6.02i Conservation of energy: mechanical energy principle

6 \includegraphics[max width=\textwidth, alt={}, center]{ae809dfc-c5af-4c0a-9c88-009949d3e9f9-4_324_1267_794_440} A particle \(P\) of mass 0.35 kg is attached to the mid-point of a light elastic string of natural length 4 m . The ends of the string are attached to fixed points \(A\) and \(B\) which are 4.8 m apart at the same horizontal level. \(P\) hangs in equilibrium at a point 0.7 m vertically below the mid-point \(M\) of \(A B\) (see diagram).

1. Find the tension in the string and hence show that the modulus of elasticity of the string is 25 N . \(P\) is now held at rest at a point 1.8 m vertically below \(M\), and is then released.
2. Find the speed with which \(P\) passes through \(M\).

7 One end of a light elastic string of natural length 3 m and modulus of elasticity 24 N is attached to a fixed point \(O\). A particle \(P\) of mass 0.4 kg is attached to the other end of the string. \(P\) is projected vertically downwards from \(O\) with initial speed \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). When the extension of the string is \(x \mathrm {~m}\) the speed of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Show that \(v ^ { 2 } = 64 + 20 x - 20 x ^ { 2 }\).
2. Find the greatest speed of the particle.
3. Calculate the greatest tension in the string.

3 \includegraphics[max width=\textwidth, alt={}, center]{18398d27-15eb-4515-8210-4f0f614d5b28-2_247_839_1375_653} A light elastic string of natural length 1.2 m and modulus of elasticity 24 N is attached to fixed points \(A\) and \(B\) on a smooth horizontal surface, where \(A B = 1.2 \mathrm {~m}\). A particle \(P\) is attached to the mid-point of the string. \(P\) is projected with speed \(0.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) along the surface in a direction perpendicular to \(A B\) (see diagram). \(P\) comes to instantaneous rest at a distance 0.25 m from \(A B\).

1. Show that the mass of \(P\) is 0.8 kg .
2. Calculate the greatest deceleration of \(P\).

4 One end of a light elastic string of natural length 0.5 m and modulus of elasticity 12 N is attached to a fixed point \(O\). The other end of the string is attached to a particle \(P\) of mass \(0.24 \mathrm {~kg} . P\) is projected vertically upwards with speed \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a position 0.8 m vertically below \(O\).

1. Calculate the speed of the particle when it is moving upwards with zero acceleration.
2. Show that the particle moves 0.6 m while it is moving upwards with constant acceleration.

4 The ends of a light elastic string of natural length 0.8 m and modulus of elasticity \(\lambda \mathrm { N }\) are attached to fixed points \(A\) and \(B\) which are 1.2 m apart at the same horizontal level. A particle of mass 0.3 kg is attached to the centre of the string, and released from rest at the mid-point of \(A B\). The particle descends 0.32 m vertically before coming to instantaneous rest.

1. Calculate \(\lambda\).
2. Calculate the speed of the particle when it is 0.25 m below \(A B\).

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 \(A B\).

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 \(P\).

3 A particle \(P\) of mass 0.2 kg is projected horizontally from a fixed point \(O\), and moves in a straight line on a smooth horizontal surface. A force of magnitude \(0.4 x \mathrm {~N}\) acts on \(P\) in the direction \(P O\), where \(x \mathrm {~m}\) is the displacement of \(P\) from \(O\).

1. Given that \(P\) comes to instantaneous rest when \(x = 2.5\), find the initial kinetic energy of \(P\).
2. Find the value of \(x\) on the first occasion when the speed of \(P\) is \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

5 A light elastic string has natural length 3 m and modulus of elasticity 45 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 on a line of greatest slope of a smooth plane inclined at \(30 ^ { \circ }\) to the horizontal. The distance \(A B\) is 4 m , and \(A\) is higher than \(B\).

1. Calculate the distance \(A P\) when \(P\) rests on the slope in equilibrium. \(P\) is released from rest at the point between \(A\) and \(B\) where \(A P = 2.5 \mathrm {~m}\).
2. Find the maximum speed of \(P\).
3. Show that \(P\) is at rest when \(A P = 1.6 \mathrm {~m}\).

2 A particle \(P\) of mass 0.4 kg is attached to one end of a light elastic string of natural length 1.2 m and modulus of elasticity 19.2 N . The other end of the string is attached to a fixed point \(A\). The particle \(P\) is released from rest at the point 2.7 m vertically above \(A\). Calculate

1. the initial acceleration of \(P\),
2. the speed of \(P\) when it reaches \(A\).

3 A particle \(P\) of mass 0.2 kg is attached to one end of a light elastic string of natural length 1.6 m and modulus of elasticity 18 N . The other end of the string is attached to a fixed point \(O\) which is 1.6 m above a smooth horizontal surface. \(P\) is placed on the surface vertically below \(O\) and then projected horizontally. \(P\) moves with initial speed \(1.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in a straight line on the surface. Show that, when \(O P = 1.8 \mathrm {~m}\),

1. \(P\) is at instantaneous rest,
2. \(P\) is on the point of losing contact with the surface.

5 \includegraphics[max width=\textwidth, alt={}, center]{c85aa042-7b8c-44cc-b579-a5deef91e7e5-3_341_529_260_808} A block \(B\) of mass 3 kg is attached to one end of a light elastic string of modulus of elasticity 70 N and natural length 1.4 m . The other end of the string is attached to a particle \(P\) of mass 0.3 kg . \(B\) is at rest 0.9 m from the edge of a horizontal table and the string passes over a small smooth pulley at the edge of the table. \(P\) is released from rest at a point next to the pulley and falls vertically. At the first instant when \(P\) is 0.8 m below the pulley and descending, \(B\) is in limiting equilibrium with the part of the string attached to \(B\) horizontal (see diagram).

1. Calculate the speed of \(P\) when \(B\) is first in limiting equilibrium.
2. Find the coefficient of friction between \(B\) and the table.

5 One end of a light elastic string \(S _ { 1 }\) of modulus of elasticity 20 N and natural length 0.5 m is attached to a fixed point \(O\). The other end of \(S _ { 1 }\) is attached to a particle \(P\) of mass \(0.4 \mathrm {~kg} . P\) hangs in equilibrium vertically below \(O\).

1. Find the distance \(O P\). The opposite ends of a light inextensible string \(S _ { 2 }\) of length \(l \mathrm {~m}\) are now attached to \(O\) and \(P\) respectively. The elastic string \(S _ { 1 }\) remains attached to \(O\) and \(P\). The particle \(P\) hangs in equilibrium vertically below \(O\).
2. Find the tension in the inextensible string \(S _ { 2 }\) for each of the following cases:
   (a) \(l < 0.5\);
   (b) \(l > 0.6\);
   (c) \(l = 0.54\). In the case \(l = 0.54\), the inextensible string \(S _ { 2 }\) suddenly breaks and \(P\) begins to descend vertically.
3. Calculate the greatest speed of \(P\) in the subsequent motion.

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.

6 \includegraphics[max width=\textwidth, alt={}, center]{9c82b387-8e5e-48b9-973d-5337b4e56a66-3_652_618_849_762} A particle \(P\) of mass 0.6 kg is attached to one end of a light elastic string of natural length 1.5 m and modulus of elasticity 9 N . The string passes through a small smooth ring \(R\) fixed at a height of 0.4 m above a rough horizontal surface. The other end of the string is attached to a fixed point \(O\) which is 1.5 m vertically above \(R\). The points \(A\) and \(B\) are on the horizontal surface, and \(B\) is vertically below \(R\). When \(P\) is on the surface between \(A\) and \(B , R P\) makes an acute angle \(\theta ^ { \circ }\) with the horizontal (see diagram).

1. Show that the normal force exerted on \(P\) by the surface has magnitude 3.6 N , for all values of \(\theta\). \(P\) is projected with speed \(2.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) towards \(B\) from its initial position at \(A\) where \(\theta = 30\). The speed of \(P\) when it passes through \(B\) is \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
2. Find the work done against friction as \(P\) moves from \(A\) to \(B\).
3. Calculate the value of the coefficient of friction between \(P\) and the surface.

2 One end of a light elastic string of natural length 0.5 m and modulus of elasticity 30 N is attached to a fixed point \(O\). The other end of the string is attached to a particle \(P\) which hangs in equilibrium vertically below \(O\), with \(O P = 0.8 \mathrm {~m}\).

1. Show that the mass of \(P\) is 1.8 kg . The particle is pulled vertically downwards and released from rest from the point where \(O P = 1.2 \mathrm {~m}\).
2. Find the speed of \(P\) at the instant when the string first becomes slack.

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 \(e\).

5 \includegraphics[max width=\textwidth, alt={}, center]{8f8492a7-8a83-4eb2-81ee-99b4a385b704-3_876_483_260_840} A uniform triangular prism of weight 20 N rests on a horizontal table. \(A B C\) is the cross-section through the centre of mass of the prism, where \(B C = 0.5 \mathrm {~m} , A B = 0.4 \mathrm {~m} , A C = 0.3 \mathrm {~m}\) and angle \(B A C = 90 ^ { \circ }\). The vertical plane \(A B C\) is perpendicular to the edge of the table. The point \(D\) on \(A C\) is at the edge of the table, and \(A D = 0.25 \mathrm {~m}\). One end of a light elastic string of natural length 0.6 m and modulus of elasticity 48 N is attached to \(C\) and a particle of mass 2.5 kg is attached to the other end of the string. The particle is released from rest at \(C\) and falls vertically (see diagram).

1. Show that the tension in the string is 60 N at the instant when the prism topples.
2. Calculate the speed of the particle at the instant when the prism topples.

2 One end of a light elastic string of natural length 0.4 m is attached to a fixed point \(O\). The other end of the string is attached to a particle of weight 5 N which hangs in equilibrium 0.6 m vertically below \(O\).

1. Find the modulus of elasticity of the string. The particle is projected vertically upwards from the equilibrium position and comes to instantaneous rest after travelling 0.3 m upwards.
2. Calculate the speed of projection of the particle.
3. Calculate the greatest extension of the string in the subsequent motion.

5 A particle of mass 0.3 kg is attached to one end of a light elastic string of natural length 0.8 m and modulus of elasticity 6 N . The other end of the string is attached to a fixed point \(O\). The particle is projected vertically downwards from \(O\) with initial speed \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Calculate the greatest speed of the particle during its descent.
2. Find the greatest distance of the particle below \(O\). \includegraphics[max width=\textwidth, alt={}, center]{2b0425b2-2f8f-491a-996c-3d3b589bd7df-12_558_554_260_794} The end \(A\) of a non-uniform rod \(A B\) of length 0.6 m and weight 8 N rests on a rough horizontal plane, with \(A B\) inclined at \(60 ^ { \circ }\) to the horizontal. Equilibrium is maintained by a force of magnitude 3 N applied to the rod at \(B\). This force acts at \(30 ^ { \circ }\) above the horizontal in the vertical plane containing the rod (see diagram).
3. Find the distance of the centre of mass of the rod from \(A\).
   The 3 N force is removed, and the rod is held in equilibrium by a force of magnitude \(P \mathrm {~N}\) applied at \(B\), acting in the vertical plane containing the rod, at an angle of \(30 ^ { \circ }\) below the horizontal.
4. Calculate \(P\).
   In one of the two situations described, the \(\operatorname { rod } A B\) is in limiting equilibrium.
5. Find the coefficient of friction at \(A\). \(7 \quad\) A particle \(P\) is projected from a point \(O\) with speed \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\). At time \(t \mathrm {~s}\) after projection the horizontal and vertically upwards displacements of \(P\) from \(O\) are \(x \mathrm {~m}\) and \(y \mathrm {~m}\) respectively. The equation of the trajectory of \(P\) is \(y = 2 x - \frac { 25 x ^ { 2 } } { V ^ { 2 } }\).
6. Write down the value of \(\tan \theta\), where \(\theta\) is the angle of projection of \(P\).
   When \(t = 4 , P\) passes through the point \(A\) where \(x = y = a\).
7. Calculate \(V\) and \(a\).
8. Find the direction of motion of \(P\) when it passes through \(A\).

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 \(P\).

7 A particle \(P\) of mass 0.5 kg is attached to a fixed point \(O\) by a light elastic string of natural length 1 m and modulus of elasticity 16 N . The particle \(P\) is projected vertically upwards from \(O\) with speed \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). A resisting force of magnitude \(0.1 x ^ { 2 } \mathrm {~N}\) acts on \(P\) when \(P\) has displacement \(x \mathrm {~m}\) above \(O\). After projection the upwards velocity of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Show that, before the string becomes taut, \(v \frac { \mathrm {~d} v } { \mathrm {~d} x } = - 10 - 0.2 x ^ { 2 }\).
2. Find the velocity of \(P\) at the instant the string becomes taut.
3. Find an expression for the acceleration of \(P\) while it is moving upwards after the string becomes taut.
4. Verify that \(P\) comes to instantaneous rest before the extension of the string is 0.5 m .
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

4 A particle \(P\) of mass 0.5 kg is attached to one end of a light elastic string of natural length 0.8 m and modulus of elasticity 16 N . The other end of the string is attached to a fixed point \(O\). The particle \(P\) is released from rest at the point 0.8 m vertically below \(O\). When the extension of the string is \(x \mathrm {~m}\), the downwards velocity of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and a force of magnitude \(25 x ^ { 2 } \mathrm {~N}\) opposes the motion of \(P\).

1. Show that, when \(P\) is moving downwards, \(v \frac { \mathrm {~d} v } { \mathrm {~d} x } = 10 - 40 x - 50 x ^ { 2 }\).
2. For the instant when \(P\) has its greatest downwards speed, find the kinetic energy of \(P\) and the elastic potential energy stored in the string.

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.

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 24 N . The other end of the string is attached to a fixed point \(O\). The particle \(P\) is released from rest at the point 0.4 m vertically below \(O\).

1. Find the greatest speed of \(P\).
2. Calculate the greatest distance of \(P\) below \(O\).
   \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{b8e52188-f9a6-46fc-90bf-97965c6dd324-10_608_611_258_767} \captionsetup{labelformat=empty} \caption{Fig. 1}
   \end{figure} Fig. 1 shows the cross-section of a solid cylinder through which a cylindrical hole has been drilled to make a uniform prism. The radius of the cylinder is \(5 r\) and the radius of the hole is \(r\). The centre of the hole is a distance \(2 r\) from the centre of the cylinder.