Find amplitude from speed conditions

4. A particle moves with simple harmonic motion along a straight line. When the particle is 3 cm from its centre of motion it has a speed of \(8 \mathrm {~cm} \mathrm {~s} ^ { - 1 }\) and an acceleration of magnitude \(12 \mathrm {~cm} \mathrm {~s} ^ { - 2 }\).

1. Show that the period of the motion is \(\pi\) seconds.
2. Find the amplitude of the motion.
3. Hence, find the greatest speed of the particle.

6. A particle \(P\) of mass 2.5 kg is moving with simple harmonic motion in a straight line between two points \(A\) and \(B\) on a smooth horizontal table. When \(P\) is 3 m from \(O\), the centre of the oscillations, its speed is \(6 \mathrm {~ms} ^ { - 1 }\). When \(P\) is 2.25 m from \(O\), its speed is \(8 \mathrm {~ms} ^ { - 1 }\).

1. Show that \(A B = 7.5 \mathrm {~m}\).
2. Find the period of the motion.
3. Find the kinetic energy of \(P\) when it is 2.7 m from \(A\).
4. Show that the time taken by \(P\) to travel directly from \(A\) to the midpoint of \(O B\) is \(\frac { \pi } { 4 }\) seconds.

1. A particle \(P\) moves in a straight line with simple harmonic motion about a fixed point \(O\). The magnitude of the greatest acceleration of \(P\) is \(18 \mathrm {~m} \mathrm {~s} ^ { - 2 }\)

When \(P\) is 0.3 m from \(O\), the speed of \(P\) is \(2.4 \mathrm {~ms} ^ { - 1 }\) The amplitude of the motion is \(a\) metres.

1. Show that \(a = 0.5\)
2. Find the greatest speed of \(P\). During one oscillation, the speed of \(P\) is at least \(2 \mathrm {~ms} ^ { - 1 }\) for \(S\) seconds.
3. Find the value of \(S\).

8 A particle, \(P\), is moving in a straight line with simple harmonic motion about a centre \(O\). When \(P\) is at the point \(A , 2 \mathrm {~m}\) from \(O\), it has speed \(4 \sqrt { 5 } \mathrm {~m} \mathrm {~s} ^ { - 1 }\). When \(P\) is at the point \(B , \sqrt { 5 } \mathrm {~m}\) from \(O\), it has speed \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Find the amplitude and period of the motion.
2. Given that \(A\) and \(B\) are on opposite sides of \(O\), find the time taken for \(P\) to travel directly from \(A\) to \(B\).

The point \(O\) is on the fixed horizontal line \(l\). Points \(A\) and \(B\) on \(l\) are such that \(OA = 0.1\) m and \(OB = 0.5\) m, with \(B\) between \(O\) and \(A\). A particle \(P\) oscillates on \(l\) in simple harmonic motion with centre \(O\). The kinetic energy of \(P\) when it is at \(A\) is twice its kinetic energy when it is at \(B\). Find the amplitude of the motion. [3]

A particle \(P\) moves with simple harmonic motion in a straight line. The centre of oscillation is \(O\). When \(P\) is at a distance 1 m from \(O\), its speed is 8 ms\(^{-1}\). When it is at a distance 2 m from \(O\), its speed is 4 ms\(^{-1}\).

1. Find the amplitude of the motion. [4 marks]
2. Show that the period of motion is \(\frac{\pi}{2}\) s. [3 marks]