Exercises: Work

The work done by a constant force on a moving body is $W = \vec{F} \cdot \vec{d} = F \cdot d \cdot \cos \alpha$, where $\alpha$ is the angle between the force and the displacement. Without performing calculations, state whether the work is positive, negative, or zero in each case, justifying your answer using the value of $\alpha$:

1. a) The pulling force of a tractor towing a trailer along a horizontal path.
2. b) The friction force opposing the motion of the trailer.
3. c) The normal force exerted by the ground on the trailer.
4. d) The force with which a worker holds a box at rest on their shoulder.

A warehouse worker pushes a pallet with a constant horizontal force of $250\ \text{N}$, moving it $12\ \text{m}$ along the floor in a straight line.

1. a) What angle does the force make with the displacement?
2. b) Calculate the work done by the worker.

A person walks their dog, pulling on the lead with a constant force of $80\ \text{N}$ at an angle of $60^\circ$ to the horizontal over a straight section of $25\ \text{m}$ ($\cos 60^\circ = 0.5$), as shown in the figure.

Calculate the work done.

You place a toolbox of $10\ \text{kg}$ onto a shelf located $2.5\ \text{m}$ above the floor (take $g = 9.8\ \text{m/s}^2$).

1. a) Calculate the weight of the box.
2. b) Calculate the work done by the weight while raising it. Remember that weight points downwards and displacement upwards.
3. c) What is the minimum work that you do against gravity?

A traveller pushes their luggage trolley through an airport terminal in a straight line on a horizontal floor.

1. a) Reason, using the expression $W = F \times d \times \cos \alpha$, why the work done by the normal force and the work done by the weight are zero during the journey.
2. b) If the traveller exerts a horizontal force of $18\ \text{N}$ over $100\ \text{m}$, what work do they do?

In the laboratory, a trolley on an air track moves $\vec{d} = (5, 0)\ \text{m}$ while acted upon by a constant force $\vec{F} = (6, 8)\ \text{N}$, as shown in the figure.

1. a) Calculate the work using the dot product $W = \vec{F} \cdot \vec{d} = F_x \cdot d_x + F_y \cdot d_y$.
2. b) Explain why the vertical component of the force does not contribute to the work.

When dragging a sofa $8\ \text{m}$ across a living room floor, the kinetic friction force is $35\ \text{N}$.

1. a) What angle does the friction force make with the displacement?
2. b) Calculate the work done by friction and interpret its sign.

The work-energy theorem states that the net work done on a body is equal to the change in its kinetic energy: $W_{neto} = \Delta E_c = \frac{1}{2} \times m \times v^2 - \frac{1}{2} \times m \times v_0^2$. A demonstration ball of $2\ \text{kg}$ increases its speed from $3\ \text{m/s}$ to $5\ \text{m/s}$ on an air track.

1. a) Calculate its initial kinetic energy and its final kinetic energy.
2. b) What is the net work done on the ball?

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 centre 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 towards the centre 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.

In a treasure hunt, a student drags a box of books $15\ \text{m}$ across the horizontal floor of the gymnasium by pushing with a horizontal force of $120\ \text{N}$, while kinetic friction opposes the motion with $40\ \text{N}$, as shown in the figure.

1. a) Calculate the work done by the student.
2. b) Calculate the work done by friction (remember it is negative: $\alpha = 180^\circ$).
3. c) Calculate the work done by weight and by the normal force, justifying their values.
4. d) Find the net work done on the box.

To load a fridge of $60\ \text{kg}$ into a van, it is pushed up a ramp of $3\ \text{m}$ at an angle of $20^\circ$ to the horizontal ($\sen 20^\circ \approx 0.34$; take $g = 10\ \text{m/s}^2$), as shown in the figure.

1. a) Calculate the component of the weight parallel to the ramp.
2. b) Calculate the work done by the weight during the ascent.
3. c) What minimum work must the person pushing do against that component of weight?

A musher pulls their team's sled using a rope at an angle of $37^\circ$ to the horizontal, exerting a constant force of $100\ \text{N}$ over $50\ \text{m}$ ($\sen 37^\circ \approx 0.6$; $\cos 37^\circ \approx 0.8$). Friction with the snow opposes with $30\ \text{N}$.

1. a) Calculate the work done by the musher.
2. b) Calculate the work done by friction.
3. c) Find the net work done on the sled.

A constant force $\vec{F} = (15, -8)\ \text{N}$ acts on a machine component while the component moves through a displacement of $\vec{d} = (4, 2)\ \text{m}$, as shown in the figure.

1. a) Calculate the work using the dot product $W = \vec{F} \cdot \vec{d} = F_x \cdot d_x + F_y \cdot d_y$.
2. b) Check that the component $F_y$ does negative work and interpret what that sign means physically.

A construction crane lowers a beam of $500\ \text{kg}$ from a height of $8\ \text{m}$ to the ground at constant speed (take $g = 9.8\ \text{m/s}^2$).

1. a) Calculate the work done by the weight during the descent.
2. b) Calculate the work done by the tension in the cable.
3. c) What is the net work? Relate this to the fact that the speed is constant, using the work-energy theorem.

A passenger car of $1200\ \text{kg}$ accelerates from rest to $108\ \text{km/h}$ when joining a dual carriageway.

1. a) Express the final velocity in $\text{m/s}$.
2. b) Calculate the change in kinetic energy of the car.
3. c) What net work has been done by the forces acting on the car? Express it in $\text{kJ}$.

A car of $1000\ \text{kg}$ travelling at $20\ \text{m/s}$ brakes to a halt over a distance of $50\ \text{m}$ along a horizontal section.

1. a) Calculate the change in kinetic energy during braking.
2. b) Applying the work-energy theorem, find the work done by the net braking force.
3. c) What is the average value of this net braking force? State its sense.

A firefighter of $80\ \text{kg}$ with their equipment climbs a fire escape from the ground to a terrace located $18\ \text{m}$ higher up (take $g = 10\ \text{m/s}^2$).

1. a) Calculate the work done by the weight during the ascent.
2. b) Calculate the work done by the firefighter against gravity.
3. c) If instead of the fire escape they had climbed a longer ramp to the same terrace, how would that work change? Justify your answer.

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.

Aoife 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?

A box of $25\ \text{kg}$ starts from rest on the horizontal floor of a warehouse, where the coefficient of kinetic friction is $\mu = 0.2$. It is pushed with a constant horizontal force of $100\ \text{N}$ over $12\ \text{m}$ (take $g = 10\ \text{m/s}^2$), as shown in the figure.

1. a) Calculate the kinetic friction force, $F_{roz} = \mu \times N$, taking into account that on a horizontal floor the normal force is $N = m \times g$.
2. b) Calculate the work done by each of the forces (push, friction, weight, and normal force) and the net work.
3. c) Apply the work-energy theorem to find the speed of the box at the end of the $12\ \text{m}$.

A block of $10\ \text{kg}$ is pulled up a frictionless inclined plane of $30^\circ$ using a force $\vec{F}$ parallel to the plane, so that the block ascends at a constant speed along $5\ \text{m}$ ($\sen 30^\circ = 0.5$; take $g = 10\ \text{m/s}^2$), as shown in the figure.

1. a) Calculate the value of $F$ for the speed to be constant.
2. b) Calculate the work done by each force: the applied force, the weight, and the normal force.
3. c) Find the net work and explain, using the work-energy theorem, why that result was to be expected.

A passenger car of $1000\ \text{kg}$ travels at $20\ \text{m/s}$ along a horizontal road and locks its wheels during emergency braking. The coefficient of friction between the tyres and the asphalt is $\mu = 0.5$ (take $g = 10\ \text{m/s}^2$).

1. a) Calculate the friction force during braking.
2. b) Apply the work-energy theorem to calculate the distance it travels until coming to a stop.
3. c) Reason, from the expression obtained, what happens to the braking distance if the initial speed is doubled: does it also double?

On a laboratory air table, a puck of $4\ \text{kg}$, initially at rest at point $A(1, 2)$, moves to point $B(9, 5)$ while a constant force $\vec{F} = (6, 8)\ \text{N}$ acts on it (the coordinates are in metres), as shown in the figure.

1. a) Determine the displacement vector $\vec{d} = \vec{AB}$.
2. b) Calculate the work done by the force using the scalar product $W = \vec{F} \cdot \vec{d}$.
3. c) Since friction is negligible, this work is the net work: apply the work-energy theorem and calculate the speed of the puck when it reaches $B$.

A cable car cabin with its passengers has a total mass of $800\ \text{kg}$. On the outbound journey, it ascends from the base station, at an altitude of $1200\ \text{m}$, to the top station, at $1750\ \text{m}$. It then partially descends to an intermediate station located at $1550\ \text{m}$ (take $g = 10\ \text{m/s}^2$), as shown in the figure.

1. a) Calculate the work done by weight during the ascent.
2. b) Calculate the work done by weight during the partial descent.
3. c) Calculate the total work done by weight over the entire journey and verify that it depends only on the net difference in altitude between the starting and ending points. What property of the gravitational force does this result highlight?

A worker pulls a $25\ \text{kg}$ box up a $20\ \text{m}$ ramp inclined at $37^\circ$ to the horizontal, pulling with a rope parallel to the ramp and at constant speed ($\sen 37^\circ \approx 0.6$; $\cos 37^\circ \approx 0.8$; coefficient of friction $\mu = 0.25$; take $g = 10\ \text{m/s}^2$), as shown in the figure.

1. a) Calculate the normal force exerted by the ramp and the friction force.
2. b) Calculate the force with which the worker pulls.
3. c) Calculate the work done by each force (pull, weight, friction, and normal force) and check that the net work is zero. Why did it have to be zero?

On an ice rink, a puck of mass $m$ is launched with an initial speed of $v_0 = 10\ \text{m/s}$, and slides in a straight line until coming to a stop due to friction ($\mu = 0.25$; take $g = 10\ \text{m/s}^2$).

1. a) Write the expression for the work done by friction and, applying the work-energy theorem, obtain the stopping distance $d$ as a function of $v_0$, $\mu$, and $g$.
2. b) Calculate this distance numerically.
3. c) Reason, from the expression obtained, whether the stopping distance depends on the mass of the puck. What practical implication does this result have for road safety?

An elevator cabin with its passengers has a mass of $600\ \text{kg}$. It starts from rest and ascends $12\ \text{m}$ with a constant acceleration of $1.2\ \text{m/s}^2$ (take $g = 10\ \text{m/s}^2$).

1. a) Calculate the force exerted by the cable applying Newton's second law: $T - m \times g = m \times a$.
2. b) Calculate the work done by the cable and the work done by weight during these $12\ \text{m}$.
3. c) Find the net work and, using the work-energy theorem, the speed of the cabin after $12\ \text{m}$. Check the result using kinematics: $v^2 = 2 \times a \times h$.

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?