Speed at given displacement

2. A particle \(P\) is moving in a straight line with simple harmonic motion. The centre of the oscillation is the fixed point \(C\), the amplitude of the oscillation is 0.5 m and the time to complete one oscillation is \(\frac { 2 \pi } { 3 }\) seconds. The point \(A\) is on the path of \(P\) and 0.2 m from \(C\). Find

1. the magnitude and direction of the acceleration of \(P\) when it passes through \(A\),
2. the speed of \(P\) when it passes through \(A\),
3. the time \(P\) takes to move directly from \(C\) to \(A\).

4 A particle \(P\) of mass \(m\) is attached to one end of a light elastic string of natural length 4a. The other end of the string is attached to a fixed point \(O\). The particle rests in equilibrium at the point \(E\), vertically below \(O\), where \(O E = 5 a\). The particle is pulled down a vertical distance \(\frac { 1 } { 2 } a\) from \(E\) and released from rest. Show that the motion of \(P\) is simple harmonic and state the period of the motion. Find the two possible values of the distance \(O P\) when the speed of \(P\) is equal to one half of its maximum speed.

\(A\) and \(B\) are two fixed points on a smooth horizontal surface, with \(AB = 3a\) m. One end of a light elastic string, of natural length \(a\) m and modulus of elasticity \(mg\) N, is attached to the point \(A\). The other end of this string is attached to a particle \(P\) of mass \(m\) kg. One end of a second light elastic string, of natural length \(ka\) m and modulus of elasticity \(2mg\) N, is attached to \(B\). The other end of this string is attached to \(P\). Given that the system is in equilibrium when \(P\) is at \(M\), the mid-point of \(AB\), find the value of \(k\). [3] The particle \(P\) is released from rest at a point between \(A\) and \(B\) where both strings are taut. Show that \(P\) performs simple harmonic motion and state the period of the motion. [5] In the case where \(P\) is released from rest at a distance \(0.2a\) m from \(M\), the speed of \(P\) is \(0.7\) m s\(^{-1}\) when \(P\) is \(0.05a\) m from \(M\). Find the value of \(a\). [3]

A particle \(P\) oscillates in simple harmonic motion between the points \(A\) and \(B\), where \(AB = 6\) m. The period of the motion is \(4\pi\) s. Find the speed of \(P\) when it is 2 m from \(B\). [3]

Answer only one of the following two alternatives. EITHER The points \(A\) and \(B\) are a distance 1.2 m apart on a smooth horizontal surface. A particle \(P\) of mass \(\frac{2}{3}\) kg is attached to one end of a light spring of natural length 0.6 m and modulus of elasticity 10 N. The other end of the spring is attached to the point \(A\). A second light spring, of natural length 0.4 m and modulus of elasticity 20 N, has one end attached to \(P\) and the other end attached to \(B\).

1. Show that when \(P\) is in equilibrium \(AP = 0.75\) m. [3]

The particle \(P\) is displaced by 0.05 m from the equilibrium position towards \(A\) and then released from rest.

1. Show that \(P\) performs simple harmonic motion and state the period of the motion. [6]
2. Find the speed of \(P\) when it passes through the equilibrium position. [2]
3. Find the speed of \(P\) when its acceleration is equal to half of its maximum value. [3]

OR The number of puncture repairs carried out each week by a small repair shop is recorded over a period of 40 weeks. The results are shown in the following table.

Number of repairs in a week	0	1	2	3	4	5	\(\geqslant 6\)
Number of weeks	6	15	9	6	3	1	0

1. Calculate the mean and variance for the number of repairs in a week and comment on the possible suitability of a Poisson distribution to model the data. [3]

Records over a longer period of time indicate that the mean number of repairs in a week is 1.6. The following table shows some of the expected frequencies, correct to 3 decimal places, for a period of 40 weeks using a Poisson distribution with mean 1.6.

Number of repairs in a week	0	1	2	3	4	5	\(\geqslant 6\)
Expected frequency	8.076	12.921	10.337	5.513	2.205	\(a\)	\(b\)

1. Show that \(a = 0.706\) and find the value of the constant \(b\). [3]
2. Carry out a goodness of fit test of a Poisson distribution with mean 1.6, using a 10% significance level. [8]