Year 4 physics: work, energy and circular motion

In mid-September 2026 Year 4 physics at the liceo classico (the Italian high school focused on Latin, Greek and the humanities) has just restarted, and the first chapter the teacher tackles is almost always circular motion and work-energy — with projectile motion often left out for lack of time. The first test arrives in five to eight weeks, so between the end of October and mid-November: very little time for a chapter that at the science high school takes up a whole year beforehand, with one more hour of lessons a week.

At the classical high school physics has two hours a week in the last three years, half the hours of maths and one hour less than the science high school. With so little time every topic is seen only once, with few exercises done in class: that is why a student who is very strong in Greek and Latin can get the first 5 in physics without having “a hole” in the syllabus — they simply have not yet seen enough exercises to automate the method.

There is also a scheduling problem nobody points out: uniform circular motion uses radians and angular velocity, but in maths trigonometry — where radians are really learned — only comes later in the year. The physics teacher introduces them from scratch, in a few lines, and from then on uses them as if they were already known.

Why in Year 4 the chapter changes register

The classical high school starts physics in Year 3, a year after the science high school, and with one hour fewer: two hours a week against the three of the science track’s last three years. The result is that the Year 4 syllabus at the classical high school resembles, in content, Year 3 at the science high school: circular motion, projectile motion, work and energy, and then — in the following weeks — momentum and gravitation. If in Year 3 you had already worked on resolving forces without yet knowing sine and cosine, a problem the classical high school shares with the human-sciences and art high schools, you will find the method in resolving forces and vectors without trigonometry: here we pick up from there.

Projectile motion, which at the science high school is always studied together with circular motion, at the classical high school is often not done at all: with two hours a week many teachers choose to skip it and go straight to work and energy. If your teacher has assigned it, the method and an exercise worked in full — range, time of flight, arrival speed — are in the guide to the first Year 3 physics test at the science high school, which covers exactly the same topic a year earlier.

Uniform circular motion: the quantities and radians

In uniform circular motion the magnitude of the velocity is constant, but its direction changes continuously: that is why there is an acceleration nonetheless, even though the velocity “does not change” in value. The quantities needed are:

where $T$ is the period (the time of one complete turn), $f=1/T$ the frequency, $r$ the radius of the path and $\omega$ the angular velocity, measured in radians per second. A radian is simply the angle for which the arc covered equals the radius: it is not an “exotic” unit, but you will certainly find it in front of you before the full theory behind the formulas arrives in maths.

The acceleration $a_c$ is centripetal: it always points towards the centre of the path, not outwards. The force that produces it — the tension of a string, the friction between tyre and road, gravity for a satellite — is called centripetal force for the same reason: it is not a new type of force, it is the name we give to the resultant when the motion is circular.

1. Angular velocity and string tension

A small ball of mass $m=0.20\mathrm{kg}$ is tied to a string of length $r=0.50\mathrm{m}$ and makes one complete turn every $T=0.40\mathrm{s}$ on a frictionless horizontal plane. Find the angular velocity, the tangential speed, the centripetal acceleration and the tension in the string.

The only horizontal force acting on the ball is the tension in the string, and it is what acts as the centripetal force:

Note that the centripetal acceleration, about $123m/{\mathrm{s}}^2$, is more than twelve times the acceleration of gravity: with a short string and a fast turn the tensions involved are large, even for a light ball.

Work and kinetic energy

The work of a constant force along a displacement $d$ is

where $\theta$ is the angle between the force and the displacement. The kinetic energy theorem links the total work — of all the forces, including those that “steal” energy such as friction — to the change in kinetic energy:

It is the shortcut that avoids solving the motion instant by instant when the acceleration is not constant along the whole path, which is exactly the case where physics stops being just “substituting numbers into a formula”.

2. A trunk pushed with friction

A trunk of mass $m=15\mathrm{kg}$, initially at rest, is pushed with a constant horizontal force $F=90\mathrm{N}$ over a floor with friction, for a stretch $d=4.0\mathrm{m}$. The coefficient of kinetic friction is $\mu =0.30$. Find the final speed of the trunk (use $g\approx 9.8m/{\mathrm{s}}^2$).

The normal force is simply the weight, because the applied force is horizontal:

The horizontal resultant force and the net work:

The trunk starts from rest, so $L_{\text{tot}}=\frac{1}{2}m{v}_f^2$:

The weight and the normal force do not appear in the work because they are perpendicular to the displacement: zero work, always, however large they are.

Conservation of mechanical energy, and how friction breaks it

If only conservative forces act on the body (weight, elastic force), the total mechanical energy does not change:

Friction is not conservative: every time it is present, a piece of mechanical energy turns into heat and “leaves” the balance. The correct way to treat it is not to ignore it, but to put it explicitly into the account:

3. A block on an inclined plane with friction

A block of mass $m=2.0\mathrm{kg}$ starts from rest at the top of an inclined plane of height $h=1.5\mathrm{m}$ and length (along the plane) $L=3.0\mathrm{m}$, with coefficient of kinetic friction $\mu =0.25$. Find the speed of the block at the bottom of the plane.

First the angle, from the triangle: $\sin \theta =h/L=1.5/3.0=0.50$, so $\theta =30^{\circ }$ and $\cos \theta \approx 0.866$.

Without friction, the same height would give $v_f=\sqrt{2gh}\approx 5.4m/s$: friction here takes away almost a quarter of the final speed, not a negligible detail.

The mistakes that cost marks

1. Confusing “uniform” with “no acceleration” in circular motion: there is always a centripetal acceleration, even at constant speed (in magnitude).
2. Forgetting that the work of a force perpendicular to the displacement is zero — it happens almost always with weight and normal force, which get calculated and added by mistake anyway.
3. Using conservation of mechanical energy even when there is friction, “forgetting” it in the balance: it always has to be put in explicitly, with the right sign.
4. The confusion that arrives in the following weeks, when you move on to momentum and collisions: it is not enough to “conserve something”. In inelastic collisions mechanical energy is not conserved, but momentum is, always. Choosing the wrong principle is the most common mistake in the next chapter, and it starts right here: energy when there are no collisions and no friction to balance, momentum when there is a collision.
5. Using degrees where the formula wants radians (in $\omega$, measured in rad/s) without converting.

How we work on this chapter

With two hours a week the time in class for exercises is very little, so the work I do in lessons always starts from the drawing: the force diagram for the inclined plane, the centripetal vector for circular motion, and the energy balance written out in full before substituting a single number. It is the quickest way to stop confusing “conserving energy” with “using any formula that contains a $v$”. The full picture of how I work on physics online is on the page about physics tutoring.

The corresponding quiz on the site is called “work and energy” and is filed in the Year 3 section, because the catalogue of tests follows the science high school’s sequence: at the classical high school the same topic arrives a year later, but the syllabus is identical. You will find it here: mock test on work and energy — fifteen questions in fifty minutes, with the correct answer free and without registration; the guided step-by-step solution unlocks with the first lesson.

After energy come momentum and collisions, which I wrote up in momentum and collisions in Year 4 at the classical high school; for circular motion with fictitious forces there is centripetal force and fictitious forces, and for conservation in several phases in another track work, power and energy.

If you are at the science high school and in Year 4 you are tackling a completely different chapter in the same period — at the liceo scientifico September-October of Year 4 is devoted to waves and sound, not to circular motion and energy, because physics there starts a year earlier — you will find the corresponding guide in Year 4 physics at the science high school: waves and sound. For the general picture of physics mistakes in high school, beyond the single chapter, there is the 5 typical physics mistakes in high school.

If you want a hand before the end-of-October test, you can start with a free 30-minute introductory call: write to me via contact.