Exercises: Work and power

The work done by a force on a moving object is calculated as $W = F \times d \times \cos \alpha$, where $\alpha$ is the angle between the force and the displacement. You pull a trolley with the same force and over the same distance in three cases: with the rope horizontal ($\alpha = 0^\circ$), at an angle of $45^\circ$, and at an angle of $90^\circ$ ($\cos 0^\circ = 1$; $\cos 45^\circ \approx 0.71$; $\cos 90^\circ = 0$). The figure shows the three cases.

Order the three amounts of work done from greatest to least and justify your answer using the formula.

A worker pushes a pallet with a horizontal force of $150\ \text{N}$ and moves it $8\ \text{m}$ along the warehouse floor. Calculate the work done.

You place a box of books of $8\ \text{kg}$ on the top shelf of a cupboard, $1.5\ \text{m}$ above the floor (take $g = 10\ \text{m/s}^2$).

1. a) What minimum force do you have to overcome to raise it?
2. b) What work do you do against gravity?

The motor of a conveyor belt in a parcel distribution center does $18000\ \text{J}$ of work in $60\ \text{s}$. Power is calculated as $P = W/t$. Calculate the power of the motor, in watts.

A satellite orbits the Earth in a circular orbit at a constant speed, as shown in the figure. The gravitational force exerted on it by the Earth points toward the center of the orbit, while its displacement is tangent to the trajectory.

1. a) What angle do the force and the displacement form?
2. b) How much work does gravity do on the satellite during each orbit? Justify your answer using the cosine.

A child pulls a cart with a rope that forms an angle of $60^\circ$ with the horizontal, exerting a force of $200\ \text{N}$, and drags it $10\ \text{m}$ ($\cos 60^\circ = 0.5$), as shown in the figure.

Calculate the work done.

Classify the following units according to whether they measure work (or energy) or power, and explain the criterion you used:

1. a) $\text{J}$
2. b) $\text{W}$
3. c) $\text{N} \times \text{m}$
4. d) $\text{kWh}$
5. e) $\text{kW}$

To drag a piano $3\ \text{m}$ along the stage floor, $900\ \text{J}$ of work was done. With what average horizontal force was it pushed? Rearrange the formula for work to solve for force first.

A mountaineer and their backpack have a combined mass of $75\ \text{kg}$ and gain $720\ \text{m}$ of elevation in $1.5\ \text{h}$ (take $g = 10\ \text{m/s}^2$).

1. a) What force is overcome when climbing?
2. b) What work is done against weight?
3. c) What average power is developed? Remember to convert time into seconds.

Aitana mows the lawn by pushing the lawnmower with a force of $80\ \text{N}$ applied to the handle, which forms an angle of $30^\circ$ with the horizontal ($\cos 30^\circ \approx 0.87$), and covers $20\ \text{m}$ of lawn, as shown in the figure.

1. a) Calculate the work done.
2. b) What work would have been done by pushing horizontally with the same force? What accounts for the difference?

A drill has a power rating of $600\ \text{W}$ and operates for $2\ \text{min}$ ($1\ \text{kWh} = 3.6\times10^{6}\ \text{J}$).

1. a) Express the time in seconds.
2. b) Calculate the work (energy) done.
3. c) Express this energy in $\text{kWh}$.

A worker pushes a container $12\ \text{m}$ along a warehouse floor with a horizontal force of $80\ \text{N}$, while friction opposes it with $45\ \text{N}$, as shown in the figure. Remember that work done by a force opposing displacement is negative ($\alpha = 180^\circ$, $\cos 180^\circ = -1$).

1. a) Calculate the work done by the worker.
2. b) Calculate the work done by friction.
3. c) What is the net work done on the container?

A swimming pool filter pump has a power rating of $750\ \text{W}$ and operates for $8\ \text{h}$ a day ($1\ \text{kWh} = 3.6\times10^{6}\ \text{J}$).

1. a) Calculate the energy consumed per day, in joules.
2. b) Express it in $\text{kWh}$, the unit used by the electricity supplier for billing.

A crane lifts a steel beam of $25\ \text{kg}$ to a height of $12\ \text{m}$ at a constant speed (take $g = 9.8\ \text{m/s}^2$).

1. a) What force does the cable exert?
2. b) What work does the crane do?

The motor of a patio awning has a power rating of $40\ \text{W}$ and takes $15\ \text{s}$ to retract it.

1. a) What work does it do?
2. b) If the awning rises by $1.5\ \text{m}$, what average force does the motor exert?

Two children drag the same sledge $50\ \text{m}$ across the snow, as shown in the figure. The first child pulls with the rope horizontal, with a force of $60\ \text{N}$; the second child pulls with the rope forming an angle of $60^\circ$ with the horizontal, with a force of $100\ \text{N}$ ($\cos 60^\circ = 0.5$).

1. a) Calculate the work done by each child.
2. b) Who does more work? Why does the second child not make full use of their force?

In a weightlifting competition, an athlete lifts a weight of $100\ \text{kg}$ from the floor to a height of $2\ \text{m}$ in $2\ \text{s}$, holds it still above her head for $3\ \text{s}$, and then walks $10\ \text{m}$ horizontally with it until leaving the platform (take $g = 10\ \text{m/s}^2$). The figure shows the three phases.

1. a) What work does she do against gravity while lifting it?
2. b) What power does she develop during this lift?
3. c) Does she do work against gravity while holding it still? What about while walking horizontally? Justify both answers using the definition of work.

The human heart does about $1.2\ \text{J}$ of work with each beat ($1\ \text{kWh} = 3.6\times10^{6}\ \text{J}$).

1. a) Calculate its average power at rest, with $60$ beats per minute.
2. b) How much energy does it pump in a full day? Express it in joules and in $\text{kWh}$.
3. c) Compare it with the power of a $10\ \text{W}$ LED light bulb and comment on the result.

A funicular railway with a mass of $2400\ \text{kg}$ including passengers climbs a height difference of $500\ \text{m}$ in $10\ \text{min}$ (take $g = 10\ \text{m/s}^2$).

1. a) Calculate the work done against gravity.
2. b) Calculate its average power, in watts and in kilowatts.
3. c) If it completed the same journey in half the time, how would the work and power change? Justify your answer.

A kayaker drags his kayak $40\ \text{m}$ across the sand by pulling a rope that forms an angle of $45^\circ$ with the horizontal, with a force of $50\ \text{N}$ ($\cos 45^\circ \approx 0.71$). Friction with the sand opposes the motion with $20\ \text{N}$.

1. a) Calculate the work done by the kayaker.
2. b) Calculate the work done by friction (remember it is negative).
3. c) What is the net work done on the kayak? How can you observe the effect of this net work?

A 4x4 car of $1800\ \text{kg}$ drives up a mountain pass gaining $100\ \text{m}$ in altitude in $2\ \text{min}$, at constant speed (take $g = 10\ \text{m/s}^2$).

1. a) Calculate the work done against gravity.
2. b) Calculate the useful power used to climb, in $\text{kW}$.
3. c) The car's engine specifies a power rating of $150\ \text{kW}$: explain at least two reasons why the useful power you calculated is much lower.

In an automated warehouse, a robot lifts packages of $5\ \text{kg}$ onto a shelf $6\ \text{m}$ high, at a rate of $120$ packages per hour (take $g = 10\ \text{m/s}^2$).

1. a) What work does it do with each package?
2. b) What total work does it do in one hour?
3. c) What average power does it develop?
4. d) The robot's motor is rated at $500\ \text{W}$, considerably more than what you calculated: give reasons why this margin is needed.

A delivery worker drags a parcel trolley in two stages. In the first stage, he pulls the rope, which forms an angle of $30^\circ$ with the horizontal, with a force of $40\ \text{N}$ for $20\ \text{m}$ ($\cos 30^\circ \approx 0.87$); in the second stage, the rope is horizontal and he pulls with $40\ \text{N}$ for $15\ \text{m}$.

1. a) Calculate the work done in each stage and the total work done.
2. b) If he takes $36\ \text{s}$ for the entire distance, what average power does he develop?

To load a box of $60\ \text{kg}$ onto a lorry, whose bed is $1.2\ \text{m}$ above the ground, you can lift it vertically or push it up a frictionless ramp $3\ \text{m}$ long (take $g = 10\ \text{m/s}^2$). The figure shows the two options.

1. a) Calculate the work required in each case and explain why it is the same.
2. b) Calculate the force that must be applied in each case.
3. c) What is the advantage of using the ramp? What is the "cost" of using it?