6.02i Conservation of energy: mechanical energy principle

A particle \(B\) of mass 0.5 kg is attached to one end of a light elastic string of natural length 0.75 m and modulus of elasticity 24.5 N. The other end of the string is attached to a fixed point \(A\). The particle is hanging in equilibrium at the point \(E\), vertically below \(A\).

1. Show that \(AE = 0.9\) m. [3]

The particle is held at \(A\) and released from rest. The particle first comes to instantaneous rest at the point \(C\).

1. Find the distance \(AC\). [5]
2. Show that while the string is taut, \(B\) is moving with simple harmonic motion with centre \(E\). [4]
3. Calculate the maximum speed of \(B\). [2]

One end \(A\) of a light elastic string \(AB\), of modulus of elasticity \(mg\) and natural length \(a\), is fixed to a point on a rough plane inclined at an angle \(\theta\) to the horizontal. The other end \(B\) of the string is attached to a particle of mass \(m\) which is held at rest on the plane. The string \(AB\) lies along a line of greatest slope of the plane, with \(B\) lower than \(A\) and \(AB = a\). The coefficient of friction between the particle and the plane is \(\mu\), where \(\mu < \tan \theta\). The particle is released from rest.

1. Show that when the particle comes to rest it has moved a distance \(2a(\sin \theta - \mu \cos \theta)\) down the plane. [6]
2. Given that there is no further motion, show that \(\mu \geqslant \frac{1}{3} \tan \theta\). [5]

A light elastic string, of natural length \(l\) and modulus of elasticity \(4mg\), has one end attached to a fixed point \(A\). The other end is attached to a particle \(P\) of mass \(m\). The particle hangs freely at rest in equilibrium at the point \(E\). The distance of \(E\) below \(A\) is \((l + e)\).

1. Find \(e\) in terms of \(l\). [2]

At time \(t = 0\), the particle is projected vertically downwards from \(E\) with speed \(\sqrt{gl}\).

1. Prove that, while the string is taut, \(P\) moves with simple harmonic motion. [5]
2. Find the amplitude of the simple harmonic motion. [3]
3. Find the time at which the string first goes slack. [4]

\includegraphics{figure_6} Two particles \(P\) and \(Q\) are attached to opposite ends of a light inextensible string which passes over a small smooth pulley at the top of a rough plane inclined at \(30°\) to the horizontal. \(P\) has mass \(0.2\text{ kg}\) and is held at rest on the plane. \(Q\) has mass \(0.2\text{ kg}\) and hangs freely. The string is taut (see diagram). The coefficient of friction between \(P\) and the plane is \(0.4\). The particle \(P\) is released.

1. State the tension in the string before \(P\) is released, and find the tension in the string after \(P\) is released. [6]

\(Q\) strikes the floor and remains at rest. \(P\) continues to move up the plane for a further distance of \(0.8\text{ m}\) before it comes to rest. \(P\) does not reach the pulley.

1. Find the speed of the particles immediately before \(Q\) strikes the floor. [5]
2. Calculate the magnitude of the contact force exerted on \(P\) by the plane while \(P\) is in motion. [3]

An eagle has caught a salmon of mass 3 kg to take to its nest. When the eagle is flying with speed \(8 \text{ m s}^{-1}\), it drops the salmon. The salmon falls a vertical distance of 13 metres back into the sea. The salmon is to be modelled as a particle. The salmon's weight is the only force that acts on it as it falls to the sea.

1. Calculate the kinetic energy of the salmon when it is dropped by the eagle. [2 marks]
2. Calculate the potential energy lost by the salmon as it falls to the sea. [2 marks]
3. 1. Find the kinetic energy of the salmon when it reaches the sea. [2 marks]
   2. Hence find the speed of the salmon when it reaches the sea. [2 marks]

An elastic string has natural length 1.5 metres and modulus of elasticity 120 newtons. One end of the string is attached to a fixed point, \(A\), on a rough plane inclined at \(20°\) to the horizontal. The other end of the elastic string is attached to a particle of mass 4 kg. The coefficient of friction between the particle and the plane is 0.8. The three points, \(A\), \(B\) and \(C\), lie on a line of greatest slope. The point \(C\) is \(x\) metres from \(A\), as shown in the diagram. The particle is released from rest at \(C\) and moves up the plane. \includegraphics{figure_8}

1. Show that, as the particle moves up the plane, the frictional force acting on the particle is 29.5 N, correct to three significant figures. [3 marks]
2. The particle comes to rest for an instant at \(B\), which is 2 metres from \(A\). The particle then starts to move back towards \(A\).
   1. Find \(x\). [8 marks]
   2. Find the acceleration of the particle as it starts to move back towards \(A\). [4 marks]

A stone, of mass \(0.3\) kg, is thrown with a speed of \(8 \text{ m s}^{-1}\) from a point at a height of \(5\) metres above a horizontal surface.

1. Calculate the initial kinetic energy of the stone. [2 marks]
2. 1. Find the kinetic energy of the stone when it hits the surface. [3 marks]
   2. Hence find the speed of the stone when it hits the surface. [2 marks]
   3. State one modelling assumption that you have made. [1 mark]

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]

A stone, of mass 0.9 kg, is projected vertically upwards with speed 10 ms\(^{-1}\) in a medium which exerts a constant resistance to motion. It comes to rest after rising a distance of 3.75 m. Find the magnitude of the non-gravitational resisting force acting on the stone. [5 marks]

A stone, of mass 1.5 kg, is projected horizontally with speed 4 ms\(^{-1}\) from a height of 7 m above horizontal ground.

1. Show that the stone travels about 4.78 m horizontally before it hits the ground. [4 marks]
2. Find the height of the stone above the ground when it has travelled half of this horizontal distance. [4 marks]
3. Calculate the potential energy lost by the stone as it moves from its point of projection to the ground. [2 marks]
4. Showing your method clearly, use your answer to part (c) to find the speed with which the stone hits the ground. [3 marks]
5. State two modelling assumptions that you have made in answering this question. [2 marks]

A particle of mass 0.5 kg is held at rest at a point \(P\), which is at the bottom of an inclined plane. The particle is given an impulse of 1.8 N s directed up a line of greatest slope of the plane.

1. Find the speed at which the particle starts to move. [2]

The particle subsequently moves up the plane to a point \(Q\), which is 0.3 m above the level of \(P\).

1. Given that the plane is smooth, find the speed of the particle at \(Q\). [4]

It is given instead that the plane is rough. The particle is now projected up the plane from \(P\) with initial speed 3 ms\(^{-1}\), and comes to rest at a point \(R\) which is 0.2 m above the level of \(P\).

1. Given that the plane is inclined at 30° to the horizontal, find the magnitude of the frictional force on the particle. [4]

A small ball of mass 0.2 kg is projected with speed \(11 \text{ ms}^{-1}\) up a line of greatest slope of a roof from a point \(A\) at the bottom of the roof. The ball remains in contact with the roof and moves up the line of greatest slope to the top of the roof at \(B\). The roof is rough and the coefficient of friction is \(\frac{1}{4}\). The distance \(AB\) is 5 m and \(AB\) is inclined at \(30°\) to the horizontal (see diagram).

1. Show that the speed of the ball when it reaches \(B\) is \(5.44 \text{ ms}^{-1}\), correct to 2 decimal places. [6]

The ball leaves the roof at \(B\) and moves freely under gravity. The point \(C\) is at the lower edge of the roof. The distance \(BC\) is 5 m and \(BC\) is inclined at \(30°\) to the horizontal.

1. Determine whether or not the ball hits the roof between \(B\) and \(C\). [7]

\(A\) 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]

Some tiles on a roof are being replaced. Each tile has a mass of 2 kg and the coefficient of friction between it and the existing roof is 0.75. The roof is at \(30°\) to the horizontal and the bottom of the roof is 6 m above horizontal ground, as shown in Fig. 4. \includegraphics{figure_4}

1. Calculate the limiting frictional force between a tile and the roof. A tile is placed on the roof. Does it slide? (Your answer should be supported by a calculation.) [5]
2. The tiles are raised 6 m from the ground, the only work done being against gravity. They are then slid 4 m up the roof and placed at the point A shown in Fig. 4.
   1. Show that each tile gains 156.8 J of gravitational potential energy. [3]
   2. Calculate the work done against friction per tile. [2]
   3. What average power is required to raise 10 tiles per minute from the ground to A? [2]
3. A tile is kicked from A directly down the roof. When the tile is at B, \(x\) m from the edge of the roof, its speed is \(4 \text{ m s}^{-1}\). It subsequently hits the ground travelling at \(9 \text{ m s}^{-1}\). In the motion of the tile from B to the ground, the work done against sliding and other resistances is 90 J. Use an energy method to find \(x\). [5]

A cyclist and her bicycle have a combined mass of 80 kg.

1. Initially, the cyclist accelerates from rest to 3 m s\(^{-1}\) against negligible resistances along a horizontal road.
   1. How much energy is gained by the cyclist and bicycle? [2]
   2. The cyclist travels 12 m during this acceleration. What is the average driving force on the bicycle? [2]
2. While exerting no driving force, the cyclist free-wheels down a hill. Her speed increases from 4 m s\(^{-1}\) to 10 m s\(^{-1}\). During this motion, the total work done against friction is 1600 J and the drop in vertical height is \(h\) m. Without assuming that the hill is uniform in either its angle or roughness, calculate \(h\). [5]
3. The cyclist reaches another horizontal stretch of road and there is now a constant resistance to motion of 40 N.
   1. When the power of the driving force on the bicycle is a constant 200 W, what constant speed can the cyclist maintain? [3]
   2. Find the power of the driving force on the bicycle when travelling at a speed of 0.5 m s\(^{-1}\) with an acceleration of 2 m s\(^{-2}\). [5]

A particle \(P\) of mass \(m\) kg is attached to one end of a light elastic string of natural length \(l\) m and modulus of elasticity \(\lambda\) N. The other end of the string is attached to a fixed point \(O\). \(P\) is released from rest at \(O\) and falls vertically downwards under gravity. The greatest distance below \(O\) reached by \(P\) is \(2l\) m.

1. Show that \(\lambda = 4mg\). [3 marks]
2. Find, in terms of \(l\) and \(g\), the speed with which \(P\) passes through the point \(A\), where \(OA = \frac{5l}{4}\) m. [6 marks]

A light elastic string, of natural length \(l\) m and modulus of elasticity \(\frac{mg}{2}\) newtons, has one end fastened to a fixed point \(O\). A particle \(P\), of mass \(m\) kg, is attached to the other end of the string. \(P\) hangs in equilibrium at the point \(E\), vertically below \(O\), where \(OE = (l + e)\) m

1. Find the numerical value of the ratio \(e : l\). [2 marks]

\(P\) is now pulled down a further distance \(\frac{3l}{2}\) m from \(E\) and is released from rest. In the subsequent motion, the string remains taut. At time \(t\) s after being released, \(P\) is at a distance \(x\) m below \(E\).

1. Write down a differential equation for the motion of \(P\) and show that the motion is simple harmonic. [4 marks]
2. Write down the period of the motion. [2 marks]
3. Find the speed with which \(P\) first passes through \(E\) again. [2 marks]
4. Show that the time taken by \(P\) after it is released to reach the point \(A\) above \(E\), where \(AE = \frac{3l}{4}\) m, is \(\frac{2\pi}{3}\sqrt{\frac{2l}{g}}\) s. [5 marks]

A particle \(P\) is attached to one end of a light inextensible string of length \(l\) m. The other end of the string is attached to a fixed point \(O\). When \(P\) is hanging at rest vertically below \(O\), it is given a horizontal speed \(u\) ms\(^{-1}\) and starts to move in a vertical circle. Given that the string becomes slack when it makes an angle of 120° with the downward vertical through \(O\),

1. show that \(u^2 = \frac{7gl}{2}\). [10 marks]
2. Find, in terms of \(l\), the greatest height above \(O\) reached by \(P\) in the subsequent motion. [7 marks]

A particle \(P\) of mass \(m\) kg is attached to one end of a light elastic string of natural length 0·5 m and modulus of elasticity \(\frac{mg}{2}\) N. The other end of the string is attached to a fixed point \(O\) and \(P\) hangs vertically below \(O\).

1. Find the stretched length of the string when \(P\) rests in equilibrium. [3 marks]
2. Find the elastic potential energy stored in the string in the equilibrium position. [2 marks]

\(P\), which is still attached to the string, is now held at rest at \(O\) and then lowered gently into its equilibrium position.

1. Find the work done by the weight of the particle as it moves from \(O\) to the equilibrium position. [2 marks]
2. Explain the discrepancy between your answers to parts (b) and (c). [1 mark]

A small bead \(P\), of mass \(m\) kg, can slide on a smooth circular ring, with centre \(O\) and radius \(r\) m, which is fixed in a vertical plane. \(P\) is projected from the lowest point \(L\) of the ring with speed \(\sqrt{(3gr)}\) ms\(^{-1}\). When \(P\) has reached a position such that \(OP\) makes an angle \(\theta\) with the downward vertical, as shown, its speed is \(v\) ms\(^{-1}\). \includegraphics{figure_5}

1. Show that \(v^2 = gr(1 + 2 \cos \theta)\). [5 marks]
2. Show that the magnitude of the reaction \(RN\) of the ring on the bead is given by $$R = mg(1 + 3 \cos \theta).$$ [4 marks]
3. Find the values of \(\cos \theta\) when
   1. \(P\) is instantaneously at rest,
   2. the reaction \(R\) is instantaneously zero. [2 marks]
4. Hence show that the ratio of the heights of \(P\) above \(L\) in cases (i) and (ii) is \(9:8\). [3 marks]

A light elastic string, of natural length 0·8 m, has one end fastened to a fixed point \(O\). The other end of the string is attached to a particle \(P\) of mass 0·5 kg. When \(P\) hangs in equilibrium, the length of the string is 1·5 m.

1. Calculate the modulus of elasticity of the string. [3 marks]

\(P\) is displaced to a point 0·5 m vertically below its equilibrium position and released from rest. \begin{enumerate}[label=(\alph*)] \setcounter{enumi}{1} \item Show that the subsequent motion of \(P\) is simple harmonic, with period 1·68 s. [5 marks] \item Calculate the maximum speed of \(P\) during its motion. [3 marks] \item Show that the time taken for \(P\) to first reach a distance 0·25 m from the point of release is 0·28 s, to 2 significant figures. [4 marks] \end{enumerate]

A particle of mass \(m\) kg is attached to the end \(B\) of a light elastic string \(AB\). The string has natural length \(l\) m and modulus of elasticity \(\lambda\) N. \includegraphics{figure_3} The end \(A\) is attached to a fixed point on a smooth plane inclined at an angle \(\alpha\) to the horizontal, as shown, and the particle rests in equilibrium with the length \(AB = \frac{5l}{4}\) m.

1. Show that \(\lambda = 4 mg \sin \alpha\). [3 marks]

The particle is now moved and held at rest at \(A\) with the string slack. It is then gently released so that it moves down the plane along a line of greatest slope.

1. Find the greatest distance from \(A\) that the particle reaches down the plane. [6 marks]

A particle of mass \(m\) kg, at a distance \(x\) m from the centre of the Earth, experiences a force of magnitude \(\frac{km}{x^2}\) N towards the centre of the Earth, where \(k\) is a constant. Given that the radius of the Earth is \(6.37 \times 10^6\) m, and that a 3 kg mass experiences a force of 30 N at the surface of the Earth,

1. calculate the value of \(k\), stating the units of your answer. [3 marks]

The 3 kg mass falls from rest at a distance \(x = 12.74 \times 10^6\) m from the centre of the Earth. Ignoring air resistance,

1. show that it reaches the surface of the Earth with speed \(7.98 \times 10^3\) ms\(^{-1}\). [7 marks]

In a simplified model, the particle is assumed to fall with a constant acceleration 10 ms\(^{-2}\). According to this model it attains the same speed as in (b), \(7.98 \times 10^3\) ms\(^{-1}\), at a distance \((12.74 - d) \times 10^6\) m from the centre of the Earth.

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

A particle \(P\) of mass 0.4 kg hangs by a light, inextensible string of length 20 cm whose other end is attached to a fixed point \(O\). It is given a horizontal velocity of 1.4 ms\(^{-1}\) so that it begins to move in a vertical circle. If in the ensuing motion the string makes an angle of \(\theta\) with the downward vertical through \(O\), show that

1. \(\theta\) cannot exceed 60°, [6 marks]
2. the tension, \(T\) N, in the string is given by \(T = 3.92(3 \cos \theta - 1)\). [4 marks]

If the string breaks when \(\cos \theta = \frac{3}{5}\) and \(P\) is ascending,

1. find the greatest height reached by \(P\) above the initial point of projection. [5 marks]