Question

Which of the following does not have the dimensions of force?
A. Gravitational Potential gradient
B. Energy gradient
C. Weight
D. Rate of change of momentum

Answer 1

Hint: Firstly, write down the formula for each physical quantity given in options thereafter write its dimensional formula. Now analyse that which quantity does not have the dimensional formula of force i.e., \[\left[ ML{{T}^{-2}} \right]\]

Complete step by step answer:
The gravitational potential gradient is defined as the rate of change of gravitational potential with distance in the field. This is equal to the gravitational field intensity \[\left( g={{E}_{G}} \right)\]at that point.
So, \[g={{E}_{G}}=-\dfrac{\Delta {{V}_{G}}}{\Delta r}\]
Where,
\[g\] = gravitational potential gradient
\[{{E}_{G}}\] = gravitational field intensity
\[{{V}_{G}}\] = gravitational potential
\[r\] = distance
Now the dimensional formula for gravitational potential gradient is given by,
\[\Rightarrow \dfrac{\left[ {{L}^{2}}{{T}^{-2}} \right]}{\left[ L \right]}=\left[ L{{T}^{-2}} \right]\]
Which does not have the dimensional formula of force.
Hence, the correct option is A, i.e., Gravitational Potential gradient.

Additional Information:
The gravitational potential \[\left( {{V}_{G}} \right)\] at a point in a field is defined as the work done in bringing unit mass to that point from infinity. Mathematically, \[{{V}_{G}}=-\dfrac{Gm}{r}\]
where \[m\] is the mass of the body, producing the field and \[r\] is the distance from its centre. The negative sign denotes that \[{{V}_{G}}\] is decreasing as \[r\] increases.

Note: Students need to memorize the basic formulae for frequently used physical quantities so that they can write dimensional formulas for these quantities instantly. Here the energy gradient has the dimensional formula same as force i.e., \[\left[ ML{{T}^{-2}} \right]\]so option B is incorrect. Weight is a force in itself so it has dimensional formula same as that of force therefore option C is also incorrect and the rate of change of momentum is the definition of force so obviously it has dimensional formula of force therefore, option D is incorrect.