Question

The dimensional formula for potential energy is:
(A) $ {M^2}{L^2}{T^{ - 2}} $
(B) $ {M^1}{L^{ - 2}}{T^{ - 2}} $
(C) $ {M^1}{L^2}{T^{ - 2}} $
(D) $ {M^1}{L^2}{T^{ - 3}} $

Answer 1

Hint: Potential energy is the energy stored in a body due to its position or state or arrangement. When an object does work the potential energy is converted into kinetic energy. The dimensional formula of a given physical quantity is an expression showing the dimensions of the fundamental quantities.

Complete Step by step solution
We know that the formula for potential energy is
 $ PE = mgh $
where $ m $ stands for the mass of the body, $ g $ stands for the acceleration due to gravity and $ h $ stands for the height at which the object is located.
The dimension of mass $ m = \left[ M \right] $
The dimension of acceleration due to gravity
Acceleration can be written as,
 $ a = \dfrac{d}{{{t^2}}} $
where $ d $ stands for the distance and $ t $ stands for the time.
The dimensional formula for acceleration can be written as,
 $ a = \dfrac{L}{{{T^2}}} = L{T^{ - 2}} $
Therefore the dimensional formula for the acceleration due to gravity is,
 $ g = \left[ {L{T^{ - 2}}} \right] $
The dimensional formula for height
 $ h = \left[ L \right] $
The dimensional formula for $ \;mgh $ can be written as,
 $ mgh = \left[ M \right]\left[ L \right]\left[ {L{T^{ - 2}}} \right] = \left[ {M{L^2}{T^{ - 2}}} \right] $

Additional information
The seven fundamental quantities are represented as, mass $ \left[ M \right] $ , length $ \left[ L \right] $ , time $ \left[ T \right] $ , electric current $ \left[ A \right] $ , thermodynamic temperature $ \left[ K \right] $ , luminous intensity $ \left[ {cd} \right] $ and amount of substance $ \left[ {mol} \right] $ . Dimensions of a physical quantity are the powers to which the base quantities are to be raised to represent that quantity. Dimensional analysis is the analysis of an equation by expressing physical quantities in terms of their base quantities by assigning the appropriate dimensions.

Note
Dimensional equations are equations connecting the physical quantity with the dimensional formula of that physical quantity. The principle of homogeneity of dimensions states that the dimensions of the equivalent fundamental quantity must be equal on either side of a dimensional equation.