Example problems. Try these for practice.

41. A gardener pulls a roller over a tennis court, with the handle of the roller inclined at 30° to the horizontal.
The length of the court is 30 m and it is necessary to traverse the court 20 times to completely cover the area.
The gardener exerts a steady force of 231N. Find
(a) the work done by the gardener to roll the court.
(b) the average rate at which the gardener works if it takes 20 minutes to roll the court.

42. The hammer of a pile driver has a mass of 1000 kg. It falls a vertical distance of 4.5 m before striking the top of the pile. Find its kinetic energy on reaching the pile.

43. A mass of 20 kg slides down a smooth plane inclined at 36°52' to the horizontal and the force of gravity moves the mass through 10 m. Given that the acceleration of gravity is 10 m.s^-2 find
(a) the work done by gravity.
(b) the speed of the mass after the 10 m.

45. A ball on a 30 m cable is used for demolition work. The maximum angle of the cable from the vertical is 20°.
Find the speed of the ball at its lowest point (neglect the mass of the cable).

46. A mass of 8 kg is moving across a smooth horizontal surface. A constant force of 28 N brings it to rest in 7 m.
Find the speed just before the force starts.

47. From what height does an automobile have to fall to gain the kinetic energy that it would have at 108 km.hr^-1.

48. The external force needed to extend a spring from its natural length is found to be proportional to the extension produced. A force of 2 N will extend a spring by 0.1 m.
Find the work done by the external force to extend the spring by 0.6m.

49. A mass of 7 kg slides down a smooth irregular surface. At a height of 1 m it is moving at 9 m.s^-1.
Find the speed that it is moving at when it is at a height of 0.05 m.

50. A 2 gram coin rests on a vertical spring. The coin is then pushed downwards compressing the spring by 10 mm. The force constant of the spring is 40 N.m^-1. Find the height above the initial position that the coin will fly if it released from the compressed position.

51. A 100 kg man jumps out of a window into a fire net 10 m below. The net stretches 1 m before bringing him to rest and tossing him back up into the air. Find the potential energy of the stretched net if there are no dissipative forces.

52. An ideal massless spring has a force constant of 1000 N.m-1. It is placed at the bottom of a smooth plane which is inclined at 30° to the horizontal. A 10 kg mass released from the top of the plane is brought to rest after compressing the spring by 0.99 m. Find
(a) the work done against the spring force to compress it.
(b) the distance the mass slides before coming (momentarily) to rest.
(c) the speed of the mass just before it hit the spring.

E1. Water passing through rapids has a speed of 4 m.s^-1 as it enters the rapids and a speed of 16 m s^-1 as it leaves. The elevation of the river changes from 200 m above sea level to 180 m above sea level. Some energy is dissipated through the rapids. Find the energy dissipated as a fraction of the potential energy change.

E2. In Ethiopia, a bucket of mass 1.5 kg is at the bottom of a well 8 m deep with 10 kg of water in it. There is a hole in the bucket so that water leaks out at a steady rate as it rises, leaving only 8 kg when it reaches the top of the well. Draw a force vs distance graph, showing the weight decreasing with distance and find the work done in raising the bucket and water to the top.

E3. A person lifts a 12 kg mass a distance of 0.5 m and does this 905 times. Fat supplies 3.8x10⁷ joules of energy per kilogram and is converted to mechanical energy with an efficiency of 20%. Find
(a) the work done by the person in lifting the mass against gravity
(b) the mass of fat that the person uses up.

E4. The St. Lawrence river has a flow of 6800 m³.s^-1 as it leaves Lake Ontario and falls 75 m to an electric generating plant at sea level. Assume no other water enters or leaves the river on the way to the sea. Find the maximum energy that could, in principle, be extracted every 24 hours.

E5. Water (density 1000 kg.m^-3) falls through a hydroelectric power station and moves a vertical distance of 150m at a volume rate of flow of 30 m³s^-1. Assume that the power station converts all the potential energy lost by the water to electrical energy at an efficiency of 60%. Find the electrical power output of the station

E6. A 1200 kg car is being towed up an 18⁰ incline by a rope attached to the rear of a tow-truck. The rope has a breaking strength of 4.6 kN and from the car to the tow-truck it is at an angle of 27⁰ upwards from the incline. Neglecting friction, find the greatest acceleration that the truck can have so that the rope doesn’t break.

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