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

5 A light elastic string has natural length \(a \mathrm {~m}\) and modulus of elasticity \(\lambda \mathrm { N }\). When the length of the string is 1.6 m the tension is 4 N . When the length of the string is 2 m the tension is 6 N .

1. Find the values of \(a\) and \(\lambda\).
   One end of the string is attached to a fixed point \(O\) on a smooth horizontal surface. The other end of the string is attached to a particle \(P\) of mass 0.2 kg . The particle \(P\) moves with constant speed on the surface in a circle with centre \(O\) and radius 1.9 m .
2. Find the speed of \(P\).

5 A particle \(P\) of mass 0.6 kg is attached to one end of a light elastic string of natural length 0.8 m and modulus of elasticity 24 N . The other end of the string is attached to a fixed point \(A\), and \(P\) hangs in equilibrium.

1. Calculate the extension of the string. \(P\) is projected vertically downwards from the equilibrium position with speed \(4.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
2. Find the distance \(A P\) when the speed of \(P\) is \(3.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(P\) is below the equilibrium position.
3. Calculate the speed of \(P\) when it is 0.5 m above the equilibrium position.

4 \includegraphics[max width=\textwidth, alt={}, center]{b8e52188-f9a6-46fc-90bf-97965c6dd324-07_337_526_262_726} \includegraphics[max width=\textwidth, alt={}, center]{b8e52188-f9a6-46fc-90bf-97965c6dd324-07_111_116_486_1308} A particle \(P\) of mass 0.3 kg is attached to a fixed point \(A\) by a light elastic string of natural length 0.8 m and modulus of elasticity 16 N . The particle \(P\) moves in a horizontal circle which has centre \(O\). It is given that \(A O\) is vertical and that angle \(O A P\) is \(60 ^ { \circ }\) (see diagram). Calculate the speed of \(P\). [6]

2 \includegraphics[max width=\textwidth, alt={}, center]{fcf239a6-6558-43ec-b404-70aa349af6a9-2_319_874_968_639} A uniform rod \(A B\), of length 2 m and mass 10 kg , is freely hinged to a fixed point at the end \(B\). A light elastic string, of modulus of elasticity 200 N , has one end attached to the end \(A\) of the rod and the other end attached to a fixed point \(O\), which is in the same vertical plane as the rod. The rod is horizontal and in equilibrium, with \(O A = 3 \mathrm {~m}\) and angle \(O A B = 150 ^ { \circ }\) (see diagram). Find

1. the tension in the string,
2. the natural length of the string.

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 \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 light elastic string has natural length 1.5 m and modulus of elasticity 60 N . The string is stretched between two fixed points \(A\) and \(B\), which are at the same horizontal level and 2 m apart.

1. Find the tension in the string. A particle of weight \(W \mathrm {~N}\) is now attached to the mid-point of the string and the particle is in equilibrium at a point 0.75 m vertically below the mid-point of \(A B\).
2. Find the value of \(W\).

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 \(O\). The other end of the string is attached to a particle \(P\) of mass \(0.25 \mathrm {~kg} . P\) hangs in equilibrium below \(O\).

1. Calculate the distance \(O P\). The particle \(P\) is raised, and is released from rest at \(O\).
2. Calculate the speed of \(P\) when it passes through the equilibrium position.
3. Calculate the greatest value of the distance \(O P\) in the subsequent motion.

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.

7 A light elastic string has natural length 3 m and modulus of elasticity 45 N . A particle \(P\) of weight 6 N is attached to the mid-point of the string. The ends of the string are attached to fixed points \(A\) and \(B\) which lie in the same vertical line with \(A\) above \(B\) and \(A B = 4 \mathrm {~m}\). The particle \(P\) is released from rest at the point 1.5 m vertically below \(A\).

1. Calculate the distance \(P\) moves after its release before first coming to instantaneous rest at a point vertically above \(B\). (You may assume that at this point the part of the string joining \(P\) to \(B\) is slack.)
2. Show that the greatest speed of \(P\) occurs when it is 2.1 m below \(A\), and calculate this greatest speed.
3. Calculate the greatest magnitude of the acceleration of \(P\).

2 A light elastic string has natural length 4 m and modulus of elasticity 60 N . A particle \(P\) of mass 0.6 kg is attached to the mid-point of the string. The ends of the string are attached to fixed points \(A\) and \(B\) which lie in the same vertical line with \(A\) at a distance of 6 m above \(B\). \(P\) is projected vertically upwards from the point 2 m vertically above \(B\). In the subsequent motion, \(P\) comes to instantaneous rest at a distance of 2 m below \(A\).

1. Calculate the speed of projection of \(P\).
2. Calculate the distance of \(P\) from \(A\) at an instant when \(P\) has its greatest kinetic energy, and calculate this kinetic energy.

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 \(A\).

5 \includegraphics[max width=\textwidth, alt={}, center]{81be887c-ab01-4327-a5df-f25c68a6fdb6-2_337_517_1749_813} Two light elastic strings each have one end attached to a fixed horizontal beam. One string has natural length 0.6 m and modulus of elasticity 12 N ; the other string has natural length 0.7 m and modulus of elasticity 21 N . The other ends of the strings are attached to a small block \(B\) of weight \(W \mathrm {~N}\). The block hangs in equilibrium \(d \mathrm {~m}\) below the beam, with both strings vertical (see diagram).

1. Given that the tensions in the strings are equal, find \(d\) and \(W\). The small block is now raised vertically to the point 0.7 m below the beam, and then released from rest.
2. Find the greatest speed of the block in its subsequent motion.

7 A particle \(P\) of mass \(M \mathrm {~kg}\) is attached to one end of a light elastic string of natural length 0.8 m and modulus of elasticity 12.5 N . The other end of the string is attached to a fixed point \(A\). The particle is released from rest at \(A\) and falls vertically until it comes to instantaneous rest at the point \(B\). The greatest speed of \(P\) during its descent is \(4.4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when the extension of the string is \(e \mathrm {~m}\).

1. Show that \(e = 0.64 M\).
2. Find a second equation in \(e\) and \(M\), and hence find \(M\).
3. Calculate the distance \(A B\).

4 \includegraphics[max width=\textwidth, alt={}, center]{6b220343-1d64-4dbc-a42d-77967eef9c6d-06_264_839_260_653} A light elastic string has natural length 2 m and modulus of elasticity 39 N . The ends of the string are attached to fixed points \(A\) and \(B\) which are at the same horizontal level and 2.4 m apart. A particle \(P\) of mass \(m \mathrm {~kg}\) is attached to the mid-point of the string and hangs in equilibrium at a point 0.5 m below \(A B\) (see diagram).

1. Show that \(m = 0.9\). \(P\) is projected vertically downwards from the equilibrium position, and comes to instantaneous rest at a point 1.6 m below \(A B\).
2. Calculate the speed of projection of \(P\).

6 One end of a light elastic string of natural length 0.4 m and modulus of elasticity 8 N is attached to a fixed point \(O\) on a smooth horizontal plane. The other end of the string is attached to a particle \(P\) of mass 0.2 kg which moves on the plane in a circular path with centre \(O\). The speed of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the extension of the string is \(x \mathrm {~m}\).

1. Given that \(v = 2.5\), find \(x\).
   It is given instead that the kinetic energy of \(P\) is twice the elastic potential energy stored in the string.
2. Form two simultaneous equations and hence find \(x\) and \(v\).

3 A particle \(P\) of mass 0.4 kg is projected horizontally along a smooth horizontal plane from a point \(O\). After projection the velocity of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and its displacement from \(O\) is \(x \mathrm {~m}\). A force of magnitude \(8 x \mathrm {~N}\) directed away from \(O\) acts on \(P\) and a force of magnitude ( \(2 \mathrm { e } ^ { - x } + 4\) ) N opposes the motion of \(P\). One end of a light elastic string of natural length 0.5 m is attached to \(O\) and the other end of the string is attached to \(P\).

1. Show that \(v \frac { \mathrm {~d} v } { \mathrm {~d} x } = 20 x - 10 - 5 \mathrm { e } ^ { - x }\) before the elastic string becomes taut.
2. Given that the initial velocity of \(P\) is \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), find \(v\) when the string first becomes taut.
   When the string is taut, the acceleration of \(P\) is proportional to \(\mathrm { e } ^ { - x }\).
3. Find the modulus of elasticity of the string.

5 A particle \(P\) 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 plane inclined at \(30 ^ { \circ }\) to the horizontal. \(O A\) is a line of greatest slope of the plane with \(A\) below the level of \(O\) and \(O A = 0.8 \mathrm {~m}\). The particle \(P\) is released from rest at \(A\).

1. Find the initial acceleration of \(P\).
2. Find the greatest speed of \(P\). \(6 \quad A\) and \(B\) are two fixed points on a vertical axis with \(A 0.6 \mathrm {~m}\) above \(B\). A particle \(P\) of mass 0.3 kg is attached to \(A\) by a light inextensible string of length 0.5 m . The particle \(P\) is attached to \(B\) by a light elastic string with modulus of elasticity 46 N . The particle \(P\) moves with constant angular speed \(8 \mathrm { rad } \mathrm { s } ^ { - 1 }\) in a horizontal circle with centre at the mid-point of \(A B\).
3. Find the speed of \(P\).
4. Calculate the tension in the string \(B P\) and hence find the natural length of this string. \includegraphics[max width=\textwidth, alt={}, center]{4cd525d5-d59b-4ab9-85a3-fc3d97fd09fc-10_540_574_260_781} \(A B C\) is the cross-section through the centre of mass of a uniform prism which rests with \(A B\) on a rough horizontal surface. \(A B = 0.4 \mathrm {~m}\) and \(C\) is 0.9 m above the surface (see diagram). The prism is on the point of toppling about its edge through \(B\).
5. Show that angle \(B A C = 48.4 ^ { \circ }\), correct to 3 significant figures.
   A force of magnitude 18 N acting in the plane of the cross-section and perpendicular to \(A C\) is now applied to the prism at \(C\). The prism is on the point of rotating about its edge through \(A\).
6. Calculate the weight of the prism.
7. Given also that the prism is on the point of slipping, calculate the coefficient of friction between the prism and the surface.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

3 A light elastic string has natural length \(a\) and modulus of elasticity 12 mg . One end of the string is attached to a fixed point \(O\). The other end of the string is attached to a particle of mass \(m\). The particle hangs in equilibrium vertically below \(O\). The particle is pulled vertically down and released from rest with the extension of the string equal to \(e\), where \(\mathrm { e } > \frac { 1 } { 3 } \mathrm { a }\). In the subsequent motion the particle has speed \(\sqrt { 2 \mathrm { ga } }\) when it has ascended a distance \(\frac { 1 } { 3 } a\). Find \(e\) in terms of \(a\). \includegraphics[max width=\textwidth, alt={}, center]{b10c65ef-dafd-4746-be5b-789130b7d030-06_488_496_269_781} A uniform lamina \(A E C F\) is formed by removing two identical triangles \(B C E\) and \(C D F\) from a square lamina \(A B C D\). The square has side \(3 a\) and \(E B = D F = h\) (see diagram).

1. Find the distance of the centre of mass of the lamina \(A E C F\) from \(A D\) and from \(A B\), giving your answers in terms of \(a\) and \(h\).
   The lamina \(A E C F\) is placed vertically on its edge \(A E\) on a horizontal plane.
2. Find, in terms of \(a\), the set of values of \(h\) for which the lamina remains in equilibrium.

2 A light elastic string has natural length \(a\) and modulus of elasticity 4 mg . One end of the string is fixed to a point \(O\) on a smooth horizontal surface. A particle \(P\) of mass \(m\) is attached to the other end of the string. The particle \(P\) is projected along the surface in the direction \(O P\). When the length of the string is \(\frac { 5 } { 4 } a\), the speed of \(P\) is \(v\). When the length of the string is \(\frac { 3 } { 2 } a\), the speed of \(P\) is \(\frac { 1 } { 2 } v\).

1. Find an expression for \(v\) in terms of \(a\) and \(g\).
2. Find, in terms of \(g\), the acceleration of \(P\) when the stretched length of the string is \(\frac { 3 } { 2 } a\). \includegraphics[max width=\textwidth, alt={}, center]{7febbd80-4cbb-4b2e-b022-d6a20e7e13aa-04_552_1059_264_502} A smooth cylinder is fixed to a rough horizontal surface with its axis of symmetry horizontal. A uniform rod \(A B\), of length \(4 a\) and weight \(W\), rests against the surface of the cylinder. The end \(A\) of the rod is in contact with the horizontal surface. The vertical plane containing the rod \(A B\) is perpendicular to the axis of the cylinder. The point of contact between the rod and the cylinder is \(C\), where \(A C = 3 a\). The angle between the rod and the horizontal surface is \(\theta\) where \(\tan \theta = \frac { 3 } { 4 }\) (see diagram). The coefficient of friction between the rod and the horizontal surface is \(\frac { 6 } { 7 }\). A particle of weight \(k W\) is attached to the rod at \(B\). The rod is about to slip. The normal reaction between the rod and the cylinder is \(N\).

6. One end of a light elastic string, of natural length 5 l and modulus of elasticity 20 mg , is attached to a fixed point \(A\). A particle \(P\) of mass \(2 m\) is attached to the free end of the string and \(P\) hangs freely in equilibrium at the point \(B\).

1. Find the distance \(A B\).
   (3) The particle is now pulled vertically downwards from \(B\) to the point \(C\) and released from rest. In the subsequent motion the string does not become slack.
2. Show that \(P\) moves with simple harmonic motion with centre \(B\).
3. Find the period of this motion. The greatest speed of \(P\) during this motion is \(\frac { 1 } { 5 } \sqrt { g l }\)
4. Find the amplitude of this motion. The point \(D\) is the midpoint of \(B C\) and the point \(E\) is the highest point reached by \(P\).
5. Find the time taken by \(P\) to move directly from \(D\) to \(E\).

1. A particle of mass 0.9 kg is attached to one end of a light elastic string, of natural length 1.2 m and modulus of elasticity 29.4 N . The other end of the string is attached to a fixed point \(A\) on a ceiling.

The particle is held at \(A\) and then released from rest. The particle first comes to instantaneous rest at the point \(B\). Find the distance \(A B\).
(5)

7. \begin{figure}[h] \end{figure} The fixed points \(A\) and \(B\) are 4.2 m apart on a smooth horizontal floor. One end of a light elastic spring, of natural length 1.8 m and modulus of elasticity 20 N , is attached to a particle \(P\) and the other end is attached to \(A\). One end of another light elastic spring, of natural length 0.9 m and modulus of elasticity 15 N , is attached to \(P\) and the other end is attached to \(B\). The particle \(P\) rests in equilibrium at the point \(O\), where \(A O B\) is a straight line, as shown in Figure 5.

1. Show that \(A O = 2.7 \mathrm {~m}\). The particle \(P\) now receives an impulse acting in the direction \(O B\) and moves away from \(O\) towards \(B\). In the subsequent motion \(P\) does not reach \(B\).
2. Show that \(P\) moves with simple harmonic motion about centre \(O\). The mass of \(P\) is 10 kg and the magnitude of the impulse is \(J \mathrm { Ns }\). Given that \(P\) first comes to instantaneous rest at the point \(C\) where \(A C = 2.9 \mathrm {~m}\),
3. 1. find the value of \(J\),
   2. find the time taken by \(P\) to travel a total distance of 0.5 m from when it first leaves \(O\).

4. \begin{figure}[h] \end{figure} The ends of a light elastic string, of natural length \(4 l\) and modulus of elasticity \(\lambda\), are attached to two fixed points \(A\) and \(B\), where \(A B\) is horizontal and \(A B = 4 l\). A particle \(P\) of mass \(2 m\) is attached to the midpoint of the string. The particle hangs freely in equilibrium at a distance \(\frac { 3 } { 2 } l\) vertically below the midpoint of \(A B\), as shown in Figure 2.

1. Show that \(\lambda = \frac { 20 } { 3 } m g\). The particle is pulled vertically downwards from its equilibrium position until the total length of the string is 6l. The particle is then released from rest.
2. Show that \(P\) comes to instantaneous rest before reaching the line \(A B\).