2.3 Work, Energy & Power

Some cargo ships use kites together with the ships engine to move heavy pieces of cargo. 

The tension in the cable that connects the kite to the ship is 475 kN. The kite is pulling on the ship at an angle of θ to the horizontal plane of the ship's deck. The ship travels at a steady speed of 14.76 km hr⁻¹ when its engines operate with a power output of 5.9 MW. The work done by the engines is 0.365 GJ when turned on for 0.09 hours. 

Calculate the efficiency of the kite.

The ship continues its journey using both the engine and the kite to maintain the same constant speed. After 0.09 hours with the engine on, the ship drives into a spring net erected between two rocky outcrops. Upon impact, the net force from the ship acts parallel to the direction of extension of twelve identical springs in the net and the ship comes to a complete stop with the springs in the net extended. The ship has a mass of 2200 metric tonnes. 

Calculate the extension of one of the springs in the net to the nearest millimetre. Assume no energy is lost upon impact. 

A golfing team are conducting investigations into the optimum angles at which golfers should hold their clubs to swing for the ball. 

Golf club A is held at an angle of 30° to the vertical at the pivot and golf club B at an angle of 45°. Both golf clubs are 1.05 m long and have the same mass. 

Compare the maximum speeds of the club heads just before they hit the ball. 

Obtain an expression to calculate the time of swing for golf club A if the time of swing for golf club B is 0.23 seconds. 

Assume no additional force is applied to the golf clubs apart from their weight during the swing. 

An object is at rest at the top of a straight slope that makes a fixed angle with the horizontal at a distance h above the ground. 

The object is released and slides down the slope from A to B with negligible friction. Assume that the potential energy is zero at B. 

Sketch and explain a graph showing:

In a theme park ride, a cage containing passengers falls freely a distance of x m from A to B and travels in a circular arc of radius 25 m. from B to C. The force required for circular motion on a passenger of mass 63.25 kg is 0.064 × 10² kN.

The equation for calculating centripetal force on an object moving in an arc is: 

Brakes are applied at C after which the cage with its passengers travels 70 m along an upward sloping ramp and comes to rest at D. The track, together with relevant distances, is shown in the diagram. CD and makes an angle θ with the horizontal. 

The total mass of the cage and passengers is 3.556 × 10² kg. The average resistive force exerted by the brakes between C and D is 4.4 × 10³ N.

Calculate the angle of the ramp to the horizontal, θ.

The diagram below shows a projectile launcher with a spring in both a compressed and uncompressed position. When the spring is compressed, in preparation for launching the projectile, the plate is held in place by a pin at three different positions, P, Q and R.  When the pin is released the sphere is launched. 

A student hypothesises that the spring constant of the spring inside the launcher has the same value for different compression distances. 

The student plans to test the hypothesis by launching the sphere using the launcher.

(ii)

Hence, or otherwise, show that k = 

Measurements can be made with equipment usually found in a school laboratory using the principle or law stated in part (i).

[3]

Design an experimental procedure to test the hypothesis. 

Assess how the experimental data can be analysed to confirm or dispel the hypothesis. 

Some cargo ships use kites working together with the ship’s engines to move the vessel.

The tension in the cable that connects the kite to the ship is 350 kN. The kite is pulling the ship at an angle of θ to the horizontal. The ship travels at a steady speed of 3.9 m s^–1 when the ship’s engines operate with a power output of 5.7 MW.

Calculate the angle θ if the work done on the ship by the kite when the ship travels for 5 minutes, before the engines are cut off, is 355 MJ.

When the ship is travelling at 3.9 m s^–1, calculate the power that the kite provides.

Hence calculate the percentage of the total power required by the ship that the kite provides.

The kite is taken down and no longer produces a force on the ship. The resistive force F that opposes the motion of the ship is related to the speed v of the ship by

 F = kv

where k is a constant.

Show that, if the power output of the engines remains at 5.7 MW, the speed of the ship will decrease to about 3.5 m s^–1. Assume that k is independent of whether the kite is in use or not.

The arrangement shown below is used to test golf club heads.

The shaft of a club is pivoted and the centre of mass of the club head is raised by a height h before being released. The club head then falls back to the vertical position where it strikes the ball.

Calculate the maximum speed of the club head achieved when h = 75 cm.

Explain why, in reality, the speed of the ball will not be the same as the maximum speed of the club head.

The optimal launch angle for a projectile, such as a golf ball, is 45º. Another experiment is carried out with a golf club that has a shaft 1.14 m long.

Calculate the maximum speed this club head achieves just before it hits the ball.

The golf club head has a mass of 200 g and has a power of 460 mW just before it hits the ball.

Show that time taken for the golf club swing is about 1.4 s.

A motor is used to lift a 50 kg mass from rest up a vertical distance of 18 m in 0.3 minutes.

Calculate the minimum power requires to the lift the mass.

Explain why the power of the motor is only a minimum.

A different motor is used to lift an identical mass through the same distance in the same amount of time with an overall efficiency of 38 %. The mass experiences a resistive force of 170 N.

Calculate the work required from the motor.

Hence, determine the average power input to the motor.

The top speed of a car with a mass of 1400 kg whose engine is delivering 490kW of power is 530 km h^–1.

Calculate the force from the engine on the car.

Assume that air resistance is negligible.

The car now travels at a new constant velocity up a hill of incline 25º.

Determine the decrease in velocity from the previous motion of the car as it travels up the hill with the same power.

The car engine cuts off whilst travelling up the incline.

Assuming there are no resistive forces, calculate the distance travelled by the car from when the engine cuts off to when it has a gravitational potential energy of 110 kJ. 

A spring in the car’s suspension system has the following force–displacement graph.

Using the area under the graph, show that the spring constant of the spring is about 49 kN m^-1.