(\frac { \pi } { 4 }\) seconds. At time \(t = 0 , P\) is at rest at the point \(A\) and the acceleration of \(P\) has magnitude \(20 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).

Find

1. the amplitude of the motion,
2. the greatest speed of \(P\) during the motion,
3. the time \(P\) takes to travel a total distance of 1.5 m after it has first left \(A\).

1. A particle \(P\) is moving in a straight line with simple harmonic motion of period 4 s . The centre of the motion is the point \(O\)

At time \(t = 0 , P\) passes through \(O\) At time \(t = 0.5 \mathrm {~s} , P\) is moving with speed \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\)

1. Show that the amplitude of the motion is \(\frac { 4 \sqrt { 2 } } { \pi } \mathrm {~m}\)
2. Find the maximum speed of \(P\)

2 A particle \(P\) is moving in simple harmonic motion with centre \(O\). When \(P\) is 5 m from \(O\) its speed is \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\), and when it is 9 m from \(O\) its speed is \(\frac { 3 } { 5 } V \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Show that the amplitude of the motion is \(\frac { 15 } { 2 } \sqrt { } 2 \mathrm {~m}\). Given that the greatest speed of \(P\) is \(3 \sqrt { } 2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), find \(V\).

7 A small, hollow, plastic ball, of mass \(m \mathrm {~kg}\) is at rest at a point \(O\) on a polished horizontal surface. The ball is attached to two identical springs. The other ends of the springs are attached to the points \(P\) and \(Q\) which are 1.8 metres apart on a straight line through \(O\). The ball is struck so that it moves away from \(O\), towards \(P\) with a speed of \(0.75 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). As the ball moves, its displacement from \(O\) is \(x\) metres at time \(t\) seconds after the motion starts. The force that each of the springs applies to the ball is \(12.5 m x\) newtons towards \(O\). The ball is to be modelled as a particle. The surface is assumed to be smooth and it is assumed that the forces applied to the ball by the springs are the only horizontal forces acting on the ball. 7

1. Find the minimum distance of the ball from \(P\), in the subsequent motion. 7
2. In practice the minimum distance predicted by the model is incorrect.
   Is the minimum distance predicted by the model likely to be too big or too small?
   Explain your answer with reference to the model.
   [0pt] [2 marks]

8 A particle P of mass \(3 m \mathrm {~kg}\) is attached to one end of a light elastic string of modulus of elasticity \(4 m g \mathrm {~N}\) and natural length 0.4 m . The other end of the string is attached to a fixed point O . The particle P rests in equilibrium at a point A with the string vertical.

1. Find the distance OA . At time \(t = 0\) seconds, P is given a speed of \(2.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) vertically downwards from A .
2. Show that P initially performs simple harmonic motion with amplitude \(a \mathrm {~m}\), where \(a\) is to be determined correct to \(\mathbf { 3 }\) significant figures.
3. Determine the smallest distance between P and O in the subsequent motion.