Hooke's law and elastic energy

A particle P of mass \(0.5\) kg is attached to a fixed point O by a light elastic string of natural length \(3\) m and modulus of elasticity \(75\) N. P is released from rest at O and is allowed to fall freely. Determine the length of the string when P is at its lowest point in the subsequent motion. [5]

5 One end of a light elastic string, of natural length 0.5 m and modulus of elasticity 140 N , is attached to a fixed point \(O\). A particle of mass 0.8 kg is attached to the other end of the string. The particle is released from rest at \(O\). By considering the energy of the system, find

1. the speed of the particle when the extension of the string is 0.1 m ,
2. the extension of the string when the particle is at its lowest point.

5 A particle \(P\) of mass 0.4 kg is attached to one end of a light elastic string of natural length 0.5 m and modulus of elasticity 6 N . The other end of the string is attached to a fixed point \(O\). The particle \(P\) is released from rest at the point \(( 0.5 + x ) \mathrm { m }\) vertically below \(O\). The particle \(P\) comes to instantaneous rest at \(O\).

1. Find \(x\).
2. Find the greatest speed of \(P\).

1. A light elastic string of modulus of elasticity 29.4 N has one end attached to a fixed point \(A\). A particle \(P\) of mass 1.5 kg is attached to the other end of the string and \(P\) hangs freely in equilibrium 0.5 m vertically below \(A\). Find the natural length of the string.
   

1 A small object of mass 5 kg is attached to one end of each of two identical parallel light elastic strings. The upper ends of both strings are attached to a horizontal ceiling.
The object hangs in equilibrium at R , with the extension of each string being 0.1 m , as shown in Fig. 1. \begin{figure}[h] \end{figure}

1. Find the stiffness of each string. One of the strings is now removed and the object initially falls downwards. The object does not return to R at any point in the subsequent motion.
2. Suggest a reason why the object does not return to \(R\).

\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]

3 A light elastic string has natural length 2.2 m and modulus of elasticity 14.3 N . A particle \(P\) of mass \(m \mathrm {~kg}\) is attached to the mid-point of the string. The ends of the string are attached to fixed points \(A\) and \(B\) which are 2.4 m apart at the same horizontal level. \(P\) is released from rest at the mid-point of \(A B\). In the subsequent motion \(P\) has its greatest speed at a point 0.5 m below \(A B\).

1. Find \(m\).
2. Calculate the greatest speed of \(P\).

7 \includegraphics[max width=\textwidth, alt={}, center]{7f8646df-a7d8-4ca1-a6ee-3ceab6bb83af-4_232_905_762_621} A light elastic string has natural length 10 m and modulus of elasticity 130 N . The ends of the string are attached to fixed points \(A\) and \(B\), which are at the same horizontal level. A small stone is attached to the mid-point of the string and hangs in equilibrium at a point 2.5 m below \(A B\), as shown in the diagram. With the stone in this position the length of the string is 13 m .

1. Find the tension in the string.
2. Show that the mass of the stone is 3 kg . The stone is now held at rest at a point 8 m vertically below the mid-point of \(A B\).
3. Find the elastic potential energy of the string in this position.
4. The stone is now released. Find the speed with which it passes through the mid-point of \(A B\).

4 A light elastic string has natural length 2.4 m and modulus of elasticity 21 N . A particle \(P\) of mass \(m \mathrm {~kg}\) is attached to the mid-point of the string. The ends of the string are attached to fixed points \(A\) and \(B\) which are 2.4 m apart at the same horizontal level. \(P\) is projected vertically upwards with velocity \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from the mid-point of \(A B\). In the subsequent motion \(P\) is at instantaneous rest at a point 1.6 m above \(A B\).

1. Find \(m\).
2. Calculate the acceleration of \(P\) when it first passes through a point 0.5 m below \(A B\).

5 A light elastic string has natural length 2 m and modulus of elasticity 1.5 N . One end of the string is attached to a fixed point \(O\) of a smooth plane which is inclined at \(30 ^ { \circ }\) to the horizontal. The other end of the string is attached to a particle \(P\) of mass \(0.075 \mathrm {~kg} . P\) is released from rest at \(O\). Find

1. the distance of \(P\) from \(O\) when \(P\) is at its lowest point,
2. the acceleration with which \(P\) starts to move up the plane immediately after it has reached its lowest point.

3 One end of a light elastic string of natural length 0.4 m and modulus of elasticity 20 N is attached to a fixed point \(A\) on a smooth plane inclined at \(30 ^ { \circ }\) to the horizontal. The other end of the string is attached to a particle \(P\) of mass 0.5 kg which rests in equilibrium on the plane.

1. Calculate the extension of the string. \(P\) is projected down the plane from the equilibrium position with speed \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The extension of the string is \(e \mathrm {~m}\) when the speed of the particle is \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) for the first time.
2. Find \(e\).

7 One end of a light elastic string of natural length 0.4 m and modulus of elasticity 20 N is attached to a particle \(P\) of mass 0.8 kg . The other end of the string is attached to a fixed point \(O\) at the top of a smooth plane inclined at \(30 ^ { \circ }\) to the horizontal. The particle rests in equilibrium on the plane.

1. Calculate the extension of the string. \(P\) is projected from its equilibrium position up the plane along a line of greatest slope. In the subsequent motion \(P\) just reaches \(O\), and later just reaches the foot of the plane. Calculate
2. the speed of projection of \(P\),
3. the length of the line of greatest slope of the plane.

3. A particle \(P\) of mass 0.5 kg is attached to one end of a light elastic spring, of natural length 2 m and modulus of elasticity 20 N . The other end of the spring is attached to a fixed point \(A\). The particle \(P\) is held at rest at the point \(B\), which is 1 m vertically below \(A\), and then released.

1. Find the acceleration of \(P\) immediately after it is released from rest. The particle comes to instantaneous rest for the first time at the point \(C\).
2. Find the distance \(B C\).

3 \includegraphics[max width=\textwidth, alt={}, center]{be83d46f-bf5b-4382-b424-bb5067626adc-2_433_446_1635_854} One end of a light elastic spring, of natural length 0.4 m and modulus of elasticity 88 N , is attached to a fixed point \(O\). A particle \(P\) of mass 0.2 kg is attached to the other end of the spring and is held, with the spring compressed, at a point 0.3 m vertically above \(O\), as shown in the diagram. \(P\) is now released from rest and moves vertically upwards.

1. Find the initial acceleration of \(P\).
2. Find the initial elastic potential energy of the spring.
3. Find the speed of \(P\) when the distance \(O P\) is 0.4 m . \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{be83d46f-bf5b-4382-b424-bb5067626adc-3_362_657_269_744} \captionsetup{labelformat=empty} \caption{Fig. 1}
   \end{figure} Fig. 1 shows a uniform lamina \(A B C D\) with dimensions \(A B = 15.5 \mathrm {~cm} , B C = 8 \mathrm {~cm}\) and \(C D = 9.5 \mathrm {~cm}\). Angles \(A B C\) and \(B C D\) are right angles.

1 A particle \(P\) of mass \(m\) is placed on a fixed smooth plane which is inclined at an angle \(\theta\) to the horizontal. A light spring, of natural length \(a\) and modulus of elasticity \(3 m g\), has one end attached to \(P\) and the other end attached to a fixed point \(O\) at the top of the plane. The spring lies along a line of greatest slope of the plane. The system is released from rest with the spring at its natural length. Find, in terms of \(a\) and \(\theta\), an expression for the greatest extension of the spring in the subsequent motion. \includegraphics[max width=\textwidth, alt={}, center]{1c53c407-25ea-43fc-a571-74ba1fffea8f-04_515_707_267_685} A particle \(P\) is attached to one end of a light inextensible string of length \(a\). The other end of the string is attached to a fixed point \(O\). The particle \(P\) is held with the string taut and making an angle \(\theta\) with the downward vertical. The particle \(P\) is then projected with speed \(\frac { 4 } { 5 } \sqrt { 5 a g }\) perpendicular to the string and just completes a vertical circle (see diagram). Find the value of \(\cos \theta\).

A light spring of natural length 0.6 metres is compressed to a length of 0.4 metres by a force of 20 newtons. The stiffness of the spring is \(k\) N m\(^{-1}\) Find \(k\) Circle your answer. [1 mark] 20 50 100 200

A light spring of natural length 0.6 metres is compressed to a length of 0.4 metres by a force of 20 newtons. The stiffness of the spring is \(k\) N m\(^{-1}\) Find \(k\) Circle your answer. [1 mark] 20 50 100 200

5. \begin{figure}[h] \end{figure} The end \(A\) of a uniform rod \(A B\), of length \(2 a\) and mass \(4 m\), is smoothly hinged to a fixed point. The end \(B\) is attached to one end of a light inextensible string which passes over a small smooth pulley, fixed at the same level as \(A\). The distance from \(A\) to the pulley is \(4 a\). The other end of the string carries a particle of mass \(m\) which hangs freely, vertically below the pulley, with the string taut. The angle between the rod and the downward vertical is \(\theta\), where \(0 < \theta < \frac { \pi } { 2 }\), as shown in Figure 1.

1. Show that the potential energy of the system is $$2 m g a ( \sqrt { } ( 5 - 4 \sin \theta ) - 2 \cos \theta ) + \text { constant }$$
2. Hence, or otherwise, show that any value of \(\theta\) which corresponds to a position of equilibrium of the system satisfies the equation $$4 \sin ^ { 3 } \theta - 6 \sin ^ { 2 } \theta + 1 = 0 .$$
3. Given that \(\theta = \frac { \pi } { 6 }\) corresponds to a position of equilibrium, determine its stability. \section*{L \(\_\_\_\_\)}

\includegraphics{figure_2} A particle of mass 0.5 kg is attached to one end of a light elastic spring of natural length 0.9 m and modulus of elasticity \(\lambda\) newtons. The other end of the spring is attached to a fixed point \(O\) on a rough plane which is inclined at an angle \(\theta\) to the horizontal, where \(\sin \theta = \frac{3}{5}\). The coefficient of friction between the particle and the plane is 0.15. The particle is held on the plane at a point which is 1.5 m down the line of greatest slope from \(O\), as shown in Figure 2. The particle is released from rest and first comes to rest again after moving 0.7 m up the plane. Find the value of \(\lambda\). [9]

4. \begin{figure}[h] \end{figure} A particle \(P\) of mass 2 kg is attached to one end of a light elastic string of natural length 1.5 m and modulus of elasticity \(\lambda\). The other end of the string is fixed to a point \(A\) on a rough plane inclined at an angle of \(30 ^ { \circ }\) to the horizontal as shown in Figure 2. The coefficient of friction between \(P\) and the plane is \(\frac { 1 } { 6 } \sqrt { 3 }\). \(P\) is held at rest at \(A\) and then released. It first comes to instantaneous rest at the point \(B , 2.2 \mathrm {~m}\) from \(A\). For the motion of \(P\) from \(A\) to \(B\),

1. show that the work done against friction is 10.78 J ,
2. find the change in the gravitational potential energy of \(P\). By using the work-energy principle, or otherwise,
3. find \(\lambda\).

A particle \(P\), of mass \(5\) kg is placed at the point \(A\) on a rough plane which is inclined at \(30°\) to the horizontal. The points \(Q\) and \(R\) are also on the surface of the inclined plane, with \(QR = 15\) metres. The point \(A\) is between \(Q\) and \(R\) so that \(AQ = 4\) metres and \(AR = 11\) metres. The three points \(Q\), \(A\) and \(R\) are on a line of greatest slope of the plane. \includegraphics{figure_8} The particle is attached to two light elastic strings, \(PQ\) and \(PR\). One of the strings, \(PQ\), has natural length \(4\) metres and modulus of elasticity \(160\) N, the other string, \(PR\), has natural length \(6\) metres and modulus of elasticity \(120\) N. The particle is released from rest at the point \(A\). The coefficient of friction between the particle and the plane is \(0.4\). Find the distance of the particle from \(Q\) when it is next at rest. [8 marks]

1. Show that, when \(P\) is in equilibrium, \(O P = 7.25 \mathrm {~m}\).
2. Verify that \(P\) and \(Q\) together just reach the safety net.
3. At the lowest point of their motion \(P\) releases \(Q\). Prove that \(P\) subsequently just reaches \(O\).
4. State two additional modelling assumptions made when answering this question.

8 Carol, a bungee jumper of mass 70 kg , is attached to one end of a light elastic cord of natural length 26 metres and modulus of elasticity 1456 N . The other end of the cord is attached to a fixed horizontal platform which is at a height of 69 metres above the ground. Carol steps off the platform at the point where the cord is attached and falls vertically. Hooke's law can be assumed to apply whilst the cord is taut. Model Carol as a particle and assume air resistance to be negligible.
When Carol has fallen \(x \mathrm {~m}\), her speed is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. By considering energy, show that $$5 v ^ { 2 } = 306 x - 4 x ^ { 2 } - 2704 \text { for } x \geqslant 26$$
2. Why is the expression found in part (a) not true when \(x\) takes values less than 26?
3. Find the maximum value of \(x\).
4. 1. Find the distance fallen by Carol when her speed is a maximum.
   2. Hence find Carol's maximum speed.

1. Show that, when \(P\) is in equilibrium, \(O P = 7.25 \mathrm {~m}\).
2. Verify that \(P\) and \(Q\) together just reach the safety net.
3. At the lowest point of their motion \(P\) releases \(Q\). Prove that \(P\) subsequently just reaches \(O\).
4. State two additional modelling assumptions made when answering this question.

The figure shows a particle \(P\), of mass 0·8 kg, attached to the ends of two light elastic strings. \(AP\) has natural length 20 cm and modulus of elasticity \(\lambda\) N. \(BP\) has natural length 20 cm and modulus of elasticity \(\mu\) N. \(A\) and \(B\) are fixed to points on the same horizontal level so that \(AB = 50\) cm. When \(P\) is suspended in equilibrium, \(AP = 30\) cm and \(BP = 40\) cm. Calculate the values of \(\lambda\) and \(\mu\). \includegraphics{figure_2} [9 marks]

6 Two identical light elastic strings, each of length \(l\) and modulus of elasticity \(\lambda m g\) are attached to a particle \(P\) of mass \(m\). The other end of the first string is attached to a fixed point A , and the other end of the second string is attached to a fixed point B . The points A and B are such that A is above and to the right of B and both strings are taut. The string attached to A makes an angle of \(30 ^ { \circ }\) with the vertical, and the string attached to B makes an angle of \(\theta ^ { \circ }\) with the horizontal, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{feb9a438-26b0-41d3-b044-6acd6efccde0-6_546_533_699_242} The system is in equilibrium in a vertical plane. The extension of the string attached to A is 0.9 l and the extension of the string attached to B is \(0.5 l\).

1. Explain how you know that APB is not a straight line.
2. Show that the elastic potential energy stored in string AP is \(k m g l\), where the value of \(k\) is to be determined correct to \(\mathbf { 3 }\) significant figures.

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 \(B\).

5 One end of a light elastic string of natural length 0.4 m and modulus of elasticity 16 N is attached to a fixed point \(O\) of a horizontal table. A particle \(P\) of mass 0.8 kg is attached to the other end of the string. The particle \(P\) is released from rest on the table, at a point which is 0.5 m from \(O\). The coefficient of friction between the particle and the table is 0.2 . By considering work and energy, find the speed of \(P\) at the instant the string becomes slack.

8 An elastic string has one end attached to a point \(O\) fixed on a rough horizontal surface. The other end of the string is attached to a particle of mass 2 kg . The elastic string has natural length 0.8 metres and modulus of elasticity 32 newtons. The particle is pulled so that it is at the point \(A\), on the surface, 3 metres from the point \(O\).

1. Calculate the elastic potential energy when the particle is at the point \(A\).
2. The particle is released from rest at the point \(A\) and moves in a straight line towards \(O\). The particle is next at rest at the point \(B\). The distance \(A B\) is 5 metres. \includegraphics[max width=\textwidth, alt={}, center]{06c3e260-8167-4616-97d4-0f360a376a0f-6_179_1055_877_497} Find the frictional force acting on the particle as it moves along the surface.
3. Show that the particle does not remain at rest at the point \(B\).
4. The particle next comes to rest at a point \(C\) with the string slack. Find the distance \(B C\).
5. Hence, or otherwise, find the total distance travelled by the particle after it is released from the point \(A\).

In this question use \(g = 9.8\) m s\(^{-2}\) A particle \(P\) of mass \(m\) is attached to two light elastic strings, \(AP\) and \(BP\). The other ends of the strings, \(A\) and \(B\), are attached to fixed points which are 4 metres apart on a rough horizontal surface at the bottom of a container. The coefficient of friction between \(P\) and the surface is 0.68 • When the extension of string \(AP\) is \(e_A\) metres, the tension in \(AP\) is \(24me_A\) • When the extension of string \(BP\) is \(e_B\) metres, the tension in \(BP\) is \(10me_B\) • The natural length of string \(AP\) is 1 metre • The natural length of string \(BP\) is 1.3 metres \includegraphics{figure_15}

1. Show that when \(AP = 1.5\) metres, the tension in \(AP\) is equal to the tension in \(BP\). [1 mark]
2. \(P\) is held at the point between \(A\) and \(B\) where \(AP = 1.9\) metres, and then released from rest. At time \(t\) seconds after \(P\) is released, \(AP = (1.5 + x)\) metres. \includegraphics{figure_15b} Show that when \(P\) is moving towards \(A\), $$\frac{\mathrm{d}^2 x}{\mathrm{d}t^2} + 34x = 6.664$$ [3 marks]
3. The container is then filled with oil, and \(P\) is again released from rest at the point between \(A\) and \(B\) where \(AP = 1.9\) metres. At time \(t\) seconds after \(P\) is released, the oil causes a resistive force of magnitude \(10mv\) newtons to act on the particle, where \(v\) m s\(^{-1}\) is the speed of the particle. Find \(x\) in terms of \(t\) when \(P\) is moving towards \(A\). [9 marks]

1 A particle of mass 0.3 kg is attached to one end of a light elastic string of natural length 0.6 m and modulus of elasticity 9 N . The other end of the string is attached to a fixed point \(O\) on a smooth horizontal surface. The particle is projected horizontally from \(O\) with speed \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find the greatest distance of the particle from \(O\).

2 A particle \(P\) of mass 0.2 kg is attached to one end of a light elastic string of natural length 0.8 m and modulus of elasticity 64 N . The other end of the string is attached to a fixed point \(A\) on a smooth horizontal surface. \(P\) is placed on the surface at a point 0.8 m from \(A\). The particle \(P\) is then projected with speed \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) directly away from \(A\).

1. Calculate the distance \(A P\) when \(P\) is at instantaneous rest.
2. Calculate the speed of \(P\) when it is 1.0 m from \(A\).

A particle of mass \(m\) kg is attached to one end of a light elastic string of natural length \(a\) metres and modulus of elasticity \(5ma\) newtons. The other end of the string is attached to a fixed point \(O\) on a smooth horizontal plane. The particle is held at rest on the plane with the string stretched to a length \(2a\) metres and then released at time \(t = 0\). During the subsequent motion, when the particle is moving with speed \(v\) m s\(^{-1}\), the particle experiences a resistance of magnitude \(4mv\) newtons. At time \(t\) seconds after the particle is released, the length of the string is \((a + x)\) metres, where \(0 \leqslant x \leqslant a\).

1. Show that, from \(t = 0\) until the string becomes slack, $$\frac{\mathrm{d}^2 x}{\mathrm{d}t^2} + 4\frac{\mathrm{d}x}{\mathrm{d}t} + 5x = 0$$ [3]
2. Hence express \(x\) in terms of \(a\) and \(t\). [6]
3. Find the speed of the particle at the instant when the string first becomes slack, giving your answer in the form \(ka\), where \(k\) is a constant to be found correct to 2 significant figures. [4]

1. Hence find the speed of \(P\) when it is \(2\text{m}\) below \(O\). [2]

1. Hence find the speed of \(P\) when it is \(2\text{m}\) below \(O\). [2]

A particle \(P\) of mass \(m\text{kg}\) is attached to one end of a light elastic string of natural length \(2\text{m}\) and modulus of elasticity \(2mg\text{N}\). The other end of the string is attached to a fixed point \(O\). The particle \(P\) hangs in equilibrium vertically below \(O\). The particle \(P\) is pulled down vertically a distance \(d\text{m}\) below its equilibrium position and released from rest.

1. Given that the particle just reaches \(O\) in the subsequent motion, find the value of \(d\). [6]

6 One end of a light elastic string, of natural length \(a\) and modulus of elasticity \(k\), is attached to a particle \(P\) of mass \(m\). The other end of the string is attached to a fixed point \(Q\). The particle \(P\) is projected vertically upwards from \(Q\). When \(P\) is moving upwards and at a distance \(\frac { 4 } { 3 } a\) directly above \(Q\), it has a speed \(\sqrt { 2 g a }\). At this point, its acceleration is \(\frac { 7 } { 3 } g\) downwards. Show that \(\mathrm { k } = 4 \mathrm { mg }\) and find in terms of \(a\) the greatest height above \(Q\) reached by \(P\).

4 One end of a light elastic string of natural length 3 m and modulus of elasticity 15 mN is attached to a fixed point \(O\). A particle \(P\) of mass \(m \mathrm {~kg}\) is attached to the other end of the string. \(P\) is released from rest at \(O\) and moves vertically downwards. When the extension of the string is \(x \mathrm {~m}\) the velocity of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Show that \(v ^ { 2 } = 5 \left( 12 + 4 x - x ^ { 2 } \right)\).
2. Find the magnitude of the acceleration of \(P\) when it is at its lowest point, and state the direction of this acceleration.

A light elastic string has modulus of elasticity \(\frac { 3 } { 2 } m g\) and natural length \(a\). A particle of mass \(m\) is attached to one end of the string. The other end of the string is attached to a fixed point \(A\). The particle is released from rest at \(A\). Show that when the particle has fallen a distance \(k a\) from \(A\), where \(k > 1\), its kinetic energy is $$\frac { 1 } { 4 } m g a \left( 10 k - 3 - 3 k ^ { 2 } \right) .$$ Show that the particle first comes to instantaneous rest at the point \(B\) which is at a distance \(3 a\) vertically below \(A\). Show that the time taken by the particle to travel from \(A\) to \(B\) is $$\sqrt { } \left( \frac { 2 a } { g } \right) + \frac { 2 \pi } { 3 } \sqrt { } \left( \frac { 2 a } { 3 g } \right)$$

5. A light elastic string of natural length \(l\) has one end attached to a fixed point \(A\). A particle \(P\) of mass \(m\) is attached tot he other end of the string and hangs in equilibrium at the point \(O\), where \(A O = \frac { 5 } { 4 } l\).

1. Find the modulus of the elasticity of the string. The particle \(P\) is then pulled down and released from rest. At time \(t\) the length of the string is \(\frac { 5 l } { 4 } + x\).
2. Prove that, while the string is taut, $$\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } = - \frac { 4 g x } { l }$$ When \(P\) is released, \(A P = \frac { 7 } { 4 } l\). The point \(B\) is a distance \(l\) vertically below \(A\).
3. Find the speed of \(P\) at \(B\).
4. Describe briefly the motion of \(P\) after it has passed through \(B\) for the first time until it next passes through \(O\).

1 A light elastic string has modulus of elasticity 5 N and natural length 1.5 m . One end of the string is attached to a fixed point \(O\) and a particle \(P\) of mass 0.1 kg is attached to the other end of the string. \(P\) is released from rest at the point 2.4 m vertically below \(O\). Calculate the speed of \(P\) at the instant the string first becomes slack.

\includegraphics{figure_3} A light elastic string has natural length \(8a\) and modulus of elasticity \(5mg\). A particle \(P\) of mass \(m\) is attached to the midpoint of the string. The ends of the string are attached to points \(A\) and \(B\) which are a distance \(12a\) apart on a smooth horizontal table. The particle \(P\) is held on the table so that \(AP = BP = L\) (see diagram). The particle \(P\) is released from rest. When \(P\) is at the midpoint of \(AB\) it has speed \(\sqrt{80ag}\).

1. Find \(L\) in terms of \(a\). [5]
2. Find the initial acceleration of \(P\) in terms of \(g\). [3]

\(A\) and \(B\) are two points on a line of greatest slope of a plane inclined at \(55°\) to the horizontal. \(A\) is below the level of \(B\) and \(AB = 4\) m. A particle \(P\) of mass 2.5 kg is projected up the plane from \(A\) towards \(B\) and the speed of \(P\) at \(B\) is \(6.7 \text{ m s}^{-1}\). The coefficient of friction between the plane and \(P\) is 0.15. Find

1. the work done against the frictional force as \(P\) moves from \(A\) to \(B\), [3]
2. the initial speed of \(P\) at \(A\). [4]

\includegraphics{figure_6} The diagram shows a smooth narrow tube formed into a fixed vertical circle with centre \(O\) and radius 0.9 m. A light elastic string with modulus of elasticity 8 N and natural length 1.2 m has one end attached to the highest point \(A\) on the inside of the tube. The other end of the string is attached to a particle \(P\) of mass 0.2 kg. The particle is released from rest at the lowest point on the inside of the tube. By considering energy, calculate

1. the speed of \(P\) when it is at the same horizontal level as \(O\), [4]
2. the speed of \(P\) at the instant when the string becomes slack. [3]

5 Each of two light elastic strings, \(S _ { 1 }\) and \(S _ { 2 }\), has modulus of elasticity 16 N . The string \(S _ { 1 }\) has natural length 0.4 m and the string \(S _ { 2 }\) has natural length 0.5 m . One end of \(S _ { 1 }\) is attached to a fixed point \(A\) of a smooth horizontal table and the other end is attached to a particle \(P\) of mass 0.5 kg . One end of \(S _ { 2 }\) is attached to a fixed point \(B\) of the table and the other end is attached to \(P\). The distance \(A B\) is 1.5 m . The particle \(P\) is held at \(A\) and then released from rest.

1. Find the speed of \(P\) at the instant that \(S _ { 2 }\) becomes slack.
2. Find the greatest distance of \(P\) from \(A\) in the subsequent motion.

6 One end of a light elastic string of natural length 1.25 m and modulus of elasticity 20 N is attached to a fixed point \(O\). A particle \(P\) of mass 0.5 kg is attached to the other end of the string. \(P\) is held at rest at \(O\) and then released. When the extension of the string is \(x \mathrm {~m}\) the speed of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Show that \(v ^ { 2 } = - 32 x ^ { 2 } + 20 x + 25\).
2. Find the maximum speed of \(P\).
3. Find the acceleration of \(P\) when it is at its lowest point.

1 \includegraphics[max width=\textwidth, alt={}, center]{6fe2c5e0-0496-4fb4-95d2-354b90607b5b-2_643_218_264_959} A particle \(P\) of mass \(m \mathrm {~kg}\) is attached to the mid-point of a light elastic string of natural length 0.8 m and modulus of elasticity 8 N . One end of the string is attached to a fixed point \(A\) and the other end is attached to a fixed point \(B\) which is 2 m vertically below \(A\). When the particle is in equilibrium the distance \(A P\) is 1.1 m (see diagram). Find the value of \(m\).

One end of a light elastic string, of natural length \(a\) and modulus of elasticity \(\frac{16}{9}Mg\), is attached to a fixed point \(O\). A particle \(P\) of mass \(4M\) is attached to the other end of the string and hangs vertically in equilibrium. Another particle of mass \(2M\) is attached to \(P\) and the combined particle is then released from rest. The speed of the combined particle when it has descended a distance \(\frac{1}{4}a\) is \(v\). Find an expression for \(v\) in terms of \(g\) and \(a\). [6]

A particle \(P\) of mass \(0.2\text{ kg}\) is attached to one end of a light elastic string of natural length \(0.75\text{ m}\) and modulus of elasticity \(21\text{ N}\). The other end of the string is attached to a fixed point \(A\) which is \(0.8\text{ m}\) vertically above a smooth horizontal surface. \(P\) rests in equilibrium on the surface.

1. Find the magnitude of the force exerted on \(P\) by the surface. [2]

\(P\) is now projected horizontally along the surface with speed \(3\text{ m s}^{-1}\).

1. Calculate the extension of the string at the instant when \(P\) leaves the surface. [3]
2. Hence find the speed of \(P\) at the instant when it leaves the surface. [3]

\includegraphics{figure_4} A particle \(P\) of mass \(0.5\text{ kg}\) is projected along a smooth horizontal surface towards a fixed point \(A\). Initially \(P\) is at a point \(O\) on the surface, and after projection, \(P\) has a displacement from \(O\) of \(x\text{ m}\) and velocity \(v\text{ m s}^{-1}\). The particle \(P\) is connected to \(A\) by a light elastic string of natural length \(0.8\text{ m}\) and modulus of elasticity \(16\text{ N}\). The distance \(OA\) is \(1.6\text{ m}\) (see diagram). The motion of \(P\) is resisted by a force of magnitude \(24x^2\text{ N}\).

1. Show that \(v\frac{\text{d}v}{\text{d}x} = 32 - 40x - 48x^2\) while \(P\) is in motion and the string is stretched. [3]
2. The maximum value of \(v\) is \(4.5\). Find the initial value of \(v\). [5]

6 A van, of mass 1400 kg , is accelerating at a constant rate of \(0.2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) as it travels up a slope inclined at an angle \(\theta\) to the horizontal. The van experiences total resistance forces of 4000 N .
When the van is travelling at a speed of \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the power output of the van's engine is 91.1 kW . Find \(\theta\).
[0pt] [9 marks]

3. A particle \(P\) of mass \(m\) moves in a straight line away from the centre of the Earth. The Earth is modelled as a sphere of radius \(R\). When \(P\) is at a distance \(x , x \geqslant R\), from the centre of the Earth, the force exerted by the Earth on \(P\) is directed towards the centre of the Earth and has magnitude \(\frac { m g R ^ { 2 } } { x ^ { 2 } }\). When \(P\) is at a distance \(2 R\) from the surface of the Earth, the speed of \(P\) is \(\sqrt { \frac { g R } { 3 } }\). Assuming that air resistance can be ignored, find the distance of \(P\) from the surface of the Earth when the speed of \(P\) is \(2 \sqrt { \frac { g R } { 3 } }\).

1. A particle \(P\) moves from the point \(A\), with position vector ( \(2 \mathbf { i } + 4 \mathbf { j } + a \mathbf { k }\) ) m , where \(a\) is a positive constant, to the point \(B\), with position vector ( \(- \mathbf { i } + a \mathbf { j } - \mathbf { k }\) ) m , under the action of a constant force \(\mathbf { F } = ( 2 \mathbf { i } + a \mathbf { j } - 3 \mathbf { k } )\) N. The work done by \(\mathbf { F }\), as it moves the particle \(P\) from \(A\) to \(B\), is 3 J . Find the value of \(a\).
   (6)

2. A particle of mass 4 kg is moving along the horizontal \(x\)-axis under the action of a single force which acts in the positive \(x\)-direction. At time \(t\) seconds the force has magnitude \(\left( 1 + 3 t ^ { \frac { 1 } { 2 } } \right) \mathrm { N }\).
When \(t = 0\) the particle has speed \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in the positive \(x\)-direction. Find the work done by the force in the interval \(0 \leqslant t \leqslant 4\)