Work

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.

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 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.

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.

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.

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.

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.

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?

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?

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.

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 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 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?

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?

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.

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.

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 $B$.

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?

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 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?

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 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?

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?

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.

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) $\text{kW}$

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 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.

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 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.