Which pair has the same dimensions?

To determine which pair has the same dimensions, we analyze the dimensional formula for each quantity based on the provided text:

1. Work and Power:

• Work is defined as force times displacement ($W = \mathbf{F} \cdot \mathbf{d}$). Its dimensions are $[ML^2T^{-2}]$ .
• Power is defined as work done per unit time ($P = W/t$). Its dimensions are $[ML^2T^{-3}]$ .
• Result: Different dimensions.

2. Density and Relative Density:

• Density is mass per unit volume ($ho = m/V$). Its dimensions are $[ML^{-3}]$ .
• Relative Density is a ratio of densities (density of substance / density of water). Being a ratio of identical quantities, it is dimensionless $[M^0L^0T^0]$ .
• Result: Different dimensions.

3. Momentum and Impulse:

• Momentum ($p$) is the product of mass and velocity ($p = mv$). Its dimensions are $[MLT^{-1}]$ .
• Impulse is defined as the product of force and time duration, which equals the change in momentum (Impulse = $\mathbf{F} \times t$). Its dimensions are $[MLT^{-1}]$ .
• Result: Same dimensions.

4. Stress and Strain:

• Stress is the restoring force per unit area. Its dimensions are $[ML^{-1}T^{-2}]$ .
• Strain is the ratio of change in dimension to the original dimension (e.g., $\Delta L / L$). It is a dimensionless quantity $[M^0L^0T^0]$ .
• Result: Different dimensions.