Work, Energy and Power: Physics 9702 (Cambridge International AS & A Level)

A warehouse worker pulls a supply crate across a horizontal floor using a rope that makes an angle of $60°$ with the floor. The tension in the rope is $80\text{ N}$, and the crate is dragged through a displacement of $5.0\text{ m}$ along the floor.

What is the work done on the crate by the tension force?

A cyclist and her bicycle, of combined mass $78\text{ kg}$, start from rest at the top of a straight downhill slope and free-wheel (do not pedal) to the bottom. The vertical height of the slope is $22\text{ m}$, and the distance travelled along the slope is $140\text{ m}$. A constant resistive force (from air resistance and rolling friction) of average magnitude $45\text{ N}$ acts on the cyclist throughout the descent.

Take $g = 9.81\text{ m s}^{-2}$.

(a) Calculate the loss in gravitational potential energy of the cyclist and bicycle during the descent. [2]

(b) Calculate the work done against the resistive force during the descent. [2]

(c) Use the principle of conservation of energy to calculate the kinetic energy of the cyclist and bicycle at the bottom of the slope. [1]

(d) Hence calculate the speed of the cyclist and bicycle at the bottom of the slope. [2]

An electric winch is used to lift a steel beam of mass $250\text{ kg}$ vertically upward at a constant speed. The beam rises through a height of $8.0\text{ m}$ in a time of $20\text{ s}$.

Take $g = 9.81\text{ m s}^{-2}$.

(a) Show that the increase in gravitational potential energy of the beam is about $1.96\times10^{4}\text{ J}$. [2]

(b) Calculate the useful output power of the winch motor as it lifts the beam. [2]

(c) The winch motor has an efficiency of $75\%$. Calculate the total power input to the motor. [2]

(d) Calculate the total energy supplied to the motor during the $20\text{ s}$ lift. [2]

A speedboat travels at a constant velocity of $12\text{ m s}^{-1}$ across a lake. At this constant velocity, the total resistive force (from water drag and air resistance) acting on the boat has magnitude $3600\text{ N}$.

What is the useful output power of the boat's engine at this constant velocity?

A book of mass $1.2\text{ kg}$ falls from rest off a shelf and falls freely (assume air resistance is negligible) through a vertical height of $1.8\text{ m}$ before it hits the floor.

Take $g = 9.81\text{ m s}^{-2}$.

(a) Calculate the loss in gravitational potential energy of the book as it falls. [2]

(b) State the kinetic energy of the book immediately before it hits the floor, giving a reason for your answer. [1]

(c) Hence calculate the speed of the book immediately before it hits the floor. [2]

A skateboarder of mass $55\text{ kg}$ is moving in a straight line at a constant speed of $4.0\text{ m s}^{-1}$.

What is the kinetic energy of the skateboarder at this speed?

A warehouse worker pushes a crate of mass $50\text{ kg}$, initially at rest, across a horizontal floor by applying a constant horizontal force of $90\text{ N}$. As the crate moves through a distance of $6.0\text{ m}$, a constant frictional force of $30\text{ N}$ acts on it, opposing its motion.

(a) Calculate the work done on the crate by the applied force. [2]

(b) Calculate the work done against the frictional force. [2]

(c) Use the work–energy principle to calculate the kinetic energy gained by the crate. [1]

(d) Calculate the final speed of the crate. [2]

An escalator motor has a total (input) power of $900\text{ W}$. The motor does useful work raising passengers, delivering a useful output power of $630\text{ W}$; the rest is dissipated as heat in the motor and its gears.

What is the efficiency of the escalator motor?

A lift (elevator) cabin and its passengers have a total mass of $850\text{ kg}$. Starting from rest, the lift accelerates uniformly upward. In a time of $5.0\text{ s}$ it rises through a vertical height of $6.0\text{ m}$ and reaches a final speed of $2.5\text{ m s}^{-1}$. Assume no energy is lost to resistive forces such as friction in the lift mechanism.

Take $g = 9.81\text{ m s}^{-2}$.

(a) Calculate the gain in gravitational potential energy of the lift during this time. [2]

(b) Calculate the gain in kinetic energy of the lift during this time. [2]

(c) Calculate the total work done by the lift motor during this time. [1]

(d) Calculate the average power output of the lift motor during this time. [2]

A car of mass $1200\text{ kg}$ is travelling at a constant speed of $20\text{ m s}^{-1}$ along a straight, level road when the driver applies the brakes. A constant braking (frictional) force brings the car to rest after it has travelled a further $50\text{ m}$.

(a) Calculate the kinetic energy of the car immediately before the brakes are applied. [2]

(b) Use the work–energy principle to calculate the magnitude of the braking force, assuming it is constant. [3]

(c) State one assumption, other than the braking force being constant, that was made in your calculation in part (b). [1]