Work, energy and power

A crate of mass 250 kg is raised vertically through a height of 12 m by a crane cable. The crate moves at a constant speed of 0.40 m s⁻¹ throughout. The crate is modelled as a particle and air resistance is ignored. Take g = 9.8 m s⁻².

A suitcase of mass 18 kg is pulled across a rough horizontal floor by its handle. The force in the handle has magnitude 30 N and acts at 40° above the horizontal. The suitcase moves in a straight line at constant speed. It is modelled as a particle. Take g = 9.8 m s⁻².

A stone of mass 0.3 kg is thrown from a bridge with speed 6 m s⁻¹. The point of release is 20 m above the surface of a river. The stone is modelled as a particle moving freely under gravity. Take g = 9.8 m s⁻².

A car of mass 1200 kg moves along a straight horizontal road. Its engine works at a constant rate of 36 kW and the resistance to motion is constant. The maximum speed of the car on this road is 40 m s⁻¹.

Figure 1 shows a sledge of mass 12 kg being pulled across horizontal snow by a rope. The tension in the rope is 48 N and the rope is inclined at 30° above the horizontal throughout. The resistance to motion is constant, of magnitude 20 N. The sledge starts from rest at A and moves in a straight line to B, where AB = 25 m. The sledge is modelled as a particle.

A small bead of mass 0.2 kg is threaded on a smooth wire which is fixed in a vertical plane. Figure 1 shows the profile of the wire, with the heights of the points A, B, C, E and D above a fixed horizontal level shown on the grid. The bead is released from rest at A. Take g = 9.8 m s⁻².

A motorboat of mass 600 kg moves in a straight line across a lake. Its engine works at a constant rate of P watts. Figure 1 shows the driving force of the engine plotted against the speed v of the boat, together with the resistance to motion, which is modelled as constant.

Figure 1 shows a child on a rope swing. The rope has length 4.5 m and is attached to a fixed point O. The child is released from rest at A, where the rope makes an angle of 40° with the vertical, swings through C, where the rope makes an angle of 25° with the vertical, and reaches the lowest point B. The rope is modelled as light and inextensible, the child as a particle of mass 28 kg, and air resistance is ignored. Take g = 9.8 m s⁻².

A child of mass 30 kg slides from rest down a straight playground slide of length 6 m, inclined at 35° to the horizontal. The coefficient of friction between the child and the slide is 0.3. At the bottom the child moves onto rough horizontal ground with no change in speed, and the coefficient of friction between the child and the ground is 0.4. The child is modelled as a particle and air resistance is ignored. Take g = 9.8 m s⁻².

A cyclist and her bicycle have a total mass of 85 kg. She rides up a straight road inclined at an angle α to the horizontal, where sin α = 130. The resistance to motion from non-gravitational forces is modelled as a constant force of magnitude 25 N. Take g = 9.8 m s⁻².

A box of mass 4 kg is projected with speed 7 m s⁻¹ up a line of greatest slope of a rough ramp. The ramp is inclined at an angle α to the horizontal, where tan α = 34. The coefficient of friction between the box and the ramp is 0.5. The box is modelled as a particle. Take g = 9.8 m s⁻².

A lorry of mass 16 000 kg is travelling at 25 m s⁻¹ when its engine is disengaged and its brakes are applied. The total resistance to motion, including braking, is modelled as a constant force of magnitude R newtons. Take g = 9.8 m s⁻².

A pump lifts water from rest at the surface of an underground tank and delivers it through a pipe whose outlet is 15 m above the surface of the water in the tank. The water leaves the outlet with speed 8 m s⁻¹. The pump delivers 30 kg of water every minute. Assume that no energy is wasted. Take g = 9.8 m s⁻².

A particle A of mass 3 kg lies on a rough horizontal table. It is attached to one end of a light inextensible string which passes over a small smooth pulley fixed at the edge of the table. A particle B of mass 2 kg hangs freely from the other end of the string. The coefficient of friction between A and the table is 0.25. The system is released from rest with the string taut, and B falls 1.2 m without reaching the floor while A stays on the table. Take g = 9.8 m s⁻².

A block of mass 5 kg is pulled from rest across a rough horizontal floor by a constant horizontal force of magnitude P newtons. After 4 s the force is removed and the block slides to rest. Figure 1 shows the speed–time graph of the motion. The frictional force on the block is constant and the block is modelled as a particle. Take g = 9.8 m s⁻².

Figure 1 shows a straight road PQ which rises 15 m over a distance of 500 m measured along the road, followed by a straight road QR which falls 10 m over a distance of 400 m measured along the road. A lorry of mass 10 000 kg travels from P to R. The resistance to motion from non-gravitational forces is constant, of magnitude 3000 N. The lorry is modelled as a particle and no energy is lost as it passes over Q. Take g = 9.8 m s⁻².

A box of mass 8 kg rests on rough horizontal ground. The coefficient of friction between the box and the ground is 0.2. The box is pulled in a straight line by a horizontal force of magnitude P newtons. Figure 1 shows how P varies with the distance x metres moved by the box from its starting point. The box is modelled as a particle. Take g = 9.8 m s⁻².

Figure 1 shows a conveyor belt of length 8.0 m inclined at 20° to the horizontal. Parcels, each of mass 5 kg, are placed at rest on the belt at the bottom end, and each leaves the top end moving with the belt at 1.5 m s⁻¹. Parcels are delivered at a steady rate of 12 per minute. Each parcel is modelled as a particle. Take g = 9.8 m s⁻².

Figure 1 shows a car of mass 1200 kg towing a trailer of mass 400 kg up a straight road inclined at an angle α to the horizontal, where sin α = 114. The trailer is attached to the car by a light rigid tow bar which is parallel to the road. The resistances to motion from non-gravitational forces are constant, 300 N on the car and 100 N on the trailer, as shown. The engine of the car works at a constant rate of 30 kW. Take g = 9.8 m s⁻².

A block of mass 2.5 kg slides down a line of greatest slope of a rough plane inclined at an angle α to the horizontal, where sin α = 0.6. Figure 1 shows the kinetic energy of the block plotted against the distance it has moved down the plane, for the first 5 m of its motion. The block is modelled as a particle. Take g = 9.8 m s⁻².

A city bus has mass 9000 kg and an engine whose greatest possible power output is 120 kW. It climbs a long hill on which the road makes an angle α with the horizontal, sin α = 150. Air resistance and friction together are taken to be a fixed 3600 N throughout. Take g = 9.8 m s⁻².

A particle of mass 2 kg is pulled up a line of greatest slope of a rough plane inclined at 20° to the horizontal. The pulling force has magnitude 30 N and acts at an angle of 15° above the line of greatest slope, in the vertical plane containing that line. The coefficient of friction between the particle and the plane is 0.3. The particle starts from rest and moves 4 m up the plane. Take g = 9.8 m s⁻².

A train of mass 80 000 kg has an engine with a maximum rate of working of 600 kW. The resistance to motion from non-gravitational forces is constant, 15 000 N. The train is modelled as a particle. Take g = 9.8 m s⁻².

A trolley of mass 120 kg runs on a fixed track. It is released from rest at a point A, 18 m vertically above the lowest point B. The section AB is smooth. From B the track is horizontal and rough for 30 m to a point C, where a constant resistance acts on the trolley. Beyond C the track is smooth and rises steadily. The trolley is modelled as a particle and air resistance is ignored. Take g = 9.8 m s⁻².

Figure 1 shows a fixed wedge whose faces are inclined at 30° and 60° to the horizontal. Particle A, of mass 3 kg, lies on the face inclined at 30°, which is rough; the coefficient of friction between A and this face is 0.25. Particle B, of mass 4 kg, lies on the face inclined at 60°, which is smooth. The particles are attached to the ends of a light inextensible string which passes over a small smooth pulley at the top of the wedge, each section of the string being parallel to a line of greatest slope of its face. B is 1.5 m from the foot of its face, measured along the face. The system is released from rest with the string taut. A does not reach the pulley. Take g = 9.8 m s⁻².

A van of mass 2000 kg has an engine whose maximum rate of working is P watts. The resistance to motion from non-gravitational forces is a constant R newtons. The van is driven with the engine working at its maximum rate up several straight roads, each inclined at an angle α to the horizontal, and its maximum speed v m s⁻¹ up each road is found. Figure 1 shows 1v plotted against sin α. Take g = 9.8 m s⁻².

A car of mass 1250 kg is driven 400 m up a straight road AB inclined at an angle α to the horizontal. Its speed at A is 10 m s⁻¹ and the journey from A to B takes 20 s. The resistance to motion from non-gravitational forces is constant. Figure 1 shows, for the journey from A to B, the work done by the engine, the gain in gravitational potential energy of the car and the work done against the resistance. The gain in kinetic energy is not shown. The car is modelled as a particle. Take g = 9.8 m s⁻².

A lift of mass 800 kg is raised vertically from rest by a cable attached to a motor at the top of the shaft. Figure 1 shows the speed–time graph of the lift for one journey, which ends with the lift at rest. A constant resistance of 300 N opposes the motion throughout. The lift is modelled as a particle and the cable as light and inextensible. Take g = 9.8 m s⁻².