Which of the following pairs of physical quantities do not have the same dimensions in the dimensional formula sense?

Introduction / Context:
Dimensional analysis is a powerful tool in physics for checking the consistency of equations and understanding relationships between quantities. Different physical quantities may or may not share the same dimensions even if they appear related in everyday language. This question asks you to identify which pair of quantities has different dimensions, helping you strengthen your grasp of basic dimensional formulas.

Given Data / Assumptions:

• The pairs given are: force and pressure; work and energy; impulse and momentum; weight and force.
• We use standard dimensional symbols: mass M, length L and time T.
• We assume Newtonian mechanics definitions for these quantities.
• We need to find the pair whose members have different dimensional formulas.

Concept / Approach:
Force is defined as mass times acceleration, so its dimension is [M L T^-2]. Pressure is defined as force per unit area, so its dimension is [M L^-1 T^-2]. Work and energy are both defined as force × distance, giving [M L^2 T^-2]. Impulse is force × time, or change in momentum, and momentum is mass × velocity; both have dimension [M L T^-1]. Weight is the gravitational force acting on a body, equal to m * g, so its dimension is the same as force, [M L T^-2]. Thus, among these pairs, only force and pressure have different dimensions because pressure includes an extra factor of 1/area.

Step-by-Step Solution:
Step 1: Write the dimensional formula for force: F = m * a, so [F] = [M][L T^-2] = [M L T^-2].Step 2: Write the dimensional formula for pressure: P = F / A, area A has dimension [L^2], so [P] = [M L T^-2] / [L^2] = [M L^-1 T^-2].Step 3: For work and energy: W = F * s, so [W] = [M L T^-2][L] = [M L^2 T^-2]; energy shares the same dimensions.Step 4: For impulse and momentum: impulse J = F * t, [J] = [M L T^-2][T] = [M L T^-1]; momentum p = m * v, [p] = [M][L T^-1] = [M L T^-1].Step 5: For weight and force: weight is a gravitational force (m * g), so it has the same dimension as force, [M L T^-2].Step 6: Conclude that only force and pressure differ in dimensions, [M L T^-2] versus [M L^-1 T^-2].

Verification / Alternative check:
Dimensional formulas are often summarised in tables. Consulting such a table shows that work and energy both use joule as a unit, confirming identical dimensions. Impulse and momentum both have units like N·s and kg·m/s, which are dimensionally equivalent. Weight is measured in newtons, just like force. By contrast, pressure uses N/m^2 (pascal), clearly including an extra per square metre factor, confirming that it differs from force.

Why Other Options Are Wrong:
Work and energy represent different concepts but are measured in the same unit (joule) and share identical dimensions, so they do not form a mismatched pair. Impulse and momentum are closely related; impulse equals change in momentum, so their dimensions must match. Weight is simply a particular force due to gravity, so dimensional formulas for weight and force are the same. Therefore, the pairs in options B, C and D all have matching dimensions.

Common Pitfalls:
One common mistake is to think that different names automatically imply different dimensions. Another is to confuse pressure with force because both can be involved in contact interactions. To avoid such errors, always fall back on the defining equations: pressure is force per area, which adds an extra L^-2 factor, distinguishing it dimensionally from force.

Final Answer:
Force and pressure do not have the same dimensions; force is [M L T^-2], while pressure is [M L^-1 T^-2].