Q.(a)

The dimensions of a physical quantity record how it is built from the base quantities — how many powers of mass, length, time (and, where needed, electric current, temperature, amount, luminous intensity) it contains. Written in square brackets, force is [M L T⁻²] and energy is [M L² T⁻²]. This single idea — that an equation must be dimensionally consistent — is one of the most powerful cross-checks in physics, and JEE Main mines it heavily.

1 — Dimensional formulae. Every derived quantity has a dimensional formula obtained from its defining relation: velocity [LT⁻¹], acceleration [LT⁻²], force [MLT⁻²], work/energy/torque [ML²T⁻²], power [ML²T⁻³], pressure/stress [ML⁻¹T⁻²], momentum/impulse [MLT⁻¹]. For constants, isolate the constant in its equation and read off its dimensions — from F = Gm₁m₂/r², [G] = [M⁻¹L³T⁻²]; from E = hν, [h] = [ML²T⁻¹]; from PV = nRT, [R] = [ML²T⁻²K⁻¹mol⁻¹]. Electrical quantities carry the base dimension of current [A]: charge [AT], potential [ML²T⁻³A⁻¹], resistance [ML²T⁻³A⁻²].

2 — The principle of homogeneity. In any valid equation, every additive term has the same dimensions. This lets you (a) test whether a given equation can be correct, (b) find a missing exponent by matching the powers of M, L and T on both sides, and (c) reject a proposed formula that is dimensionally inconsistent. A crucial corollary: the argument of any sin, cos, log or exponential — and any exponent — must be dimensionless. So in y = A sin(ωt), ωt is dimensionless, forcing [ω] = [T⁻¹].

3 — Deriving a relation. When a quantity depends on a few others, assume a power-law y = k·x₁ᵃ x₂ᵇ x₃ᶜ, write the dimensions of both sides, and equate the exponents of M, L, T to solve for a, b, c. This recovers the form of many results — the pendulum's T ∝ √(L/g), the speed of a wave on a string v ∝ √(T/μ), Stokes' drag F ∝ ηrv. The dimensionless constant k (like the 2π in the pendulum) is what dimensional analysis cannot supply.

4 — Converting between systems of units. Because a physical quantity is unit-independent, n₁u₁ = n₂u₂. Using the dimensional formula, n₂ = n₁ [M₁/M₂]ᵃ [L₁/L₂]ᵇ [T₁/T₂]ᶜ. This is how 1 N = 10⁵ dyne, 1 J = 10⁷ erg, and how the numerical value of a constant like G changes from SI to CGS. The same machinery lets you express a quantity when a new set of quantities (say force, velocity, time) is chosen as fundamental. …