Advanced Concepts
The JEE-Advanced / NEET-hard concepts that separate top rankers — each a titled nugget with a real-world story, the idea in plain words, and a memory trick. Works even when the internet doesn't.
400 advanced concepts
“An aeroplane wing, a hydraulic jack and a dripping tap all obey the same fluid rules.”
Bernoulli's equation. Use it when ideal, steady flow along a streamline.
P + 1/2 rho v^2 + rho g h = constant
“An aeroplane wing, a hydraulic jack and a dripping tap all obey the same fluid rules.”
Terminal velocity of a sphere (Stokes). Use it when small sphere, viscous laminar flow.
v_t = 2 r^2 (rho - sigma) g / (9 eta)
“An aeroplane wing, a hydraulic jack and a dripping tap all obey the same fluid rules.”
Capillary rise. Use it when narrow tube, contact angle theta.
h = 2 S cos(theta) / (r rho g)
“An aeroplane wing, a hydraulic jack and a dripping tap all obey the same fluid rules.”
Excess pressure from surface tension. Use it when soap bubble has two surfaces; a drop has one.
Excess P = 4S/r (bubble), 2S/r (drop)
“An aeroplane wing, a hydraulic jack and a dripping tap all obey the same fluid rules.”
Laminar flow is smooth and layered (low Reynolds number); turbulent flow is chaotic and mixing (high Reynolds number).
“An aeroplane wing, a hydraulic jack and a dripping tap all obey the same fluid rules.”
A liquid drop has one surface (excess pressure 2S/r); a soap bubble has two surfaces (excess pressure 4S/r).
“An aeroplane wing, a hydraulic jack and a dripping tap all obey the same fluid rules.”
Stress is force per unit area applied; strain is the fractional deformation produced; their ratio is a modulus.
“An aeroplane wing, a hydraulic jack and a dripping tap all obey the same fluid rules.”
It assumes ideal, steady, non-viscous flow along a streamline; real viscous or turbulent flows deviate.
“GPS satellites must correct for both special and general relativity, or your maps would drift by kilometres a day.”
For two masses the potential energy is negative, U = -G M m / r, taken zero at infinity; mgh is only its near-surface approximation.
Memory trick: U = -GMm/r; mgh only works near the surface.
“GPS satellites must correct for both special and general relativity, or your maps would drift by kilometres a day.”
A frequent error is using mgh for large heights or between astronomical bodies. In reality, for two masses the potential energy is negative, U = -G M m / r, taken zero at infinity; mgh is only its near-surface approximation.
Memory trick: U = -GMm/r; mgh only works near the surface.
“GPS satellites must correct for both special and general relativity, or your maps would drift by kilometres a day.”
Escape velocity is sqrt(2) times the orbital velocity at the same radius.
Memory trick: v_escape = sqrt(2) * v_orbital.
“GPS satellites must correct for both special and general relativity, or your maps would drift by kilometres a day.”
A frequent error is confusing the two or forgetting the sqrt(2) factor. In reality, escape velocity is sqrt(2) times the orbital velocity at the same radius.
Memory trick: v_escape = sqrt(2) * v_orbital.
“GPS satellites must correct for both special and general relativity, or your maps would drift by kilometres a day.”
A satellite's total energy is negative, E = -G M m / 2r, half the (negative) potential energy.
Memory trick: E = -GMm/2r; raising the orbit adds energy.
“GPS satellites must correct for both special and general relativity, or your maps would drift by kilometres a day.”
A frequent error is thinking a higher orbit has less total energy; it has MORE (less negative). In reality, a satellite's total energy is negative, E = -G M m / 2r, half the (negative) potential energy.
Memory trick: E = -GMm/2r; raising the orbit adds energy.
“GPS satellites must correct for both special and general relativity, or your maps would drift by kilometres a day.”
Orbits are ellipses (1st), equal areas are swept in equal times (2nd, from angular-momentum conservation), and T^2 ~ a^3 (3rd).
Memory trick: planets move fastest at perihelion (equal areas).
“GPS satellites must correct for both special and general relativity, or your maps would drift by kilometres a day.”
A frequent error is assuming orbital speed is constant around an ellipse. In reality, orbits are ellipses (1st), equal areas are swept in equal times (2nd, from angular-momentum conservation), and T^2 ~ a^3 (3rd).
Memory trick: planets move fastest at perihelion (equal areas).
“GPS satellites must correct for both special and general relativity, or your maps would drift by kilometres a day.”
G decreases with altitude as 1/(R+h)^2 and decreases with depth roughly linearly to zero at the centre.
Memory trick: g is maximum at the surface, falling both up and down.
“GPS satellites must correct for both special and general relativity, or your maps would drift by kilometres a day.”
A frequent error is thinking g keeps rising as you go deeper. In reality, g decreases with altitude as 1/(R+h)^2 and decreases with depth roughly linearly to zero at the centre.
Memory trick: g is maximum at the surface, falling both up and down.
“GPS satellites must correct for both special and general relativity, or your maps would drift by kilometres a day.”
The field is the negative gradient of the potential; inside a uniform shell the field is zero but the potential is constant and non-zero.
Memory trick: shell: field 0 inside, potential constant = value at surface.
“GPS satellites must correct for both special and general relativity, or your maps would drift by kilometres a day.”
A frequent error is thinking zero field means zero potential inside a shell. In reality, the field is the negative gradient of the potential; inside a uniform shell the field is zero but the potential is constant and non-zero.
Memory trick: shell: field 0 inside, potential constant = value at surface.
“GPS satellites must correct for both special and general relativity, or your maps would drift by kilometres a day.”
A satellite orbiting once per 24 h over the equator appears fixed in the sky, at about 36,000 km altitude.
Memory trick: geostationary needs T = 24 h, equatorial, one specific radius.
“GPS satellites must correct for both special and general relativity, or your maps would drift by kilometres a day.”
A frequent error is thinking any high orbit is geostationary. In reality, a satellite orbiting once per 24 h over the equator appears fixed in the sky, at about 36,000 km altitude.
Memory trick: geostationary needs T = 24 h, equatorial, one specific radius.
“GPS satellites must correct for both special and general relativity, or your maps would drift by kilometres a day.”
Gravitational potential energy of two masses. Use it when point masses, zero at infinity.
U = -G M m / r
“GPS satellites must correct for both special and general relativity, or your maps would drift by kilometres a day.”
Speed of a circular orbit of radius r. Use it when circular orbit.
v_orbit = sqrt(G M / r)