Which of the following have same dimensions ?

The correct option is torque and potential energy.

To determine which physical quantities have the same dimensions, let us find the dimensional formula for torque and potential energy step-by-step:

1. Dimensions of Torque:
Torque is defined as the product of force and the perpendicular distance from the axis of rotation:

$\mathrm{Torque}=\mathrm{Force}\times \mathrm{Distance}$

The dimensional formula of force is $\left[ML{T}^{-2}\right]$ and distance is $\left[L\right]$.

Substituting these dimensions, we get:

$\left[\mathrm{Torque}\right]=\left[ML{T}^{-2}\right]\times \left[L\right]=\left[M{L}^2{T}^{-2}\right]$

2. Dimensions of Potential Energy:
Potential energy is the energy possessed by an object due to its position or state. For gravitational potential energy near Earth's surface:

$\mathrm{Potential\; Energy}=\mathrm{Mass}\times \mathrm{Acceleration\; due\; to\; gravity}\times \mathrm{Height}$

The dimensions of mass are $\left[M\right]$, acceleration due to gravity are $\left[L{T}^{-2}\right]$, and height are $\left[L\right]$.

Substituting these dimensions, we get:

$\left[\mathrm{Potential\; Energy}\right]=\left[M\right]\times \left[L{T}^{-2}\right]\times \left[L\right]=\left[M{L}^2{T}^{-2}\right]$

Comparing the two results, we find that both torque and potential energy have the identical dimensional formula of $\left[M{L}^2{T}^{-2}\right]$.