Fundamentals
The core facts every aspirant should own — each a titled nugget with a real-world story, the concept in plain words, and a memory trick. Works even when the internet doesn't.
1,200 fundamentals
“A figure skater spins faster by pulling their arms in — physics on ice, live.”
L = Iω, conserved when the net external torque is zero.
Memory trick: a skater spins faster by pulling their arms in — conservation of L.
“A figure skater spins faster by pulling their arms in — physics on ice, live.”
A frequent error is forgetting that pulling mass inward lowers I and therefore speeds up rotation. In reality, L = Iω, conserved when the net external torque is zero.
Memory trick: a skater spins faster by pulling their arms in — conservation of L.
“A figure skater spins faster by pulling their arms in — physics on ice, live.”
Combined translation and rotation, so total KE = ½mv² + ½Iω².
Memory trick: a rolling ball is always 'heavier' to accelerate than a sliding one.
“A figure skater spins faster by pulling their arms in — physics on ice, live.”
A frequent error is counting only translational KE and ignoring the rotational part. In reality, combined translation and rotation, so total KE = ½mv² + ½Iω².
Memory trick: a rolling ball is always 'heavier' to accelerate than a sliding one.
“A figure skater spins faster by pulling their arms in — physics on ice, live.”
Rotational Newton's second law (torque = moment of inertia × angular acceleration). Use it when about a fixed axis.
τ = Iα
“A figure skater spins faster by pulling their arms in — physics on ice, live.”
Angular momentum of a rigid body. Use it when rotation about a fixed axis.
L = Iω
“A figure skater spins faster by pulling their arms in — physics on ice, live.”
Parallel-axis theorem for shifting the axis by distance d. Use it when the two axes are parallel.
I = I_cm + Md²
“A figure skater spins faster by pulling their arms in — physics on ice, live.”
Force causes linear acceleration; torque causes angular acceleration and depends on where the force is applied.
“A figure skater spins faster by pulling their arms in — physics on ice, live.”
Mass resists linear acceleration; moment of inertia resists angular acceleration and also depends on the shape and axis.
“A figure skater spins faster by pulling their arms in — physics on ice, live.”
A rolling body has both translational and rotational KE and (with friction) doesn't slip; a sliding body has only translational KE.
“A figure skater spins faster by pulling their arms in — physics on ice, live.”
A rolling object stores energy in rotation too — don't leave out ½Iω².
“The same gravity that drops an apple keeps the Moon endlessly 'falling' around the Earth.”
Every two masses attract with F = G·m₁m₂/r² along the line joining them.
Memory trick: it's an inverse-square law, so distance matters a lot.
“The same gravity that drops an apple keeps the Moon endlessly 'falling' around the Earth.”
A frequent error is forgetting the force depends on 1/r² — doubling the distance quarters the force. In reality, every two masses attract with F = G·m₁m₂/r² along the line joining them.
Memory trick: it's an inverse-square law, so distance matters a lot.
“The same gravity that drops an apple keeps the Moon endlessly 'falling' around the Earth.”
G = GM/R² at a planet's surface; it decreases with both altitude and depth.
Memory trick: g falls off with height and also drops as you go below the surface.
“The same gravity that drops an apple keeps the Moon endlessly 'falling' around the Earth.”
A frequent error is treating g as a fixed 9.8 everywhere, including in orbit. In reality, g = GM/R² at a planet's surface; it decreases with both altitude and depth.
Memory trick: g falls off with height and also drops as you go below the surface.
“The same gravity that drops an apple keeps the Moon endlessly 'falling' around the Earth.”
For large distances PE = −GMm/r (negative, zero at infinity), not mgh.
Memory trick: mgh is only the near-surface approximation of −GMm/r.
“The same gravity that drops an apple keeps the Moon endlessly 'falling' around the Earth.”
A frequent error is using mgh far from the surface, where it no longer holds. In reality, for large distances PE = −GMm/r (negative, zero at infinity), not mgh.
Memory trick: mgh is only the near-surface approximation of −GMm/r.
“The same gravity that drops an apple keeps the Moon endlessly 'falling' around the Earth.”
A frequent error is confusing it with orbital velocity. In reality, the minimum speed to leave a planet's gravity, v = √(2GM/R) ≈ 11.2 km/s for Earth.
Memory trick: escape velocity is √2 times the orbital velocity at the surface.
“The same gravity that drops an apple keeps the Moon endlessly 'falling' around the Earth.”
Orbits are ellipses; a planet sweeps equal areas in equal times; T² ∝ r³.
Memory trick: equal-areas means faster at perihelion, slower at aphelion.
“The same gravity that drops an apple keeps the Moon endlessly 'falling' around the Earth.”
A frequent error is assuming orbital speed is constant — planets move faster near the Sun. In reality, orbits are ellipses; a planet sweeps equal areas in equal times; T² ∝ r³.
Memory trick: equal-areas means faster at perihelion, slower at aphelion.
“The same gravity that drops an apple keeps the Moon endlessly 'falling' around the Earth.”
The gravitational force between two masses. Use it when point masses (or spheres) separated by r.
F = G m₁m₂/r²
“The same gravity that drops an apple keeps the Moon endlessly 'falling' around the Earth.”
Surface gravity of a body of mass M and radius R. Use it when at the surface.
g = GM/R²
“The same gravity that drops an apple keeps the Moon endlessly 'falling' around the Earth.”
Speed needed to escape a planet's gravity entirely. Use it when ignoring air resistance.
v_escape = √(2GM/R)
“The same gravity that drops an apple keeps the Moon endlessly 'falling' around the Earth.”
Speed of a satellite in a circular orbit of radius r. Use it when circular orbit.
v_orbit = √(GM/r)