Which one of the following pairs of physical quantities has the same dimensions in the SI system?

Introduction / Context:
Dimensional analysis is a powerful tool in physics for checking the correctness of formulas and for comparing different physical quantities. Two quantities that share the same dimensions can often be added or compared meaningfully, while quantities with different dimensions cannot. This question asks which pair of quantities has identical dimensions in the SI system, which is a classic concept tested in exams.

Given Data / Assumptions:

• We have four pairs of physical quantities.
• We must determine which pair has the same dimensions.
• Standard SI base dimensions are used: mass (M), length (L), time (T).

Concept / Approach:
To answer this, we express each quantity in terms of the base dimensions M, L, and T. If two quantities reduce to the same dimensional formula, then they are dimensionally equivalent. Momentum is defined as mass times velocity, and impulse is defined as force times time, or change in momentum. Therefore, their dimensions are closely related. For other pairs like power and Young modulus or energy and angular momentum, we must check their formulas and compare dimensions carefully.

Step-by-Step Solution:
Step 1: Write dimensions of momentum: p = m * v, so [p] = M * L T^-1 = M L T^-1. Step 2: Write dimensions of impulse: J = F * t, force F has dimensions M L T^-2, so [J] = M L T^-2 * T = M L T^-1. Step 3: Since momentum and impulse both have [M L T^-1], they share the same dimensions. Step 4: Check one incorrect pair as example. Power P = work per unit time, so [P] = (M L^2 T^-2) / T = M L^2 T^-3. Young modulus Y = stress / strain, so [Y] = (force per area) = (M L T^-2) / L^2 = M L^-1 T^-2. These are clearly different.

Verification / Alternative check:
For energy and angular momentum: energy E has dimensions M L^2 T^-2, while angular momentum L has dimensions M L^2 T^-1. They differ by one power of time. For force constant of a spring k (force per unit extension) dimensions are M T^-2, whereas moment of inertia I has dimensions M L^2, again clearly different. Only momentum and impulse match, confirming our choice.

Why Other Options Are Wrong:
Power and Young modulus: Power is related to rate of doing work, Young modulus to stiffness; their dimensional formulas are different. Energy and angular momentum: Both involve M and L^2 but differ in the power of T, so they are not dimensionally equal. Force constant and moment of inertia: Force constant has no factor of L, while moment of inertia has L^2, so they differ.

Common Pitfalls:
Many students assume that if two quantities are related in mechanics, they must have different dimensions, or they misremember the formulas. Another common mistake is to forget that impulse is defined as change in momentum, which strongly suggests that their dimensions are identical. Always build the dimensional formula from the basic definition, rather than relying only on memory.

Final Answer:
The only pair with the same dimensions is momentum and impulse.