Mechanics for the Physics Olympiad — NSEP and INPhO

Weightage: Mechanics is the largest and most reliably tested area in NSEP and INPhO. The syllabus is described as built on the Class XI and XII curriculum with extension beyond it, so check the current HBCSE syllabus. The method is fixed: choose a conservation law or a clean free-body diagram, and use it before reaching for Newton's equations of motion.

1. Choose the right law

Before writing any equation, ask which quantity is conserved.

• Energy is conserved if only conservative forces do work, so use it for problems about speeds and heights.
• Linear momentum is conserved along any direction with no external force, so use it for collisions and explosions.
• Angular momentum about a point is conserved if the external torque about that point is zero, so use it for orbits and spinning systems.

If none applies cleanly, draw a free-body diagram for each body and write Newton's second law in components.

2. Collisions and variable mass

In a one-dimensional elastic collision of masses $m_1$ moving with speed $u_1$ into $m_2$ at rest:

$v_1=\frac{m_1-m_2}{m_1+m_2}{u}_1,v_2=\frac{2m_1}{m_1+m_2}{u}_1$

In a perfectly inelastic collision the bodies move together with the common velocity from momentum conservation, and the energy lost is the maximum possible. Use the centre-of-mass frame to simplify: there, the total momentum is zero, and an elastic collision only reverses the relative velocity.

For a rocket expelling gas at speed $u$ relative to itself, momentum balance gives the rocket equation:

$\Delta v=u\ln \frac{m_0}{m}$

Worked example. A rocket with exhaust speed $2.5$ km/s reduces its mass to one-third of its initial value. Then $\Delta v=2.5\ln 3=2.75$ km/s.

3. Rotation and rolling

The rotational analogue of $F=ma$ is $\tau =I\alpha$. Moments of inertia to know: a thin rod about its end $\frac{1}{3}m{L}^2$, a disc about its axis $\frac{1}{2}m{R}^2$, a solid sphere $\frac{2}{5}m{R}^2$, a hollow sphere $\frac{2}{3}m{R}^2$, and a ring $m{R}^2$. The parallel axis theorem is $I=I_{cm}+m{d}^2$.

For rolling without slipping $v=\omega R$, and the kinetic energy is $\frac{1}{2}m{v}^2+\frac{1}{2}I{\omega }^2$. On an incline of angle $\theta$:

$a=\frac{g\sin \theta }{1+I/(m{R}^2)}$

Worked example. A solid sphere has $I=\frac{2}{5}m{R}^2$, so $a=\frac{g\sin \theta }{1+2/5}=\frac{5}{7}g\sin \theta$. A ring accelerates at only $\frac{1}{2}g\sin \theta$, so the sphere reaches the bottom first.

When a problem involves friction on a rolling or sliding body, decide first whether the point of contact slips, because the friction is kinetic and equal to $\mu N$ if it does, and an unknown static force if it does not.

4. Gravitation and orbits

For a circular orbit of radius $r$ around mass $M$, $v=\sqrt{GM/r}$ and $T^2\propto r^3$ (Kepler's third law). Total orbital energy is $E=-\frac{GMm}{2a}$ for semi-major axis $a$, so it is negative for a bound orbit. The escape speed from radius $R$ is $\sqrt{2GM/R}$, about $11.2$ km/s for the Earth's surface.

Angular momentum conservation gives Kepler's second law: the radius vector sweeps equal areas in equal times. At the extreme points of an elliptical orbit, $r_1{v}_1=r_2{v}_2$, and energy conservation gives the second relation.

5. Oscillations

Simple harmonic motion satisfies $\ddot{x}=-{\omega }^{2}x$. Find $\omega$ by finding the net restoring force or torque for a small displacement, or from the energy.

For a physical pendulum pivoted at distance $d$ from its centre of mass:

$T=2\pi \sqrt{\frac{I}{mgd}}$

Worked example. A uniform rod of length $L$ pivoted at one end has $I=\frac{1}{3}m{L}^2$ and $d=L/2$, so $T=2\pi \sqrt{\frac{2L}{3g}}$.

Two springs of constants $k_1,k_2$ in parallel give $k=k_1+k_2$, and in series $\frac{1}{k}=\frac{1}{k_1}+\frac{1}{k_2}$. For damped oscillations the amplitude decays as $e^{-bt/2m}$.

6. Non-inertial frames

In a frame accelerating with ${\mathbf{a}}_0$, add a pseudo force $-m{\mathbf{a}}_0$. In a frame rotating at $\omega$, there is a centrifugal force $m{\omega }^{2}r$ outward and a Coriolis force $-2m\mathbf{\omega }\times \mathbf{v}$. Often the cleanest solution to a problem on a rotating platform or an accelerating cart is to move into that frame, so that equilibrium becomes statics.

7. Fluids

Pressure at depth $h$ in a liquid is $P_0+\rho gh$, and buoyancy equals the weight of the fluid displaced. Bernoulli's equation for steady, incompressible, non-viscous flow is $P+\frac{1}{2}\rho v^2+\rho gh=\text{constant}$, together with the continuity equation $A_1{v}_1=A_2{v}_2$. The speed of efflux from a hole at depth $h$ is $\sqrt{2gh}$ (Torricelli). Surface tension gives a pressure difference $2T/R$ across a spherical film surface, and a capillary rise $h=\frac{2T\cos \theta }{\rho gr}$.

8. Method for olympiad problems

1. Draw a clear diagram and define symbols for every given and unknown.
2. Choose a law, and justify why it applies.
3. Solve symbolically to the end, and substitute numbers last.
4. Check dimensions and test limiting cases, such as $\theta \to 0$ or $m\to \infty$.
5. State the answer with its units, and compare it to a plausible estimate.

Common traps

• Using energy conservation in a collision that is not elastic.
• Applying $v=\omega R$ when the body is slipping.
• Forgetting that a pseudo force acts at the centre of mass in a non-inertial frame.
• Using Kepler's third law with the wrong radius. It uses the semi-major axis.
• Substituting numbers too early and losing the structure.

Memory aids

• "Energy for speeds, momentum for collisions, angular momentum for orbits."
• "One over one plus I over mR squared": acceleration of a rolling body.
• "Dimensions, limits, estimate": the final check.

Summary

Olympiad mechanics begins with the choice of conservation law. Collisions, variable mass and rolling are solved by momentum, the rocket equation and the rolling formula, while orbits and oscillations follow from angular momentum, energy and the restoring force.

Non-inertial frames and fluid mechanics extend the same ideas, and a disciplined method of symbols, justification and checks prevents most errors.

Exam protocol

• Name the conservation law and why it applies.
• Solve algebraically, then substitute numbers.
• Check dimensions and a limiting case.
• Confirm the current syllabus and format from HBCSE.