Physics Olympiad Guide: Basic Calculus, Five Core Fields, and the OSN Rounds

Most olympiad mechanics problems are won or lost in the first few minutes, while the free-body diagram is being drawn. A school problem usually has one object and three forces. A provincial OSN-P problem might stack two blocks on a wedge that is itself free to slide, joined by a string over a pulley that has mass. Students who sketch the whole system as one picture end up mixing internal and external forces, and their equations never balance.

The habit to build is easy to describe and takes many repetitions to own. Isolate each body, draw every force acting on it along with its action-reaction partner, and fix a positive direction before writing a single number. The choice of axes matters a great deal too. On an incline, axes parallel to the surface leave the normal force with one component. For circular motion, polar coordinates make the centripetal term appear on its own. When the wedge itself accelerates, working in its non-inertial frame with a fictitious force often cuts the algebra in half.

A useful drill at this stage is to take ten pulley and incline problems and stop each one as soon as the equations are written down. Students then compare their diagrams with the worked solutions. That keeps attention on the step where errors usually start, and it saves the time normally spent grinding through algebra.

This is where calculus stops being a separate subject and starts doing real work inside a problem. Three uses come up again and again at senior high olympiad level: finding the centre of mass of a continuous object, calculating a moment of inertia, and computing the work done by a force that changes with position.

For a uniform rod of mass m and length L, students learn to slice it into small elements dm, write dm as mass per unit length times dx, and integrate. The moment of inertia comes out as mL squared over 12 about the centre and mL squared over 3 about one end, and the gap between the two leads straight into the parallel axis theorem. Students who have derived these numbers themselves rarely mix them up under pressure, whereas students who memorised them often attach the wrong denominator.

Spring forces, gravity over large distances and drag proportional to speed all call for integration. Integrating gravitational work from Earth's radius out to infinity, for instance, produces the escape velocity formula. A sensible progression starts with simple polynomial integrals, moves to trigonometric ones for rings and discs, and saves spherical coordinates for solid spheres at the advanced stage.

A reliable sign of progress: the student derives the moment of inertia of a disc without opening a book and can explain, in plain words, why the mass element was chosen as a thin ring.

Electricity and magnetism often feel like the most abstract field to students fresh out of mechanics, since the central quantity, the field, cannot be seen. Symmetry offers the smoothest way in. Gauss's law makes the fields of a charged sphere, a long wire and a wide plate quick to compute, provided the student can choose a Gaussian surface shaped to match the symmetry. Practising that choice matters more than practising the integral itself.

Next come DC circuits with Kirchhoff's rules, followed by a capacitor charging through a resistor. That charging process is many students' first first-order differential equation, and the time constant RC measures how quickly the charge approaches its final value. Olympiad setters love symmetric resistor networks, such as a cube with a resistor along every edge, which fall apart quickly once you identify the points that sit at equal potential.

Faraday induction rounds off the chapter: the induced emf equals the negative rate of change of magnetic flux. The classic problem has a conducting rod sliding along rails in a magnetic field, and it turns out to combine Newton's laws, Ohm's law and energy conservation in one go. A student who can show that the mechanical power lost equals the electrical power dissipated in the resistor has understood this chapter well.

Higher rounds also bring in Maxwell's equations at an introductory level and charged particles moving through a uniform magnetic field, helical paths included.

Waves, optics and modern physics are usually studied later because they lean on more mature mathematics. Geometric optics makes a friendly opening: refraction with Snell's law, thin lenses, curved mirrors and total internal reflection. The key is a sign convention. Students should pick one and use it consistently, because mixing conventions from two different textbooks is one of the most common sources of error.

Wave optics starts from Young's double slit, where the spacing between bright fringes equals wavelength times screen distance divided by slit separation. From there students move to thin-film interference, diffraction gratings and resolving power. Olympiad problems like to change one condition, such as immersing the whole setup in water or covering one slit with a thin sheet of glass, and then ask which way the pattern shifts.

Modern physics at the advanced rounds covers the photoelectric effect, where the electron's maximum kinetic energy equals hf minus the metal's work function, the de Broglie wavelength, the Bohr model of the atom, and special relativity with time dilation, length contraction and relativistic energy-momentum conservation. This material feels strange because everyday intuition stops working. What helps is writing down which reference frame every quantity belongs to, and then checking the result at low speeds, where it must reduce to classical physics.

In the 2025 OSN-K Physics round, senior high students answered 40 short-answer questions on a computer in two hours, which works out to about three minutes each. That format rewards speed and accuracy on core problems. It is a very different test from OSN-P and the national OSN, which use long written solutions and, at national level, a practical exam. Since the competition committee may revise these formats between seasons, the official guidance published for the current round stays the reference.

The right practice for a short-answer format is timed sets drawn from past papers, done with the clock running and the answer key closed. Students log how long each question took. Patterns tend to show up quickly in that log, usually a few questions that were held onto far too long when they should have been skipped and revisited. A two-pass strategy, sweeping the questions whose path is obvious and then returning to the hard ones, needs practising like any other skill.

An error log becomes the main tool at this stage. Every wrong answer gets traced to its root: a concept misunderstood, a question misread, a sign flipped or a unit left unconverted. After a few weeks the log reveals two or three kinds of mistake that keep coming back, and the next round of practice can be aimed squarely at them. After each mock, the missed questions are reworked to completion without looking at the solutions.

A readiness signal students can check for themselves: several years of past OSN-K papers completed under exam conditions, with the two-hour pace feeling normal.