Which one of the following does not have the same dimensions?

To determine which pair has different dimensions, we analyze each option:

1. Work and Energy: Work ($W = F \cdot d$) has dimensions $[ML^2T^{-2}]$ . Energy ($E$) has dimensions $[ML^2T^{-2}]$ .

• Result: Same dimensions.

2. Angle and Strain: Angle ($\theta = \text{arc}/\text{radius}$) is a ratio of lengths ($L/L$). It is dimensionless $[M^0L^0T^0]$ . Strain ($\Delta L/L$) is a ratio of lengths. It is dimensionless $[M^0L^0T^0]$ .

• Result: Same dimensions.

3. Relative Density and Refractive Index: Relative Density is the ratio of the density of a substance to the density of water. It is dimensionless . Refractive Index ($n = c/v$) is the ratio of the speed of light in a vacuum to the speed of light in a medium. It is dimensionless $[M^0L^0T^0]$ .

• Result: Same dimensions.

4. Planck Constant and Energy: Planck constant ($h$) is defined by $E = h\nu$. Its dimensions are Energy divided by Frequency: $\frac{[ML^2T^{-2}]}{[T^{-1}]} = [ML^2T^{-1}]$ . Energy ($E$) has dimensions $[ML^2T^{-2}]$ .

• Result: Different dimensions.

Therefore, the Planck constant and Energy do not have the same dimensions.