6.02i Conservation of energy: mechanical energy principle

\includegraphics{figure_6_1} **Fig. 6.1** Fig. 6.1 shows particles \(A\) and \(B\), of masses 4 kg and 3 kg respectively, attached to the ends of a light inextensible string that passes over a small smooth pulley. The pulley is fixed at the top of a plane which is inclined at an angle of 30° to the horizontal. \(A\) hangs freely below the pulley and \(B\) is on the inclined plane. The string is taut and the section of the string between \(B\) and the pulley is parallel to a line of greatest slope of the plane.

1. It is given that the plane is rough and the particles are in limiting equilibrium. Find the coefficient of friction between \(B\) and the plane. [6]
2. \includegraphics{figure_6_2} **Fig. 6.2** It is given instead that the plane is smooth and the particles are released from rest when the difference in the vertical heights of the particles is 1 m (see Fig. 6.2). Use an energy method to find the speed of the particles at the instant when the particles are at the same horizontal level. [6]

A block of mass 15 kg slides down a line of greatest slope of an inclined plane. The top of the plane is at a vertical height of 1.6 m above the level of the bottom of the plane. The speed of the block at the top of the plane is 2 m s\(^{-1}\) and the speed of the block at the bottom of the plane is 4 m s\(^{-1}\). Find the work done against the resistance to motion of the block. [4]

\includegraphics{figure_3} A block of mass 10 kg is at rest on a rough plane inclined at an angle of 30° to the horizontal. A force of 120 N is applied to the block at an angle of 20° above a line of greatest slope (see diagram). There is a force resisting the motion of the block and 200 J of work is done against this force when the block has moved a distance of 5 m up the plane from rest. Find the speed of the block when it has moved a distance of 5 m up the plane from rest. [5]

A particle \(A\) of mass 0.5 kg is projected vertically upwards from horizontal ground with speed 25 m s\(^{-1}\).

1. Find the speed of \(A\) when it reaches a height of 20 m above the ground. [2]

When \(A\) reaches a height of 20 m, it collides with a particle \(B\) of mass 0.3 kg which is moving downwards in the same vertical line as \(A\) with speed 32.5 m s\(^{-1}\). In the collision between the two particles, \(B\) is brought to instantaneous rest.

1. Show that the velocity of \(A\) immediately after the collision is 4.5 m s\(^{-1}\) downwards. [2]
2. Find the time interval between \(A\) and \(B\) reaching the ground. You should assume that \(A\) does not bounce when it reaches the ground. [4]

\includegraphics{figure_2} A particle of mass \(7.5\) kg, starting from rest at \(A\), slides down an inclined plane \(AB\). The point \(B\) is \(12.5\) metres vertically below the level of \(A\), as shown in the diagram.

1. Given that the plane is smooth, use an energy method to find the speed of the particle at \(B\). [2]
2. It is given instead that the plane is rough and the particle reaches \(B\) with a speed of \(8 \text{ ms}^{-1}\). The plane is \(25\) m long and the constant frictional force has magnitude \(F\) N. Find the value of \(F\). [3]

A car of mass \(1200 \text{ kg}\) travels along a horizontal straight road. The power provided by the car's engine is constant and equal to \(20 \text{ kW}\). The resistance to the car's motion is constant and equal to \(500 \text{ N}\). The car passes through the points \(A\) and \(B\) with speeds \(10 \text{ m s}^{-1}\) and \(25 \text{ m s}^{-1}\) respectively. The car takes \(30.5 \text{ s}\) to travel from \(A\) to \(B\).

1. Find the acceleration of the car at \(A\). [4]
2. By considering work and energy, find the distance \(AB\). [8]

\includegraphics{figure_2} Particles \(A\) and \(B\), of masses 0.35 kg and 0.15 kg respectively, are attached to the ends of a light inextensible string. \(A\) is held at rest on a smooth horizontal surface with the string passing over a small smooth pulley fixed at the edge of the surface. \(B\) hangs vertically below the pulley at a distance \(h\) m above the floor (see diagram). \(A\) is released and the particles move. \(B\) reaches the floor and \(A\) subsequently reaches the pulley with a speed of \(3 \text{ m s}^{-1}\).

1. Explain briefly why the speed with which \(B\) reaches the floor is \(3 \text{ m s}^{-1}\). [1]
2. Find the value of \(h\). [4]

A lorry of mass 12 000 kg moves up a straight hill of length 500 m, starting at the bottom with a speed of \(24 \text{ m s}^{-1}\) and reaching the top with a speed of \(16 \text{ m s}^{-1}\). The top of the hill is 25 m above the level of the bottom of the hill. The resistance to motion of the lorry is 7500 N. Find the driving force of the lorry. [6]

A car has mass \(1250 \text{ kg}\).

1. The car is moving along a straight level road at a constant speed of \(36 \text{ m s}^{-1}\) and is subject to a constant resistance of magnitude \(850 \text{ N}\). Find, in kW, the rate at which the engine of the car is working. [2]
2. The car travels at a constant speed up a hill and is subject to the same resistance as in part (i). The hill is inclined at an angle of \(\theta°\) to the horizontal, where \(\sin \theta° = 0.1\), and the engine is working at \(63 \text{ kW}\). Find the speed of the car. [3]
3. The car descends the same hill with the engine of the car working at a constant rate of \(20 \text{ kW}\). The resistance is not constant. The initial speed of the car is \(20 \text{ m s}^{-1}\). Eight seconds later the car has speed \(24 \text{ m s}^{-1}\) and has moved \(176 \text{ m}\) down the hill. Use an energy method to find the total work done against the resistance during the eight seconds. [5]

\includegraphics{figure_7} The diagram shows a triangular block with sloping faces inclined to the horizontal at \(45°\) and \(30°\). Particle \(A\) of mass \(0.8 \text{ kg}\) lies on the face inclined at \(45°\) and particle \(B\) of mass \(1.2 \text{ kg}\) lies on the face inclined at \(30°\). The particles are connected by a light inextensible string which passes over a small smooth pulley \(P\) fixed at the top of the faces. The parts \(AP\) and \(BP\) of the string are parallel to lines of greatest slope of the respective faces. The particles are released from rest with both parts of the string taut. In the subsequent motion neither particle reaches the pulley and neither particle reaches the bottom of a face.

1. Given that both faces are smooth, find the speed of \(A\) after each particle has travelled a distance of \(0.4 \text{ m}\). [6]
2. It is given instead that both faces are rough. The coefficient of friction between each particle and a face of the block is \(\mu\). Find the value of \(\mu\) for which the system is in limiting equilibrium. [6]

\includegraphics{figure_4} Two particles \(A\) and \(B\), of masses \(0.8\text{ kg}\) and \(1.6\text{ kg}\) respectively, are connected by a light inextensible string. Particle \(A\) is placed on a smooth plane inclined at an angle \(\theta\) to the horizontal, where \(\sin \theta = \frac{3}{5}\). The string passes over a small smooth pulley \(P\) fixed at the top of the plane, and \(B\) hangs freely (see diagram). The section \(AP\) of the string is parallel to a line of greatest slope of the plane. The particles are released from rest with both sections of the string taut. Use an energy method to find the speed of the particles after each particle has moved a distance of \(0.5\text{ m}\), assuming that \(A\) has not yet reached the pulley. [6]

A constant resistance to motion of magnitude 350 N acts on a car of mass 1250 kg. The engine of the car exerts a constant driving force of 1200 N. The car travels along a road inclined at an angle of \(\theta\) to the horizontal, where \(\sin \theta = 0.05\). Find the speed of the car when it has moved 100 m from rest in each of the following cases. • The car is moving up the hill. • The car is moving down the hill. [7]

\includegraphics{figure_5} Two particles \(A\) and \(B\), of masses 0.4 kg and 0.2 kg respectively, are connected by a light inextensible string which passes over a fixed smooth pulley. Both \(A\) and \(B\) are 0.5 m above the ground. The particles hang vertically (see diagram). The particles are released from rest. In the subsequent motion \(B\) does not reach the pulley and \(A\) remains at rest after reaching the ground.

1. For the motion before \(A\) reaches the ground, show that the magnitude of the acceleration of each particle is \(\frac{10}{3}\) m s\(^{-2}\) and find the tension in the string. [4]
2. Find the maximum height of \(B\) above the ground. [4]

A particle of mass \(0.4\) kg is projected with a speed of \(12\) m s\(^{-1}\) up a line of greatest slope of a smooth plane inclined at \(30°\) to the horizontal.

1. Find the initial kinetic energy of the particle. [1]
2. Use an energy method to find the distance the particle moves up the plane before coming to instantaneous rest. [3]

A car of mass \(900\) kg is moving on a straight horizontal road \(ABCD\). There is a constant resistance of magnitude \(800\) N in the sections \(AB\) and \(BC\), and a constant resistance of magnitude \(R\) N in the section \(CD\). The power of the car's engine is a constant \(36\) kW.

1. The car moves from \(A\) to \(B\) at a constant speed in \(120\) s. Find the speed of the car and the distance \(AB\). [3]
2. The distance \(BC\) is \(450\) m. Find the speed of the car at \(C\). [3]
3. The car comes to rest at \(D\). The distance \(AD\) is \(6637.5\) m. Find the deceleration of the car and the value of \(R\). [4]

The car's engine is switched off at \(B\).

\includegraphics{figure_6} Two particles of masses \(1.2\) kg and \(0.8\) kg are connected by a light inextensible string that passes over a fixed smooth pulley. The particles hang vertically. The system is released from rest with both particles \(0.64\) m above the floor (see diagram). In the subsequent motion the \(0.8\) kg particle does not reach the pulley.

1. Show that the acceleration of the particles is \(2\) m s\(^{-2}\) and find the tension in the string. [4]
2. Find the total distance travelled by the \(0.8\) kg particle during the first second after the particles are released. [8]

\includegraphics{figure_7} The diagram shows the vertical cross-section \(PQR\) of a slide. The part \(PQ\) is a straight line of length \(8\) m inclined at angle \(α\) to the horizontal, where \(\sin α = 0.8\). The straight part \(PQ\) is tangential to the curved part \(QR\) at \(Q\), and \(R\) is \(h\) m above the level of \(P\). The straight part \(PQ\) of the slide is rough and the curved part \(QR\) is smooth. A particle of mass \(0.25\) kg is projected with speed \(15\) m s\(^{-1}\) from \(P\) towards \(Q\) and comes to rest at \(R\). The coefficient of friction between the particle and \(PQ\) is \(0.5\).

1. Find the work done by the friction force during the motion of the particle from \(P\) to \(Q\). [4]
2. Hence find the speed of the particle at \(Q\). [4]
3. Find the value of \(h\). [3]

\includegraphics{figure_4} The diagram shows the vertical cross-section of a surface. \(A\) and \(B\) are two points on the cross-section, and \(A\) is 5 m higher than \(B\). A particle of mass \(0.35\) kg passes through \(A\) with speed \(7 \text{ m s}^{-1}\), moving on the surface towards \(B\).

1. Assuming that there is no resistance to motion, find the speed with which the particle reaches \(B\). [3]
2. Assuming instead that there is a resistance to motion, and that the particle reaches \(B\) with speed \(11 \text{ m s}^{-1}\), find the work done against this resistance as the particle moves from \(A\) to \(B\). [3]

A car of mass \(1200\) kg is driving along a straight horizontal road at a constant speed of \(15\) m s\(^{-1}\). There is a constant resistance to motion of \(350\) N.

1. Find the power of the car's engine. [1]

The car comes to a hill inclined at \(1°\) to the horizontal, still travelling at \(15\) m s\(^{-1}\).

1. The car starts to descend the hill with reduced power and with an acceleration of \(0.12\) m s\(^{-2}\). Given that there is no change in the resistance force, find the new power of the car's engine at the instant when it starts to descend the hill. [3]
2. When the car is travelling at \(20\) m s\(^{-1}\) down the hill, the power is cut off and the car gradually slows down. Assuming that the resistance force remains \(350\) N, find the distance travelled from the moment when the power is cut off until the speed of the car is reduced to \(18\) m s\(^{-1}\). [4]

A particle of mass \(0.3\) kg is released from rest above a tank containing water. The particle falls vertically, taking \(0.8\) s to reach the water surface. There is no instantaneous change of speed when the particle enters the water. The depth of water in the tank is \(1.25\) m. The water exerts a force on the particle resisting its motion. The work done against this resistance force from the instant that the particle enters the water until it reaches the bottom of the tank is \(1.2\) J.

1. Use an energy method to find the speed of the particle when it reaches the bottom of the tank. [4]

When the particle reaches the bottom of the tank, it bounces back vertically upwards with initial speed \(7\) m s\(^{-1}\). As the particle rises through the water, it experiences a constant resistance force of \(1.8\) N. The particle comes to instantaneous rest \(t\) seconds after it bounces on the bottom of the tank.

1. Find the value of \(t\). [7]

The total mass of a cyclist and her bicycle is 75 kg. The cyclist ascends a straight hill of length 0.7 km inclined at 1.5° to the horizontal. Her speed at the bottom of the hill is 10 m s\(^{-1}\) and at the top it is 5 m s\(^{-1}\). There is a resistance to motion, and the work done against this resistance as the cyclist ascends the hill is 2000 J. The cyclist exerts a constant force of magnitude \(F\) N in the direction of motion. Find \(F\). [5]

A lorry of mass 25 000 kg travels along a straight horizontal road. There is a constant force of 3000 N resisting the motion.

1. Find the power required to maintain a constant speed of 30 m s\(^{-1}\). [2]

The lorry comes to a straight hill inclined at 2° to the horizontal. The driver switches off the engine of the lorry at the point \(A\) which is at the foot of the hill. Point \(B\) is further up the hill. The speeds of the lorry at \(A\) and \(B\) are 30 m s\(^{-1}\) and 25 m s\(^{-1}\) respectively. The resistance force is still 3000 N.

1. Use an energy method to find the height of \(B\) above the level of \(A\). [5]

A particle of mass 0.5 kg starts from rest and slides down a line of greatest slope of a smooth plane. The plane is inclined at an angle of 30° to the horizontal.

1. Find the time taken for the particle to reach a speed of 2.5 m s\(^{-1}\). [3]
2. Find the distance that the particle travels along the ground before it comes to rest. [3]

When the particle has travelled 3 m down the slope from its starting point, it reaches rough horizontal ground at the bottom of the slope. The frictional force acting on the particle is 1 N.

\includegraphics{figure_5} A small bead \(Q\) can move freely along a smooth horizontal straight wire \(AB\) of length 3 m. Three horizontal forces of magnitudes \(F\) N, 10 N and 20 N act on the bead in the directions shown in the diagram. The magnitude of the resultant of the three forces is \(R\) N in the direction shown in the diagram.

1. Find the values of \(F\) and \(R\). [5]
2. Initially the bead is at rest at \(A\). It reaches \(B\) with a speed of 11.7 m s\(^{-1}\). Find the mass of the bead. [3]

\includegraphics{figure_6} 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 \(AB\) (see diagram).

1. Find the tension in the string and hence show that the modulus of elasticity of the string is 25 N. [4]

\(P\) is now held at rest at a point 1.8 m vertically below \(M\), and is then released.

1. Find the speed with which \(P\) passes through \(M\). [6]