Work, power and energy: multi-stage conservation

At the liceo artistico (the Italian art high school) Year 4 physics opens with work, power and energy. The test arrives in five to eight weeks, so between late October and mid-November. With 2 hours a week and the 12 hours of specialist assignments — drawing boards, projects, materials — taking up the rest of the time, the only way to arrive ready is to have a fixed method, above all for problems in which energy is transformed several times along the way.

Work done by a force

where $\theta$ is the angle between the force $F$ and the displacement $s$. The cosine decides everything:

• $\theta =0°$: $\cos \theta =1$, driving work, maximum and positive (the force pushes in the direction of motion).
• $\theta =90°$: $\cos \theta =0$, zero work. A force perpendicular to the displacement does no work, ever — the normal reaction of a plane, the centripetal tension in circular motion.
• $\theta =180°$: $\cos \theta =-1$, resisting work, negative — friction opposing motion.

The unit is the joule: $1\mathrm{J}=1\mathrm{N}\cdot \mathrm{m}$.

Power and efficiency

Power is the work done per unit of time:

If the force and the velocity are constant, it can also be written as:

The SI unit is the watt ($1\mathrm{W}=1J/s$); in common use you also find the horsepower ($1\mathrm{CV}\approx 735\mathrm{W}$).

No real machine turns all the power it takes in into useful power: some is lost in friction, heat, vibration. Efficiency $\eta$ is the ratio between useful power and input power:

always less than 1 (or, as a percentage, less than 100%). It is the part that in the Year 4 test is asked more often than you would expect, and that many online summaries skip.

Kinetic energy

The work-energy theorem links work and kinetic energy: the total work done on a body (by all the forces together) equals the change in its kinetic energy.

Gravitational and elastic potential energy

Gravitational, with $h$ measured from a reference level of your choice:

Elastic, for a spring with spring constant $k$ compressed or stretched by $x$ from its rest position:

Conservation of mechanical energy

If only conservative forces (weight, elastic force) act on the body, the total mechanical energy — kinetic plus potential — stays constant:

If instead there is friction, mechanical energy is not conserved: part of it turns into heat. The correct balance is:

that is, the work (in absolute value) done by friction equals the amount of mechanical energy lost between the start and the end of the rough stretch.

The method for multi-stage problems

The Year 4 test rarely stops at a single step: a spring launching a block, a block climbing a plane, a plane with friction, a block falling back — four different stages in the same exercise. The method that always works is just one:

Choose two instants at a time, write the energy balance only between those two, and never try to write a single equation from the beginning to the end of the problem.

In practice:

1. Number the stages of the motion (wherever something changes: friction begins, the spring ends, the body leaves the plane…).
2. For each pair of consecutive instants, write separately $E_k+U=\text{constant}$ (if there is no friction in that stretch) or $L_{\text{friction}}=\Delta E_{\text{mech}}$ (if there is).
3. The result of one stage (a speed, an energy) becomes the starting data of the next stage.
4. Only at the end, if needed, put together the results of the individual stages.

Anyone who tries to write a single equation for the whole path almost always loses a term — usually the friction of just one of the stages — because they are trying to hold too many variables in their head at once. Stage by stage, each balance has only two or three terms and is almost impossible to get wrong.

The typical mistake: adding friction and then conserving energy

The most frequent mistake in this chapter is not one of calculation, it is conceptual: you write $E_k+U=\text{constant}$ for the whole problem and then, to be safe, you also subtract the work of friction in the same equation. It makes no sense: either mechanical energy is conserved (no friction in the stretch), or it is not conserved and then the correct balance is $L_{\text{friction}}=\Delta E_{\text{mech}}$. The two forms must never be used together on the same stretch.

The second mistake, just as common, is forgetting that the work of friction must be calculated with the real length travelled in the rough stretch, not with the overall displacement of the problem: if the block crosses the rough stretch there and back, friction acts twice, not once. And that is why it always pays to isolate the stages one at a time, as in the method above: mixing smooth and rough stretches in the same equation is the number one cause of a wrong energy balance.

Exercise 1 — work, power and efficiency

A motor has to lift a load of mass $m=50\mathrm{kg}$ through a height $h=6.0\mathrm{m}$ in a time $t=5.0\mathrm{s}$. The motor has an efficiency $\eta =80\%$. Find the useful work, the useful power and the power absorbed by the motor.

Useful work (against the weight, with $g=9.8m/{\mathrm{s}}^2$):

Useful power:

Input power, from the efficiency $\eta =P_{\text{useful}}/P_{\text{input}}$:

The motor takes in $735\mathrm{W}$ to actually deliver only $588\mathrm{W}$ to the load: the difference, $147\mathrm{W}$, is lost in the motor’s internal friction.

If you want to check for yourself whether you have understood the method, in the Year 3 physics test — work and energy you will find more exercises of this type: the correct answer is free, the guided solution is unlocked with the first lesson. It is listed under Year 3 because at the liceo scientifico this block is done in Year 3; at the art high school it comes in Year 4, the content is the same.

Exercise 2 — conservation of energy in three stages

A spring of spring constant $k=200N/m$, compressed by $x=0.30\mathrm{m}$, pushes a block of mass $m=0.50\mathrm{kg}$ on a horizontal plane. The first stretch is smooth; after the spring, the block crosses a rough stretch of length $d=2.0\mathrm{m}$ with coefficient of friction $\mu =0.25$. Find the speed of the block just as it leaves the spring and the speed with which it leaves the rough stretch.

I apply the method: two instants at a time, never a single equation for the whole path.

Stage 1 — from the compressed spring to the block leaving it. No friction in this stretch: all the elastic energy of the spring turns into kinetic energy of the block.

Stage 2 — from leaving the spring to the start of the rough stretch. The plane stays smooth and horizontal: there is no change in potential energy (the height does not change) and no friction, so the kinetic energy stays the same. The block reaches the start of the rough stretch still at $v_1=6.0m/s$, with $E_k=9.0\mathrm{J}$.

Stage 3 — across the rough stretch. Here the balance is $L_{\text{friction}}=\Delta E_{\text{mech}}$. The work of friction, in absolute value:

The final kinetic energy is the initial one minus the energy lost to friction:

Hence the final speed:

The block leaves the rough stretch at about $5.1m/s$, slower than the $6.0m/s$ with which it started: friction has “consumed” $2.45\mathrm{J}$ of the initial $9.0\mathrm{J}$. Note that each stage used a single balance, with two or three terms: trying to write a single equation from the spring to the exit of the rough stretch would have required holding elastic energy, kinetic energy and the work of friction in the same line — exactly where a term gets lost.

What is not in this chapter

If your problem involves a collision between two bodies or momentum, this is not the chapter: at the art high school those topics come later, in November-December, and you will find them covered for Year 3 in work, energy and collisions. If instead your course covers work and energy together with circular motion, that is the version for the classical high school, with circular motion: you will find it in physics in Year 4 at the classical high school.

If you want to practise on other problems of this type, you will find more exercises in the Year 3 physics test — work and energy — same label as before, same content in Year 4 at the art high school.

The Year 3 physics post for the same course, with the method for measurements and significant figures, is physics at the art high school: measurements and significant figures.

For how we work on physics at the art high school, the reference page is physics.

If you would like a hand, the first chat is free → contact.