Principle of dimensional homogeneity

“Building a LEGO model, every brick must fit the same-size slot; mismatched sizes make the structure fall.”

Every term in a physical equation must carry identical dimensions; otherwise the equation cannot represent a real relation. This check catches unit mismatches early. However it cannot reveal missing dimensionless constants (e.g., 2π) or validate equations that involve sums of like-dimensional terms but are physically wrong.

Memory trick: Like-units only, like-LEGO only

Impulse-momentum theorem

“When you slam a door shut, the quick push you feel is the impulse that changes the door’s momentum.”

Impulse J = ∫F dt equals the vector change in momentum Δp. For constant mass Δp = mΔv, but if mass varies (e.g., rocket) you must use p = mv at each instant. A common slip is treating impulse as a scalar and ignoring its direction, which flips signs in vector problems.

J = F*t = dp

Memory trick: Impulse = Force×Time = Δ(mv) – keep direction!

Angular impulse

“When you spin a gyroscope and tap its rim, the twist you feel is the torque’s impulse changing its angular momentum about the same axle.”

Angular impulse is the vector integral ∫τ dt = ΔL. The torque and the angular momentum must be taken about the same axis; otherwise components differ and the sign may seem to flip. A frequent trap is mixing axes and assuming the sign automatically reverses.

Memory trick: Same axis → same sign; don’t swap axes.

Areal velocity is constant

“A planet’s line to the Sun sweeps equal areas each second, like a sprinkler’s rotating arm covering the same lawn area no matter the radius.”

Areal velocity dA/dt = L/2m stays constant because the torque about the focus is zero for any central force, so angular momentum L is conserved. This holds for any central potential, not only the inverse-square law. The exam trap: forgetting that a non-central force (e.g., drag) would break the constancy.

dA/dt = L/2m

Memory trick: Central force → zero torque → L const → dA/dt const

Time period of a liquid in a U-tube

“Watching the water slosh back and forth in a U-tube is like the bob of a pendulum tracing a tiny arc.”

The liquid column acts as a single mass whose restoring force comes from the weight difference between the two arms, giving SHM with period T=2π√(L/2g). The effective length is half the total column, not the full length. Forgetting the ½ factor gives a wrong period.

Memory trick: U-tube: T ∝ √(L/2g) – remember the half-L in the root.

Speed of a wave on a string

“When you tighten a guitar string, the note jumps higher because the wave travels faster along the tighter rope.”

The wave speed on a stretched string is v=√(T/μ), with T the uniform tension and μ the linear mass density. Higher tension or lighter string raises v, while a heavier string lowers it. A common trap is using the string’s total mass instead of μ; you must divide by its length.

v = sqrt(T/mu)

Memory trick: v ∝ √T ÷ √μ → tighter, lighter = faster

Equipartition of energy

“Like popcorn kernels, gas molecules jiggle; each way they move stores ½kT of energy.”

Equipartition gives an average energy ½kT for every quadratic degree of freedom. Monatomic gases have 3 translational DOF; diatomic add 2 rotational (rotational modes freeze at very low T). Vibrational modes contribute ½kT per kinetic and potential term only when hν ≪ kT; otherwise they are frozen and add nothing. Ignoring the hν ≪ kT condition overestimates heat capacities.

Memory trick: ½kT per quadratic DOF; vibrational only if hν ≪ kT.

Energy density of an electric field

“A charged capacitor’s interior stores energy like a stretched rubber band stores potential in every tiny segment.”

Energy density u = ½ ε0 E² follows from work needed to build the field; the factor ½ comes from integrating dW = ε0 E dE·dV. In dielectrics replace ε0 by ε. A common trap is omitting the ½ or using ε instead of ε0 for vacuum.

u = eps0 E²/2

Memory trick: u = ½ ε0 E² → ‘Half-Epsilon-E-squared’

Temperature dependence of resistance

“Heating a copper wire like a kitchen pan makes its atoms jiggle, slowing electron traffic.”

For metals R≈R0(1+αΔT) with α>0, valid only for small ΔT where α is nearly constant. Semiconductors have α<0. Using the linear formula over large temperature ranges or ignoring the sign of α gives wrong resistance.

R = R0(1 + alpha dT)

Memory trick: Metal α>0, Semiconductor α<0; linear only for small ΔT.

Torque on a current loop

“A compass needle on a rotating turntable aligns with Earth’s field just like a current loop feels torque.”

τ = N I A B sinθ gives the magnitude. The loop’s magnetic moment μ = N I A points normal to the plane (right-hand rule). τ = μ × B, so direction follows the right-hand rule and is zero at 0° or 180°. The common trap is forgetting the vector direction of μ and omitting the factor N when multiple turns are present.

tau = NIAB sin(theta)

Memory trick: μ = N I A, τ = μ B sinθ → ‘mu-B-sin’

Motional EMF

“When a metal ruler slides across a magnetic rail of a train, the tiny voltage you can measure is Blv.”

A rod of length l moving with velocity v perpendicular to a uniform B field develops an emf ε = B l v. Use the component of v ⟂ B; if motion is at angle θ, ε = B l v sinθ. Trap: applying Blv without checking the angle gives a wrong sign or magnitude.

EMF = Blv

Memory trick: ε = B l v⊥ (use right-hand rule)

Lens maker's formula

“Adjusting a camera's zoom mimics changing the two radii of a lens, which follows the maker's formula.”

For a thin lens in air, 1/f = (n-1)(1/R₁ − 1/R₂). R is taken positive when the surface is convex toward the incoming light and negative when concave. Missing this sign rule flips the focal length sign, a frequent JEE slip.

Memory trick: Convex = + , Concave = –  for R₁,R₂

Threshold wavelength

“Just like a lock needs a minimum key length, a metal needs photons shorter than λ₀ to free electrons.”

λ₀ = hc/φ is the longest wavelength that can eject electrons. Photons with λ > λ₀ have insufficient energy regardless of intensity. Trap: increasing intensity cannot overcome the work-function limit.

lambda₀ = hc/phi

Memory trick: λ₀ ∝ 1/φ; shorter λ = higher photon energy.

Displacement current

“While charging a capacitor, the growing electric field between its plates produces a changing flux that acts like a current in the circuit.”

Maxwell added the displacement term to Ampère’s law: ∮B·dl = μ0(I_conduction+ε0 dΦ_E/dt). The displacement current density is J_d = ε0 ∂E/∂t. It ensures continuity where ∂E/∂t≠0, e.g., between capacitor plates. Trap: using I=ε0 dE/dt ignores the electric-flux area; the correct scalar form uses dΦ_E/dt.

Memory trick: J_d=ε0·∂E/∂t; I_disp=ε0·dΦ_E/dt

Reynolds number

“When you watch water swirl smoothly in a garden hose, the diameter of the hose decides the Re that tells if the flow stays laminar.”

Re = ρ v L/μ, where L is the characteristic length. For a circular pipe L = pipe diameter; for non-circular ducts use hydraulic diameter D_h = 4A/P. Laminar flow persists up to Re≈2300 in smooth pipes, but the critical value shifts with roughness and shape. A common trap: using the radius instead of diameter halves the Re and may misclassify the regime.

Memory trick: Re = ρvD/μ → remember D = Diameter, not radius

Resolving a vector into components

“Like splitting the wind on a kite into pull along the string and lift upward.”

A vector can be expressed as perpendicular x- and y-components: A_x = A cosθ, A_y = A sinθ. The sign of each component comes from the actual direction of the arrow, not from the trig function. A common trap is using the angle measured from the +x-axis without checking the quadrant, which flips the sign of cos or sin.

Memory trick: Cos = adjacent (next to angle), Sin = opposite (across). Sign from arrow direction.

Heat versus temperature

“A cup of tea at 70 °C pours heat into a colder spoon, even though the spoon feels cooler.”

Temperature is a state variable that sets the direction of heat flow; heat is energy transferred due to a temperature difference. Heat has units of joule and depends on path, while temperature is a property of the body. Exam trap: never add heat values as if they were temperatures.

Memory trick: Temp = where, Heat = how much moves

Microscopic origin of gas pressure

“Like a crowd of ping-pong balls bouncing off a table, gas molecules constantly hit the container walls, creating pressure.”

Pressure is the rate of momentum transfer to the wall: each collision changes molecular momentum and the sum over area gives force per area. More frequent or harder hits raise pressure. Trap: forgetting that pressure depends on momentum change, not just collision count.

Memory trick: P = (collision rate)×Δp ÷ area

Potential as energy per charge

“Like the voltage of a battery tells how much energy each coulomb can deliver, V = U/q.”

Electric potential V at a point equals the potential energy U of a test charge q placed there divided by q. Because V is a scalar, contributions from several source charges add algebraically, keeping each source’s sign. A trap: dropping the sign of a negative source when summing potentials.

Memory trick: V = U per coulomb → keep sign, add scalars.

Magnetic dipole moment

“A fridge magnet’s hidden south face to north face defines its dipole moment, just as a tiny compass needle aligns in a field.”

A bar magnet is modeled as a magnetic dipole with moment m directed from its internal south pole to north pole. In a uniform external field B, the torque is τ = m × B, and the potential energy is –m·B. The classic trap is to reverse m because field lines outside run north→south; remember the moment points opposite to the external line direction inside the magnet.

Memory trick: Moment: S→N inside, like arrow from hidden south to north.

Rutherford scattering and nuclear atom

“Like a bullet through a chicken wire, most α-particles pierce the gold foil, but a few bounce back sharply, revealing a tiny, dense nucleus.”

Alpha particles mostly go straight because the atom is mostly empty. The rare large-angle deflections follow Rutherford’s 1/sin⁴(θ/2) law and prove that the positive charge and most mass lie in a nucleus ≈10⁻¹⁵ m. Trap: assuming all scattering angles are equally likely – the probability falls sharply with angle.

Memory trick: Few turn, many go straight.

Nuclear size and density

“When you pack identical grapes into a sphere, the grape-to-sphere volume ratio stays the same, so the mass per volume of the bunch never changes.”

Nuclear radius R = R0 A^(1/3) (R0≈1.2 fm). Mass density ρ = (A·m)/(4/3πR³)=3m/(4πR0³)≈2.3×10¹⁷ kg m⁻3, independent of A. The trap: ρ≈3A/(4πR³) is nucleon number density, not mass density, and very heavy nuclei show a slight fall-off.

Memory trick: R∝A1/3 ⇒ A cancels in ρ = (A·m)/(4/3πR³) → ρ constant

Least count and instrument reading

“Reading a vernier caliper is like counting steps on a ladder; you can’t see a rung smaller than the step size.”

Least count is the smallest division an instrument can resolve. The true reading = main-scale reading + (coinciding division × least-count) – zero error (add if zero error is negative). A common trap is to add a positive zero error instead of subtracting it.

Memory trick: Positive zero error → subtract; negative → add.

Instantaneous velocity

“A runner’s GPS logs his position each second; the instantaneous velocity at any moment is the slope of the position-time curve right at that point.”

Instantaneous velocity v(t) is the derivative dx/dt, i.e., the slope of the x-t graph at that exact point. It exists only if the position function is differentiable there; a cusp or corner gives no defined velocity. Beware: exams often ask for speed magnitude, so give |v| when only speed is required.

Memory trick: V = dx/dt → think V as the graph’s slope