Find period from given information

3. A particle \(P\) is moving in a straight line with simple harmonic motion between two points \(A\) and \(B\), where \(A B\) is \(2 a\) metres. The point \(C\) lies on the line \(A B\) and \(A C = \frac { 1 } { 2 } a\) metres. The particle passes through \(C\) with speed \(\frac { 3 a \sqrt { 3 } } { 2 } \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Find the period of the motion. The maximum magnitude of the acceleration of \(P\) is \(45 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). Find
2. the value of \(a\),
3. the maximum speed of \(P\). The point \(D\) lies on \(A B\) and \(P\) takes a quarter of one period to travel directly from \(C\) to \(D\).
4. Find the distance CD.

2. A particle \(P\) is moving in a straight line with simple harmonic motion about the fixed point \(O\) as centre. When \(P\) is a distance 0.02 m from \(O\), the speed of \(P\) is \(0.3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the magnitude of the acceleration of \(P\) is \(0.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\)

1. Find the period of the motion. The amplitude of the motion is \(a\) metres. Find
2. the value of \(a\),
3. the total length of time during each complete oscillation for which \(P\) is within \(\frac { 1 } { 2 } a\) metres of \(O\). metres of \(O\).

4. A piston \(P\) in a machine moves in a straight line with simple harmonic motion about a point \(O\), which is the centre of the oscillations. The period of the oscillations is \(\pi \mathrm { s }\). When \(P\) is 0.5 m from \(O\), its speed is \(2.4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find

1. the amplitude of the motion,
2. the maximum speed of \(P\) during the motion,
3. the maximum magnitude of the acceleration of \(P\) during the motion,
4. the total time, in s to 2 decimal places, in each complete oscillation for which the speed of \(P\) is greater than \(2.4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

2. A particle \(P\) moves in a straight line with simple harmonic motion of period 2.4 s about a fixed origin \(O\). At time \(t\) seconds the speed of \(P\) is \(v \mathrm {~ms} ^ { - 1 }\). When \(t = 0 , P\) is at \(O\). When \(t = 0.4 , v = 4\). Find

1. the greatest speed of \(P\),
2. the magnitude of the greatest acceleration of \(P\).

5. A particle \(P\) is moving in a straight line with simple harmonic motion on a smooth horizontal floor. The particle comes to instantaneous rest at points \(A\) and \(B\) where \(A B\) is 0.5 m . The mid-point of \(A B\) is \(O\). The mid-point of \(O A\) is \(C\). The mid-point of \(O B\) is \(D\). The particle takes 0.2 s to travel directly from \(C\) to \(D\). At time \(t = 0 , P\) is moving through \(O\) towards \(A\).

1. Show that the period of the motion is \(\frac { 6 } { 5 } \mathrm {~s}\).
2. Find the distance of \(P\) from \(B\) when \(t = 2 \mathrm {~s}\).
3. Find the maximum magnitude of the acceleration of \(P\).
4. Find the maximum speed of \(P\).

1 A particle oscillates in simple harmonic motion with centre \(O\). When its distance from \(O\) is 3 m its speed is \(16 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), and when its distance from \(O\) is 4 m its speed is \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find the period and amplitude of the motion.

3 A particle moves on a straight line \(A O B\) in simple harmonic motion, where \(A B = 2 a \mathrm {~m}\). The centre of the motion is \(O\) and the particle is instantaneously at rest at \(A\) and \(B\). The point \(M\) is the mid-point of \(O B\). The particle passes through \(M\) moving towards \(O\) and next achieves its maximum speed one second later. Find the period of the motion. Find the distance of the particle from \(O\) when its speed is equal to one half of its maximum speed. At an instant 2.5 seconds after the particle passes through \(M\) moving towards \(O\), the distance of the particle from \(O\) is \(\sqrt { } 2 \mathrm {~m}\). Find, in metres, the amplitude of the motion.

3 A particle \(P\) is performing simple harmonic motion with amplitude 0.25 m . During each complete oscillation, \(P\) moves with a speed that is less than or equal to half of its maximum speed for \(\frac { 4 } { 3 }\) seconds. Find the period of the motion and the maximum speed of \(P\).

2 The piston in a large engine rises and falls in simple harmonic motion. When the piston is 1.6 m below its highest level, the rate of change of its height is \(\frac { 3 } { 5 } \pi\) metres per second. When the piston is 0.2 m below its highest level, the rate of change of its height is \(\frac { 1 } { 4 } \pi\) metres per second. Find the amplitude and period of the motion.

1

1. A particle P is executing simple harmonic motion, and the centre of the oscillations is at the point O . The maximum speed of P during the motion is \(5.1 \mathrm {~ms} ^ { - 1 }\). When P is 6 m from O , its speed is \(4.5 \mathrm {~ms} ^ { - 1 }\). Find the period and the amplitude of the motion.
2. The force \(F\) of gravitational attraction between two objects of masses \(m _ { 1 }\) and \(m _ { 2 }\) at a distance \(d\) apart is given by \(F = \frac { G m _ { 1 } m _ { 2 } } { d ^ { 2 } }\), where \(G\) is the universal gravitational constant.
   1. Find the dimensions of \(G\). Three objects, each of mass \(m\), are moving in deep space under mutual gravitational attraction. They move round a single circle with constant angular speed \(\omega\), and are always at the three vertices of an equilateral triangle of side \(R\). You are given that \(\omega = k G ^ { \alpha } m ^ { \beta } R ^ { \gamma }\), where \(k\) is a dimensionless constant.
   2. Find \(\alpha , \beta\) and \(\gamma\). For three objects of mass 2500 kg at the vertices of an equilateral triangle of side 50 m , the angular speed is \(2.0 \times 10 ^ { - 6 } \mathrm { rad } \mathrm { s } ^ { - 1 }\).
   3. Find the angular speed for three objects of mass \(4.86 \times 10 ^ { 14 } \mathrm {~kg}\) at the vertices of an equilateral triangle of side 30000 m .

A particle \(P\) is describing simple harmonic motion of amplitude 5 m. Its speed is 6 m s\(^{-1}\) when it is 3 m from the centre of the motion. Find, in terms of \(\pi\), the period of the motion. [2] Find also

1. the maximum speed of \(P\), [2]
2. the magnitude of the maximum acceleration of \(P\). [2]

An elastic string spring of modulus \(2mg\) and natural length \(l\) is fixed at one end. To the other end is attached a mass \(m\) which is allowed to hang in equilibrium. The mass is then pulled vertically downwards through a distance \(l\) and released from rest. The air resistance is modelled as having magnitude \(2m\omega v\), where \(v\) is the speed of the particle and \(\omega = \sqrt{\frac{g}{l}}\). The particle is at distance \(x\) from its equilibrium position at time \(t\).

1. Show that \(\frac{\mathrm{d}^2 x}{\mathrm{d} t^2} + 2\omega \frac{\mathrm{d} x}{\mathrm{d} t} + 2\omega^2 x = 0\). [7]
2. Find the general solution of this differential equation. [4]
3. Hence find the period of the damped harmonic motion. [1]

The displacement of a particle from its equilibrium position is \(x\) metres at time \(t\) seconds. The motion of the particle obeys the differential equation $$\frac{d^2x}{dt^2} = -9x$$ Calculate the period of its motion in seconds. Circle your answer. [1 mark] \(\frac{\pi}{9}\) \(\quad\) \(\frac{2\pi}{9}\) \(\quad\) \(\frac{\pi}{3}\) \(\quad\) \(\frac{2\pi}{3}\)