Which of the following two quantities have the same dimensions?

Correct — B. Write each quantity as a force times a length.

Work = force × displacement = [M L T⁻²] × [L] = [M L² T⁻²]
Torque = force × perpendicular distance = [M L T⁻²] × [L] = [M L² T⁻²]

The two formulas come out identical, so the pair in option (b) matches.

They stay physically different — work is a scalar measured in joules, torque a turning effect measured in newton metres — but a dimensional formula cannot see that.

Why the others are wrong

• (a)
  Power and radius of circular motion — Power is [M L² T⁻³] and a radius is a plain [L]. Two quantities cannot share dimensions when one carries mass and time and the other is only a length.
• (c)
  Power and moment of inertia — Moment of inertia is mass × distance², so [M L²] with no time term, while power is [M L² T⁻³]. The time exponents alone rule the pair out.
• (d)
  Work and angular displacement — Angular displacement is arc divided by radius, a ratio of two lengths, so it is dimensionless. It cannot match work, which is [M L² T⁻²].

A dimensional formula strips a quantity to its powers of mass, length and time. It says nothing about direction, and nothing about what the quantity means.

That is why work and torque share [M L² T⁻²] while behaving quite differently: work transfers energy along a line, torque turns a body about an axis.

Matching dimensions never prove two quantities are the same thing. The reverse test is the one that earns marks — different dimensions prove quantities cannot be equal, and that removes three options here in a single pass.

Key facts

• Work and torque both have the dimensional formula [M L² T⁻²].
• Work is measured in joules and torque in newton metres, although the two units are dimensionally identical.
• Angular displacement is dimensionless because the radian is a ratio of arc length to radius.
• Power has the dimensional formula [M L² T⁻³] and moment of inertia has [M L²].

Common traps

• Treating a shared dimensional formula as proof that two quantities are the same physical thing
• Forgetting that angle, strain and refractive index are dimensionless despite carrying names

No directly related past PYQ was found.