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

- frequency and angular frequency
- angular velocity and velocity gradient
- velocity gradient and angular frequency
- angular frequency and potential energy gradient

**Solution**

Frequency and angular frequency is measured in per second or per minute (like 3 rounds per minute).

Dimensional Formula of Frequency = M^{0}L^{0}T^{-1} and SI unit of Frequency is the hertz.

Angular Velocity is defined as change in angular displacement per unit time.

Dimensional Formula of Angular Velocity = M^{0}L^{0}T^{-1}. This is same as dimensional formula of frequency and angular frequency.

Velocity Gradient is defined as rate of change in velocity per unit of distance.

Mathematically, Velocity Gradient = velocity/distance

Dimensional Formula of velocity = M^{0}L^{1}T^{-1}

Dimensional Formula of Velocity Gradient = M^{0}L^{0}T^{-1}. Again this is same as dimensional formula of frequency, angular frequency and angular velocity.

Potential energy can be defined as the energy possessed by the body by virtue of its configuration or position.

Mathematically, Potential energy (PE) = Mass(M) X Gravitational acceleration (g) X Altitude (h)

Dimensional formula of Mass = M^{1}L^{0}T^{0}

Dimensional formula of gravitational acceleration = M^{0}L^{1}T^{-2}

Dimensional formula of altitude (length) = M^{0}L^{1}T^{0}

So, Dimensional Formula of Potential energy = M^{1}L^{2}T^{-2}. Potential energy gradient is rate of change of potential energy per unit distance.

Dimensional Formula of Potential energy gradient = M^{1}L^{1}T^{-2}. This is different from dimensions of angular frequency.

**Correct option is D.**