Mechanics Study Guide

Level High SchoolReview time ~20 hours

Mechanics is the foundation of all physics: if you can describe motion, explain it with forces, and track it with energy and momentum, every later subject gets easier. This guide organizes the whole subject into checkable topics, the formulas that matter, the mistakes that cost marks, and the tactics that earn them.

Key formulas

Centripetal Force — F = mv²/r

The inward force needed for circular motion equals mass times speed squared divided by radius.

Gravitational Potential Energy — PE = mgh

Gravitational potential energy equals mass times gravity times height.

Hooke's Law — F = kx

The restoring force of a spring equals its stiffness times the displacement from equilibrium.

Kinetic Energy Formula — KE = ½mv²

Kinetic energy equals one-half times mass times velocity squared.

Momentum Formula — p = mv

Momentum equals mass times velocity.

Newton's Second Law — F = ma

The net force on an object equals its mass times its acceleration.

Power Formula — P = W/t

Power equals work done divided by the time taken.

Work Formula — W = Fd

Work equals force times distance moved in the direction of the force.

Topic checklist

Kinematics: describing motion

Distinguish displacement from distance and velocity from speed.
Use $v = \Delta x / \Delta t$ and $a = \Delta v / \Delta t$ with correct signs.
Read position-time and velocity-time graphs: slope and area meanings.
Choose a consistent sign convention and stick to it for the whole problem.

Newton's laws and free-body diagrams

State each law in words and identify action-reaction pairs.
Draw complete free-body diagrams before writing any equation.
Apply $F = ma$ along each axis separately.
Recognize that constant velocity means zero net force, not zero force.

Friction

Know when static friction ( $f_s \le \mu_s N$ ) versus kinetic friction ( $f_k = \mu_k N$ ) applies.
Remember friction opposes relative motion, not necessarily motion itself.
Include the normal force correctly on inclines ( $N = mg\cos\theta$ ).

Work, energy, and power

Compute work for constant forces and interpret negative work.
Track energy: $KE_i + PE_i + W_{nc} = KE_f + PE_f$ .
Use $P = W/t$ for rates and $1\ \mathrm{hp} \approx 746\ \mathrm{W}$ .
Recognize conservative versus non-conservative forces.

Momentum and collisions

Apply conservation of momentum to collisions and explosions.
Distinguish elastic (KE conserved) from inelastic collisions.
Use impulse $J = F\Delta t = \Delta p$ for short impacts.

Circular motion, rotation, and springs

Use $F_c = mv^2/r$ toward the center for uniform circular motion.
Relate torque $\tau = rF\sin\theta$ to angular acceleration.
Apply $F = kx$ and $PE = \tfrac{1}{2}kx^2$ for springs.

Common mistakes

Mistake: Using mass and weight interchangeably.

Fix:

Mass resists acceleration; weight is the gravitational force $mg$ on that mass. On the Moon your mass is unchanged but your weight drops. Always check which one the problem asks for.

Mistake: Plugging the applied force into $F = ma$ instead of the net force.

Fix:

Net force means the vector sum after cancellation. Draw the free-body diagram, add components, and only then multiply by mass. A 5000 N engine push against 2000 N of drag gives 3000 N of net force.

Mistake: Forgetting the 1/2 in kinetic energy.

Fix:

The $\tfrac{1}{2}$ is not optional. Derive it once from $W = Fd$ with $v^2 = u^2 + 2as$ so you never drop it again — doubling speed quadruples energy, and the half keeps the books balanced.

Mistake: Assuming normal force always equals mg.

Fix:

Normal force equals $mg$ only on flat, unaccelerated surfaces. On an incline it is $mg\cos\theta$ ; in an elevator it is $m(g \pm a)$ . Always derive $N$ from the perpendicular force balance.

Mistake: Applying conservation laws without checking the conditions.

Fix:

Momentum is conserved in collisions only when external impulses are negligible during the impact. Energy is conserved only in elastic collisions. State your system and your assumption before calculating.

Exam tips

Always start with a diagram: free-body diagrams for forces, energy bars for energy problems.
Write the general equation first, then substitute numbers — you earn method marks even with arithmetic slips.
Check units in your final answer: $\mathrm{kg\cdot m/s^2}$ must collapse to newtons.
For multi-part problems, carry symbols (not rounded numbers) between parts.
If an answer looks absurd (a car at 3000 m/s), recheck — physics answers must be plausible.