6.02h Elastic PE: 1/2 k x^2

8 A bungee jumper, of mass 49 kg , is attached to one end of a light elastic cord of natural length 22 metres and modulus of elasticity 1078 newtons. The other end of the cord is attached to a horizontal platform, which is at a height of 60 metres above the ground. The bungee jumper steps off the platform at the point where the cord is attached, and falls vertically. The bungee jumper can be modelled as a particle. Assume that Hooke's Law applies whilst the cord is taut and that air resistance is negligible throughout the motion. When the bungee jumper has fallen \(x\) metres, his speed is \(v \mathrm {~ms} ^ { - 1 }\).

1. By considering energy, show that, when \(x\) is greater than 22, $$5 v ^ { 2 } = 318 x - 5 x ^ { 2 } - 2420$$
2. Explain why \(x\) must be greater than 22 for the equation in part (a) to be valid. ( 1 mark)
3. Find the maximum value of \(x\).
4. 1. Show that the speed of the bungee jumper is a maximum when \(x = 31.8\).
   2. Hence find the maximum speed of the bungee jumper.

8

1. Hooke's law states that the tension in a stretched string of natural length \(l\) and modulus of elasticity \(\lambda\) is \(\frac { \lambda x } { l }\) when its extension is \(x\). Using this formula, prove that the work done in stretching a string from an unstretched position to a position in which its extension is \(e\) is \(\frac { \lambda e ^ { 2 } } { 2 l }\).
   (3 marks)
2. A particle, of mass 5 kg , is attached to one end of a light elastic string of natural length 0.6 metres and modulus of elasticity 150 N . The other end of the string is fixed to a point \(O\).
   1. Find the extension of the elastic string when the particle hangs in equilibrium directly below \(O\).
   2. The particle is pulled down and held at the point \(P\), which is 0.9 metres vertically below \(O\). Show that the elastic potential energy of the string when the particle is in this position is 11.25 J .
   3. The particle is released from rest at the point \(P\). In the subsequent motion, the particle has speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when it is \(x\) metres above \(\boldsymbol { P }\). Show that, while the string is taut, $$v ^ { 2 } = 10.4 x - 50 x ^ { 2 }$$
   4. Find the value of \(x\) when the particle comes to rest for the first time after being released, given that the string is still taut.

6 A block, of mass 5 kg , is attached to one end of a length of elastic string. The other end of the string is fixed to a vertical wall. The block is placed on a horizontal surface. The elastic string has natural length 1.2 m and modulus of elasticity 180 N . The block is pulled so that it is 2 m from the wall and is then released from rest. Whilst taut, the string remains horizontal. It may be assumed that, after the string becomes slack, it does not interfere with the movement of the block. \includegraphics[max width=\textwidth, alt={}, center]{9cfa110c-ee11-447a-b21a-3f436432e27d-5_396_960_660_534}

1. Calculate the elastic potential energy when the block is 2 m from the wall.
2. If the horizontal surface is smooth, find the speed of the block when it hits the wall.
3. The surface is in fact rough and the coefficient of friction between the block and the surface is \(\mu\). Find \(\mu\) if the block comes to rest just as it reaches the wall.

6. A light elastic spring \(A B\) has natural length \(2 a\) and modulus of elasticity \(2 m n ^ { 2 } a\), where \(n\) is a constant. A particle \(P\) of mass \(m\) is attached to the end \(A\) of the spring. At time \(t = 0\), the spring, with \(P\) attached, lies at rest and unstretched on a smooth horizontal plane. The other end \(B\) of the spring is then pulled along the plane in the direction \(A B\) with constant acceleration \(f\). At time \(t\) the extension of the spring is \(x\).

1. Show that $$\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + n ^ { 2 } x = f .$$
2. Find \(x\) in terms of \(n , f\) and \(t\). Hence find
3. the maximum extension of the spring,
4. the speed of \(P\) when the spring first reaches its maximum extension.
   1. \hspace{0pt} [In this question \(\mathbf { i }\) and \(\mathbf { j }\) are unit vectors due east and due north respectively]
   A man cycles at a constant speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) on level ground and finds that when his velocity is \(u \mathbf { j } \mathrm {~m} \mathrm {~s} ^ { - 1 }\) the velocity of the wind appears to be \(v ( 3 \mathbf { i } - 4 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\), where \(v\) is a positive constant. When the man cycles with velocity \(\frac { 1 } { 5 } u ( - 3 \mathbf { i } + 4 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\), the velocity of the wind appears to be \(w \mathbf { i } \mathrm {~m} \mathrm {~s} ^ { - 1 }\), where \(w\) is a positive constant. Find, in terms of \(u\), the true velocity of the wind.
   

7. \begin{figure}[h] \end{figure} Figure 3 shows a framework \(A B C\), consisting of two uniform rods rigidly joined together at \(B\) so that \(\angle A B C = 90 ^ { \circ }\). The \(\operatorname { rod } A B\) has length \(2 a\) and mass \(4 m\), and the \(\operatorname { rod } B C\) has length \(a\) and mass \(2 m\). The framework is smoothly hinged at \(A\) to a fixed point, so that the framework can rotate in a fixed vertical plane. One end of a light elastic string, of natural length \(2 a\) and modulus of elasticity \(3 m g\), is attached to \(A\). The string passes through a small smooth ring \(R\) fixed at a distance \(2 a\) from \(A\), on the same horizontal level as \(A\) and in the same vertical plane as the framework. The other end of the string is attached to \(B\). The angle \(A R B\) is \(\theta\), where \(0 < \theta < \frac { \pi } { 2 }\).

1. Show that the potential energy \(V\) of the system is given by $$V = 8 a m g \sin 2 \theta + 5 a m g \cos 2 \theta + \text { constant }$$
2. Find the value of \(\theta\) for which the system is in equilibrium.
3. Determine the stability of this position of equilibrium. A smooth uniform sphere \(S\), of mass \(m\), is moving on a smooth horizontal plane when it collides obliquely with another smooth uniform sphere \(T\), of the same radius as \(S\) but of mass \(2 m\), which is at rest on the plane. Immediately before the collision the velocity of \(S\) makes an angle \(\alpha\), where \(\tan \alpha = \frac { 3 } { 4 }\), with the line joining the centres of the spheres. Immediately after the collision the speed of \(T\) is \(V\). The coefficient of restitution between the spheres is \(\frac { 3 } { 4 }\).

1. Find, in terms of \(V\), the speed of \(S\)
   1. immediately before the collision,
   2. immediately after the collision.
2. Find the angle through which the direction of motion of \(S\) is deflected as a result of the collision.
   

8 In this question use \(g = 9.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) Omar, a bungee jumper of mass 70 kg , has his ankles attached to one end of an elastic cord. The other end of the cord is attached to a bridge which is 80 metres above the surface of a river. Omar steps off the bridge at the point where the cord is attached and falls vertically downwards. The cord can be modelled as a light elastic string of natural length \(L\) metres and modulus of elasticity 2800 N Model Omar as a particle. 8

1. Given that Omar just reaches the surface of the river before being pulled back up, find the value of \(L\) Fully justify your answer.
   8
2. If Omar is not modelled as a particle, explain the effect of revising this assumption on your answer to part (a).

1 A spring has natural length 0.4 metres and modulus of elasticity 55 N
Calculate the elastic potential energy stored in the spring when the extension of the spring is 0.08 metres. Circle your answer. \(0.176 \mathrm {~J} \quad 0.44 \mathrm {~J} \quad 0.88 \mathrm {~J} \quad 1.76 \mathrm {~J}\)

3 A right circular cone \(C\) of height 4 m and base radius 3 m has its base fixed to a horizontal plane. One end of a light elastic string of natural length 2 m and modulus of elasticity 32 N is fixed to the vertex of \(C\). The other end of the string is attached to a particle \(P\) of mass 2.5 kg . \(P\) moves in a horizontal circle with constant speed and in contact with the smooth curved surface of \(C\). The extension of the string is 1.5 m .

1. Find the tension in the string.
2. Find the speed of \(P\).

1 A bungee jumper of mass 80 kg steps off a high bridge with an elastic rope attached to her ankles. She is assumed to fall vertically from rest and the air resistance she experiences is modelled as a constant force of 32 N . The rope has natural length 4 m and modulus of elasticity 470 N . By considering energy, determine the total distance she falls before first coming to instantaneous rest.

4 One end of a light elastic string of natural length \(l \mathrm {~m}\) and modulus of elasticity \(\lambda \mathrm { N }\) is attached to a particle \(A\) of mass \(m \mathrm {~kg}\). The other end of the string is attached to a fixed point \(O\) which is on a horizontal surface. The surface is modelled as being smooth and \(A\) moves in a circular path around \(O\) with constant speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The extension of the string is denoted by \(x \mathrm {~m}\).

1. Show that \(x\) satisfies \(\lambda x ^ { 2 } + \lambda l x - l m v ^ { 2 } = 0\).
2. By solving the equation in part (a) and using a binomial series, show that if \(\lambda\) is very large then \(\lambda x \approx m v ^ { 2 }\).
3. By considering the tension in the string, explain how the result obtained when \(\lambda\) is very large relates to the situation when the string is inextensible. The nature of the horizontal surface is such that the modelling assumption that it is smooth is justifiable provided that the speed of the particle does not exceed \(7 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). In the case where \(m = 0.16\) and \(\lambda = 260\), the extension of the string is measured as being 3.0 cm .
4. Estimate the value of \(v\).
5. Explain whether the value of \(v\) means that the modelling assumption is necessarily justifiable in this situation.

4 One end of a light elastic string of natural length 0.2 m and modulus of elasticity 100 N is attached to a fixed point \(A\). The other end is attached to a particle of mass 5 kg . The particle moves with angular speed \(\omega\) radians per second in a horizontal circle with the centre vertically below \(A\). The string makes an angle \(\theta\) with the vertical.

1. By considering the horizontal component of the tension in the string, show that the tension in the string is \(( 1 + 5 x ) \omega ^ { 2 } \mathrm {~N}\), where \(x\) is the extension, in metres, of the string.
2. (a) By considering vertical forces and also Hooke's law, deduce that \(\cos \theta = \frac { 1 } { 10 x }\).
   (b) Show that \(\omega > \frac { 10 \sqrt { 3 } } { 3 }\).
3. When the value of \(\omega\) is \(5 \sqrt { 2 }\), find the radius of the circular motion.

5 A particle of mass \(m\) is attached by a light elastic string of natural length \(l\) and modulus of elasticity \(\lambda\) to a fixed point \(A\), from which it is allowed to fall freely. The particle first comes to rest, instantaneously, at \(B\), where \(A B = 2 l\). Prove that

1. \(\lambda = 4 m g\),
2. while the string is taut, \(\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } = - \frac { 4 g } { l } x\), where \(x\) is the displacement from the equilibrium position at time \(t\),
3. the time taken between the first occasion when the string becomes taut and the next occasion when it becomes slack is $$\left[ \frac { 1 } { 2 } \pi + \sin ^ { - 1 } \left( \frac { 1 } { 3 } \right) \right] \sqrt { \frac { l } { g } }$$

9 A light string, of natural length 0.5 m and modulus of elasticity 4 N , has one end attached to the ceiling of a room. A particle of mass 0.2 kg is attached to the free end of the string and hangs in equilibrium.

1. Find the extension of the string when the particle is in the equilibrium position. The particle is pulled down a further 0.5 m from the equilibrium position and released from rest. At time \(t\) seconds the displacement of the particle from the equilibrium position is \(x \mathrm {~m}\).
2. Show that, while the string is taut, the equation of motion is \(\ddot { x } = - 40 x\).
3. Find the time taken for the string to become slack for the first time.
4. Show that the particle comes to instantaneous rest 0.125 m below the ceiling.

8 \includegraphics[max width=\textwidth, alt={}, center]{86cc07e7-ea69-4480-96c8-82b818445199-4_182_803_264_671} A light spring of modulus of elasticity 8 N and natural length 0.4 m has one end fixed to a smooth horizontal table at a fixed point \(L\). A particle of mass 0.2 kg is attached to the other end of the spring and pulled out horizontally to a point \(M\) on the table, so that the spring is extended by 0.2 m . The particle is then released from rest. The mid-point of \(L M\) is \(N\) and the point \(O\) is on \(L M\) such that \(L O = 0.4 \mathrm {~m}\) (see diagram).

1. Show that the particle moves in simple harmonic motion with centre \(O\) and state the exact period of its motion.
2. Find the exact time taken for the particle to move directly from \(M\) to \(N\).

14 One end of a light elastic string of natural length 0.5 m and modulus of elasticity 3 N is attached to a ceiling at a point \(P\). A particle of mass 0.3 kg is attached to the other end of the string.

1. Find the extension of the string when the particle hangs vertically in equilibrium. The particle is released from rest at \(P\) so that it falls vertically. Find
2. the maximum extension of the string,
3. the equation of motion for the particle when the string is stretched, in terms of the displacement \(x \mathrm {~m}\) below the equilibrium position,
4. the time between the string first becoming stretched and next becoming unstretched again.

13 Two light strings, each of natural length \(l\) and modulus of elasticity \(6 m g\), are attached at their ends to a particle \(P\) of mass \(m\). The other ends of the strings are attached to two fixed points \(A\) and \(B\), which are at a distance \(6 l\) apart on a smooth horizontal table. Initially \(P\) is at rest at the mid-point of \(A B\). The particle is now given a horizontal impulse in the direction perpendicular to \(A B\). At time \(t\) the displacement of \(P\) from the line \(A B\) is \(x\).

1. Show that the tension in each string is \(\frac { 6 m g } { l } \left( \sqrt { 9 l ^ { 2 } + x ^ { 2 } } - l \right)\).
2. Show that $$\ddot { x } = - \frac { 12 g x } { l } \left( 1 - \frac { l } { \sqrt { 9 l ^ { 2 } + x ^ { 2 } } } \right) .$$
3. Given that throughout the motion \(\frac { x ^ { 2 } } { l ^ { 2 } }\) is small enough to be negligible, show that the equation of motion is approximately $$\ddot { x } = - \frac { 8 g x } { l } .$$
4. Given that the initial speed of \(P\) is \(\sqrt { \frac { g l } { 200 } }\), find the time taken for the particle to travel a distance of \(\frac { 1 } { 80 } l\).

8 A light elastic string of natural length 0.2 m and modulus of elasticity 8 N has one end fixed to a point \(P\) on a horizontal ceiling. A particle of mass 0.4 kg is attached to the other end of the string.

1. Find the extension of the string when the particle hangs in equilibrium vertically below \(P\).
2. The particle is held at rest, with the string stretched, at a point \(x \mathrm {~m}\) vertically below \(P\) and is then released. Find the smallest value of \(x\) for which the particle will reach the ceiling.

13 Two light strings, each of natural length \(l\) and modulus of elasticity \(6 m g\), are attached at their ends to a particle \(P\) of mass \(m\). The other ends of the strings are attached to two fixed points \(A\) and \(B\), which are at a distance \(6 l\) apart on a smooth horizontal table. Initially \(P\) is at rest at the mid-point of \(A B\). The particle is now given a horizontal impulse in the direction perpendicular to \(A B\). At time \(t\) the displacement of \(P\) from the line \(A B\) is \(x\).

1. Show that the tension in each string is \(\frac { 6 m g } { l } \left( \sqrt { 9 l ^ { 2 } + x ^ { 2 } } - l \right)\).
2. Show that $$\ddot { x } = - \frac { 12 g x } { l } \left( 1 - \frac { l } { \sqrt { 9 l ^ { 2 } + x ^ { 2 } } } \right)$$
3. Given that throughout the motion \(\frac { x ^ { 2 } } { l ^ { 2 } }\) is small enough to be negligible, show that the equation of motion is approximately $$\ddot { x } = - \frac { 8 g x } { l } .$$
4. Given that the initial speed of \(P\) is \(\sqrt { \frac { g l } { 200 } }\), find the time taken for the particle to travel a distance of \(\frac { 1 } { 80 } l\).

13 Two light strings, each of natural length \(l\) and modulus of elasticity \(6 m g\), are attached at their ends to a particle \(P\) of mass \(m\). The other ends of the strings are attached to two fixed points \(A\) and \(B\), which are at a distance \(6 l\) apart on a smooth horizontal table. Initially \(P\) is at rest at the mid-point of \(A B\). The particle is now given a horizontal impulse in the direction perpendicular to \(A B\). At time \(t\) the displacement of \(P\) from the line \(A B\) is \(x\).

1. Show that the tension in each string is \(\frac { 6 m g } { l } \left( \sqrt { 9 l ^ { 2 } + x ^ { 2 } } - l \right)\).
2. Show that $$\ddot { x } = - \frac { 12 g x } { l } \left( 1 - \frac { 1 } { \sqrt { 9 l ^ { 2 } + x ^ { 2 } } } \right)$$
3. Given that throughout the motion \(\frac { x ^ { 2 } } { l ^ { 2 } }\) is small enough to be negligible, show that the equation of motion is approximately $$\ddot { x } = - \frac { 8 g x } { l } .$$
4. Given that the initial speed of \(P\) is \(\sqrt { \frac { g l } { 200 } }\), find the time taken for the particle to travel a distance of \(\frac { 1 } { 80 } l\).

\includegraphics{figure_5} A light elastic band, of total natural length \(a\) and modulus of elasticity \(\frac{1}{2}mg\), is stretched over two small smooth pins fixed at the same horizontal level and at a distance \(a\) apart. A particle of mass \(m\) is attached to the lower part of the band and when the particle is in equilibrium the sloping parts of the band each make an angle \(\beta\) with the vertical (see diagram). Express the tension in the band in terms of \(m\), \(g\) and \(\beta\), and hence show that \(\beta = \frac{1}{4}\pi\). [4] The particle is given a velocity of magnitude \(\sqrt{(ag)}\) vertically downwards. At time \(t\) the displacement of the particle from its equilibrium position is \(x\). Show that, neglecting air resistance, $$\ddot{x} = -\frac{2g}{a}x.$$ [3] Show that the particle passes through the level of the pins in the subsequent motion, and find the time taken to reach this level for the first time. [6]

Answer only one of the following two alternatives. EITHER A particle \(P\) of mass \(m\) is attached to one end of a light elastic string of modulus of elasticity \(4mg\) and natural length \(l\). The other end of the string is attached to a fixed point \(O\). The particle rests in equilibrium at the point \(E\), vertically below \(O\). The particle is pulled down a vertical distance \(\frac{3l}{4}\) from \(E\) and released from rest. Show that the motion of \(P\) is simple harmonic with period \(\pi\sqrt{\left(\frac{l}{g}\right)}\). [4] At an instant when \(P\) is moving vertically downwards through \(E\), the string is cut. When \(P\) has descended a further distance \(\frac{3l}{4}\) under gravity, it strikes a fixed smooth plane which is inclined at 30° to the horizontal. The coefficient of restitution between \(P\) and the plane is \(\frac{1}{3}\). Show that the speed of \(P\) immediately after the impact is \(\frac{1}{3}\sqrt{(5gl)}\). [8] OR A new restaurant \(S\) has recently opened in a particular town. In order to investigate any effect of \(S\) on an existing restaurant \(R\), the daily takings, \(x\) and \(y\) in thousands of dollars, at \(R\) and \(S\) respectively are recorded for a random sample of 8 days during a six-month period. The results are shown in the following table.

1. Calculate the product moment correlation coefficient for this sample. [4]
2. Stating your hypotheses, test, at the 2.5% significance level, whether there is negative correlation between daily takings at the two restaurants and comment on your result in the context of the question. [5]

Another sample is taken over \(N\) randomly chosen days and the product moment correlation coefficient is found to be \(-0.431\). A test, at the 5% significance level, shows that there is evidence of negative correlation between daily takings in the two restaurants.

1. Find the range of possible values of \(N\). [3]

Answer only one of the following two alternatives. EITHER A particle \(P\) of mass \(m\) is attached to one end of a light elastic string of modulus of elasticity \(4mg\) and natural length \(l\). The other end of the string is attached to a fixed point \(O\). The particle rests in equilibrium at the point \(E\), vertically below \(O\). The particle is pulled down a vertical distance \(\frac{3l}{4}\) from \(E\) and released from rest. Show that the motion of \(P\) is simple harmonic with period \(\pi\sqrt{\left(\frac{l}{g}\right)}\). [4] At an instant when \(P\) is moving vertically downwards through \(E\), the string is cut. When \(P\) has descended a further distance \(\frac{5l}{4}\) under gravity, it strikes a fixed smooth plane which is inclined at 30° to the horizontal. The coefficient of restitution between \(P\) and the plane is \(\frac{1}{3}\). Show that the speed of \(P\) immediately after the impact is \(\frac{1}{3}\sqrt{(5gl)}\). [8] OR A new restaurant \(S\) has recently opened in a particular town. In order to investigate any effect of \(S\) on an existing restaurant \(R\), the daily takings, \(x\) and \(y\) in thousands of dollars, at \(R\) and \(S\) respectively are recorded for a random sample of 8 days during a six-month period. The results are shown in the following table.

Day	1	2	3	4	5	6	7	8
\(x\)	1.2	1.4	0.9	1.1	0.8	1.0	0.6	1.5
\(y\)	0.3	0.4	0.6	0.6	0.25	0.75	0.6	0.35

1. Calculate the product moment correlation coefficient for this sample. [4]
2. Stating your hypotheses, test, at the 2.5\% significance level, whether there is negative correlation between daily takings at the two restaurants and comment on your result in the context of the question. [5]

Another sample is taken over \(N\) randomly chosen days and the product moment correlation coefficient is found to be \(-0.431\). A test, at the 5\% significance level, shows that there is evidence of negative correlation between daily takings in the two restaurants.

1. Find the range of possible values of \(N\). [3]

Answer only one of the following two alternatives. EITHER A particle \(P\) of mass \(3m\) is attached to one end of a light elastic spring of natural length \(a\) and modulus of elasticity \(kmg\). The other end of the spring is attached to a fixed point \(O\) on a smooth plane that is inclined to the horizontal at an angle \(\alpha\), where \(\sin \alpha = \frac{3}{5}\). The system rests in equilibrium with \(P\) on the plane at the point \(E\). The length of the spring in this position is \(\frac{5}{4}a\).

1. Find the value of \(k\). [3]

The particle \(P\) is now replaced by a particle \(Q\) of mass \(2m\) and \(Q\) is released from rest at the point \(E\).

1. Show that, in the resulting motion, \(Q\) performs simple harmonic motion. State the centre and the period of the motion. [6]
2. Find the least tension in the spring and the maximum acceleration of \(Q\) during the motion. [5]

OR A shop is supplied with large quantities of plant pots in packs of six. These pots can be damaged easily if they are not packed carefully. The manager of the shop is a statistician and he believes that the number of damaged pots in a pack of six has a binomial distribution. He chooses a random sample of 250 packs and records the numbers of damaged pots per pack. His results are shown in the following table.

1. Show that the mean number of damaged pots per pack in this sample is 1.656. [1]

The following table shows some of the expected frequencies, correct to 2 decimal places, using an appropriate binomial distribution.

1. Find the values of \(a\) and \(b\), correct to 2 decimal places [5]
2. Use a goodness-of-fit test at the 1% significance level to determine whether the manager's belief is justified. [8]

Answer only one of the following two alternatives. EITHER A particle \(P\) of mass \(3m\) is attached to one end of a light elastic spring of natural length \(a\) and modulus of elasticity \(kmg\). The other end of the spring is attached to a fixed point \(O\) on a smooth plane that is inclined to the horizontal at an angle \(\alpha\), where \(\sin\alpha = \frac{3}{5}\). The system rests in equilibrium with \(P\) on the plane at the point \(E\). The length of the spring in this position is \(\frac{5}{4}a\).

1. Find the value of \(k\). [3]
2. The particle \(P\) is now replaced by a particle \(Q\) of mass \(2m\) and \(Q\) is released from rest at the point \(E\). Show that, in the resulting motion, \(Q\) performs simple harmonic motion. State the centre and the period of the motion. [6]
3. Find the least tension in the spring and the maximum acceleration of \(Q\) during the motion. [5]

OR A shop is supplied with large quantities of plant pots in packs of six. These pots can be damaged easily if they are not packed carefully. The manager of the shop is a statistician and he believes that the number of damaged pots in a pack of six has a binomial distribution. He chooses a random sample of 250 packs and records the numbers of damaged pots per pack. His results are shown in the following table.

Number of damaged pots per pack \((x)\)	0	1	2	3	4	5	6
Frequency	48	69	78	32	22	1	0

1. Show that the mean number of damaged pots per pack in this sample is 1.656. [1]
2. The following table shows some of the expected frequencies, correct to 2 decimal places, using an appropriate binomial distribution.

   Number of damaged pots per pack \((x)\)	0	1	2	3	4	5	6
   Expected frequency	36.01	82.36	\(a\)	39.89	\(b\)	1.74	0.11

   Find the values of \(a\) and \(b\), correct to 2 decimal places [5]
3. Use a goodness-of-fit test at the 1% significance level to determine whether the manager's belief is justified. [8]

\includegraphics{figure_4} A uniform rod \(AB\) of length \(3a\) and weight \(W\) is freely hinged to a fixed point at the end \(A\). The end \(B\) is below the level of \(A\) and is attached to one end of a light elastic string of natural length \(4a\). The other end of the string is attached to a point \(O\) on a vertical wall. The horizontal distance between \(A\) and the wall is \(5a\). The string and the rod make angles \(\theta\) and \(2\theta\) respectively with the horizontal (see diagram). The system is in equilibrium with the rod and the string in the same vertical plane. It is given that \(\sin \theta = \frac{3}{5}\) and you may use the fact that \(\cos 2\theta = \frac{7}{25}\).

1. Find the tension in the string in terms of \(W\). [3]
2. Find the modulus of elasticity of the string in terms of \(W\). [4]
3. Find the angle that the force acting on the rod at \(A\) makes with the horizontal. [3]