6.02g Hooke's law: T = k*x or T = lambda*x/l

1. Throughout this question, use \(\boldsymbol { g } = \mathbf { 1 0 m ~ s } ^ { \mathbf { - 2 } }\)

A light elastic string has natural length 1.25 m and modulus of elasticity 25 N .
A particle \(P\) of mass 0.5 kg is attached to one end of the string. The other end of the string is attached to a fixed point \(A\). Particle \(P\) hangs freely in equilibrium with \(P\) vertically below \(A\) The particle is then pulled vertically down to a point \(B\) and released from rest.

1. Show that, while the string is taut, \(P\) moves with simple harmonic motion with period \(\frac { \pi } { \sqrt { 10 } }\) seconds. The maximum kinetic energy of \(P\) during the subsequent motion is 2.5 J .
2. Show that \(A B = 2 \mathrm {~m}\) The particle returns to \(B\) for the first time \(T\) seconds after it was released from rest at \(B\)
3. Find the value of \(T\)

8. \begin{figure}[h] \end{figure} The fixed points \(A\) and \(B\) lie on a smooth horizontal surface with \(A B = 6 \mathrm {~m}\).
A particle \(P\) has mass 0.3 kg .
One end of a light elastic string, of natural length 2 m and modulus of elasticity 20 N , is attached to \(P\), and the other end is attached to \(A\). One end of another light elastic string, of natural length 2 m and modulus of elasticity 40 N , is attached to \(P\) and the other end is attached to \(B\). The particle \(P\) is at rest in equilibrium at the point \(E\) on the surface, as shown in Figure 7.

1. Show that \(E B = \frac { 8 } { 3 } \mathrm {~m}\). The particle \(P\) is now held at the midpoint of \(A B\) and released from rest.
2. Show that \(P\) oscillates with simple harmonic motion about the point \(E\). The time between the instant when \(P\) is released and the instant when it first returns to the point \(E\) is \(S\) seconds.
3. Find the exact value of \(S\).
4. Find the length of time during one oscillation for which the speed of \(P\) is more than \(2 \mathrm {~ms} ^ { - 1 }\)

1. Two points \(A\) and \(B\) are 6 m apart on a smooth horizontal surface.

A light elastic string of natural length 2 m and modulus of elasticity 20 N , has one end attached to the point \(A\). A second light elastic string of natural length 2 m and modulus of elasticity 50 N , has one end attached to the point \(B\). A particle \(P\) of mass 3.5 kg is attached to the free end of each string.
The particle \(P\) is held at the point on \(A B\) which is 2 m from \(B\) and then released from rest.
In the subsequent motion both strings remain taut.

1. Show that \(P\) moves with simple harmonic motion about its equilibrium position.
2. Find the maximum speed of \(P\).
3. Find the length of time within each oscillation for which \(P\) is closer to \(A\) than to \(B\).

3 A particle \(P\) of mass 3.5 kg is attached to one end of a light elastic string of natural length 0.8 m and modulus of elasticity 75 N . The other end of the string is attached to a fixed point \(O\). The particle rotates in a horizontal circle with a constant angular velocity of \(3 \mathrm { rad } \mathrm { s } ^ { - 1 }\). The centre of the circle is vertically below \(O\). The magnitude of the tension in the string is \(T \mathrm {~N}\) and the length of the extended string is \(L \mathrm {~m}\) (see diagram). \includegraphics[max width=\textwidth, alt={}, center]{a8c9d007-e67f-4637-9e74-630ba9a91442-3_460_424_447_817}

1. By considering the acceleration of \(P\), show that \(T = 31.5 L\).
2. Write down another relationship between \(T\) and \(L\).
3. Find the value of \(T\) and the value of \(L\).
4. Find the angle that the string makes with the downwards vertical through \(O\).

4 A ball \(B\) of mass 1.7 kg is connected to one end of a light elastic spring of natural length 1.2 m . The other end of the spring is attached to a point \(O\) on the ceiling of a large room. The modulus of elasticity of the spring is 50 N . The ball is held 3.2 m vertically below \(O\) and projected upwards with an initial speed of \(0.5 \mathrm {~ms} ^ { - 1 }\). In order to model the motion of \(B\) (before any collision with the ceiling) the following assumptions are made.

• Air resistance is ignored.
• \(B\) is small.
• The fully compressed length of the spring is negligible. \begin{enumerate}[label=(\roman*)]
• Determine whether, according to the model, \(B\) reaches \(O\).
• Without doing any further calculations, explain whether the answer to part (i) could change in each of the following different cases.
  1. A new model is used in which air resistance is taken into account.
  2. The spring is replaced by an elastic string with the same natural length and modulus of elasticity.
  3. \(\quad B\) is initially projected downwards rather than upwards.

7 A particle, of mass 10 kg , is attached to one end of a light elastic string of natural length 0.4 metres and modulus of elasticity 100 N . The other end of the string is fixed to the point \(O\).

1. Find the length of the elastic string when the particle hangs in equilibrium directly below \(O\).
2. The particle is pulled down and held at a point \(P\), which is 1 metre vertically below \(O\). Show that the elastic potential energy of the string when the particle is in this position is 45 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 below \(\boldsymbol { O }\).
   1. Show that, while the string is taut, $$v ^ { 2 } = 39.6 x - 25 x ^ { 2 } - 14.6$$
   2. 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.

8 Two small blocks, \(A\) and \(B\), of masses 0.8 kg and 1.2 kg respectively, are stuck together. A spring has natural length 0.5 metres and modulus of elasticity 49 N . One end of the spring is attached to the top of the block \(A\) and the other end of the spring is attached to a fixed point \(O\).

1. The system hangs in equilibrium with the blocks stuck together, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{88aec6ab-af83-4d5e-84b6-5fd84c16a6c9-017_385_239_669_881} Find the extension of the spring.
2. Show that the elastic potential energy of the spring when the system is in equilibrium is 1.96 J .
3. The system is hanging in this equilibrium position when block \(B\) falls off and block \(A\) begins to move vertically upwards. Block \(A\) next comes to rest when the spring is compressed by \(x\) metres.
   1. Show that \(x\) satisfies the equation $$x ^ { 2 } + 0.16 x - 0.008 = 0$$
   2. Find the value of \(x\).

8 Two small blocks, \(A\) and \(B\), of masses 0.8 kg and 1.2 kg respectively, are stuck together. A spring has natural length 0.5 metres and modulus of elasticity 49 N . One end of the spring is attached to the top of the block \(A\) and the other end of the spring is attached to a fixed point \(O\).

1. The system hangs in equilibrium with the blocks stuck together, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{480a817d-074f-440d-829e-c8f8a9746151-8_385_239_669_881} Find the extension of the spring.
2. Show that the elastic potential energy of the spring when the system is in equilibrium is 1.96 J .
3. The system is hanging in this equilibrium position when block \(B\) falls off and block \(A\) begins to move vertically upwards. Block \(A\) next comes to rest when the spring is compressed by \(x\) metres.
   1. Show that \(x\) satisfies the equation $$x ^ { 2 } + 0.16 x - 0.008 = 0$$
   2. Find the value of \(x\).

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.

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).

13 Two light elastic strings each have one end attached to a particle \(B\) of mass \(3 c \mathrm {~kg}\), which rests on a smooth horizontal table. The other ends of the strings are attached to the fixed points \(A\) and \(C\), which are 8 metres apart. \(A B C\) is a horizontal line. \includegraphics[max width=\textwidth, alt={}, center]{44e22a98-6424-4fb1-8a37-c965773cb7b6-26_92_910_635_566} String \(A B\) has a natural length of 4 metres and a stiffness of \(5 c\) newtons per metre.
String \(B C\) has a natural length of 1 metre and a stiffness of \(c\) newtons per metre.
The particle is pulled a distance of \(\frac { 1 } { 3 }\) metre from its equilibrium position towards \(A\), and released from rest. 13

1. Show that the particle moves with simple harmonic motion.
   13
2. Find the speed of the particle when it is at a point \(P\), a distance \(\frac { 1 } { 4 }\) metre from the equilibrium position. Give your answer to two significant figures.
   [0pt] [4 marks]

11 Two light strings, each of natural length \(a\) 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 \(6 a\) 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 $$\ddot { x } = - \frac { 12 g x } { a } \left( 1 - \frac { a } { \sqrt { 9 a ^ { 2 } + x ^ { 2 } } } \right) .$$
2. Given that \(\frac { x } { a }\) is small throughout the motion, show that the equation of motion is approximately $$\ddot { x } = - \frac { 8 g x } { a }$$ and state the period of the simple harmonic motion that this equation represents.
3. Given that the initial speed of \(P\) is \(\sqrt { \frac { g a } { 200 } }\), show that the amplitude of the simple harmonic motion is \(\frac { 1 } { 40 } a\).

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.

14 \includegraphics[max width=\textwidth, alt={}, center]{22640c3b-792f-4003-a4f8-78220efd73b0-5_86_1589_1297_278} A particle of mass 0.05 kg is attached to two identical light elastic strings, each of natural length 1.2 m and modulus of elasticity 0.6 N . The other ends of the strings are attached to points \(A\) and \(E\) on a smooth horizontal table. The distance \(A E\) is 2 m and points \(B , C\) and \(D\) lie between \(A\) and \(E\) so that \(A B = 0.7 \mathrm {~m} , B C = 0.1 \mathrm {~m} , C D = 0.4 \mathrm {~m}\) and \(D E = 0.8 \mathrm {~m}\) (see diagram). Initially the particle is held at \(B\) and it is then released. In the subsequent motion the displacement of the particle from \(C\), in the direction of \(A\), is denoted by \(x \mathrm {~m}\).

1. Find the equation of motion for the particle when it is between \(B\) and \(C\).
2. Find the velocity of the particle when it is at \(C\).
3. Find the total time that elapses before the particle first returns to \(B\).

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 \(3m\) 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]

\includegraphics{figure_6} A particle \(P\) of mass \(0.2 \text{ kg}\) is attached to one end of a light inextensible string of length \(0.6 \text{ m}\). The other end of the string is attached to a particle \(Q\) of mass \(0.3 \text{ kg}\). The string passes through a small hole \(H\) in a smooth horizontal surface. A light elastic string of natural length \(0.3 \text{ m}\) and modulus of elasticity \(15 \text{ N}\) joins \(Q\) to a fixed point \(A\) which is \(0.4 \text{ m}\) vertically below \(H\). The particle \(P\) moves on the surface in a horizontal circle with centre \(H\) (see diagram).

1. Calculate the greatest possible speed of \(P\) for which the elastic string is not extended. [4]
2. Find the distance \(HP\) given that the angular speed of \(P\) is \(8 \text{ rad s}^{-1}\). [5]

One end of a light elastic string is attached to a fixed point \(O\). The other end of the string is attached to a particle \(P\) of mass 0.24 kg. The string has natural length 0.6 m and modulus of elasticity 24 N. The particle is released from rest at \(O\). Find the two possible values of the distance \(OP\) for which the particle has speed 1.5 m s\(^{-1}\). [6]

One end of a light elastic string of natural length \(0.6 \text{ m}\) and modulus of elasticity \(24 \text{ N}\) is attached to a fixed point \(O\). The other end of the string is attached to a particle \(P\) of mass \(0.4 \text{ kg}\) which hangs in equilibrium vertically below \(O\).

1. Calculate the extension of the string. [2]

\(P\) is projected vertically downwards from the equilibrium position with speed \(5 \text{ m s}^{-1}\).

1. Calculate the distance \(P\) travels before it is first at instantaneous rest. [4]

When \(P\) is first at instantaneous rest a stationary particle of mass \(0.4 \text{ kg}\) becomes attached to \(P\).

1. Find the greatest speed of the combined particle in the subsequent motion. [4]