Force = mass x acceleration, so its dimensional formula is [M L T^-2]. Work and energy are force x distance, so both share [M L^2 T^-2]. Power is work per time, so it is [M L^2 T^-3]. Memory hook: start from force [MLT-2], multiply by L to get energy, then divide by T to get power.
Force [MLT-2] is the base. Multiply by a length to reach work/energy [ML2T-2], then divide by time to reach power [ML2T-3].
Your doubts, answered
Do work and energy really have the same dimensions?
Yes. Work is force times displacement, W = F x d = [MLT^-2] x [L] = [M L^2 T^-2]. Energy is the capacity to do work, so it must have the same dimensions [M L^2 T^-2]. Kinetic energy (1/2)mv^2 = [M][LT^-1]^2 = [M L^2 T^-2] and potential energy mgh = [M][LT^-2][L] = [M L^2 T^-2] both match. Same dimensions means the same physical nature, which is why they can be added and share the unit joule.
Why does power have T to the power minus 3?
Power = work / time = [M L^2 T^-2] / [T] = [M L^2 T^-3]. You start with energy [M L^2 T^-2] and divide by one more factor of time, so the time exponent goes from -2 to -3. This single extra division by T is the only difference between energy and power dimensions.
How do I derive the dimensional formula of force step by step?
Force = mass x acceleration. Mass is [M]. Acceleration is change in velocity per time = [LT^-1]/[T] = [LT^-2]. Multiply: F = [M] x [LT^-2] = [M L T^-2]. This is the base building block, because work, energy, torque and pressure are all built from force.
What is the difference between the dimensions of energy and power?
Energy [M L^2 T^-2] measures a total amount of work stored or transferred. Power [M L^2 T^-3] measures how fast that energy is delivered (energy per second). Power has exactly one more T^-1 than energy. If two quantities differ only by a factor of time, one is a 'rate' of the other, and this shows up as the extra T^-1.
Are kinetic energy and potential energy dimensions different?
No, they are identical: both are [M L^2 T^-2]. Kinetic energy (1/2)mv^2 uses velocity squared, and potential energy mgh uses g x h; both reduce to [M L^2 T^-2]. This is expected because all forms of energy must have the same dimensions so they can be added in the conservation of energy equation.
⚠️ The NEET trap ✗ Writing the dimensions of power as [M L^2 T^-2], the same as energy. ✓ Power is energy per unit time, so it is [M L^2 T^-3] with T to the power -3, not -2. 🧠 Energy is the total; power is per second. Every 'per second' adds one more T^-1. Force MLT-2, times L is energy ML2T-2, divide by T is power ML2T-3.
Real NEET questions
NEET 2021
If force [F], acceleration [A] and time [T] are chosen as the fundamental physical quantities, find the dimensions of energy.
A · [F][A][T^-1]
B · [F][A^-1][T]
C · [F][A][T]
D · [F][A][T^2] ✓
Solution: Energy = force x distance, so we need distance in terms of F, A, T. Distance from motion: d = (1/2) A T^2, so [distance] = [A][T^2]. Therefore energy = [F] x [A][T^2] = [F][A][T^2]. Option D. Trick: since F, A, T are the chosen base quantities here, do not fall back to M, L, T; express distance using acceleration and time.
NEET 2024
A force defined by F = alpha t + beta t^2 acts on a particle at a given time t. The factor which is dimensionless, if alpha and beta are constants, is:
A · alpha/(beta t) ✓
B · alpha beta / t
C · alpha beta t
D · beta t / alpha
Solution: By the principle of homogeneity each term must have dimensions of force [M L T^-2]. From alpha t = F, [alpha] = [F]/[t] = [M L T^-3]. From beta t^2 = F, [beta] = [F]/[t^2] = [M L T^-4]. Now test alpha/(beta t) = [M L T^-3] / ([M L T^-4][T]) = [M L T^-3]/[M L T^-3] = dimensionless. Option A.
NEET 2021
If E and G respectively denote energy and gravitational constant, then E/G has the dimensions of:
A · [M][L^0][T^0]
B · [M^2][L^-2][T^-1]
C · [M^2][L^-1][T^0] ✓
D · [M][L^-1][T^-1]
Solution: Energy E = [M L^2 T^-2]. Gravitational constant G = [M^-1 L^3 T^-2] (from F = G m1 m2 / r^2). Divide: E/G = [M L^2 T^-2] / [M^-1 L^3 T^-2] = [M^(1-(-1)) L^(2-3) T^(-2-(-2))] = [M^2 L^-1 T^0]. Option C. This uses the energy dimension [M L^2 T^-2] directly.
Solved Units And Measurements NEET PYQs
Try the real previous-year questions from this chapter — each with the answer and a full solution.
Force = mass x acceleration = [M] x [L T^-2] = [M L T^-2]. Its SI unit is the newton (N).
What are the dimensions of work and energy?
Both are [M L^2 T^-2] because work = force x displacement and energy is the capacity to do work. Their SI unit is the joule (J).
What is the dimensional formula of power?
Power = work / time = [M L^2 T^-2] / [T] = [M L^2 T^-3]. Its SI unit is the watt (W).
Why do force, work and power all start with M?
All three depend on mass through force (F = ma). Work and power are built by multiplying force by length or dividing by time, but the single mass factor [M] carries through all of them.
Is torque the same dimension as work?
Yes, torque = force x perpendicular distance = [M L^2 T^-2], the same as work and energy. But torque is a vector and work is a scalar, so they are different physical quantities despite equal dimensions. This is a common NEET trap.
