Question

The potential energy of a particle varies the distance x from a fixed origin as $U = \dfrac{{A\sqrt x }}{{x + B}}$ , where A and B are dimensional constants, then find the dimensional formula for AB.
A) $M{L^{\dfrac{5}{2}}}{T^{ - 2}}$
B) $M{L^{ - 2}}{T^{ - 2}}$
C) ${M^{\dfrac{3}{2}}}{L^{\dfrac{3}{2}}}{T^{ - 2}}$
D) $M{L^{\dfrac{7}{2}}}{T^{ - 2}}$

Answer 1

Hint: Dimensional formula of any quantity is the representation of formula in terms of 7 fundamental units of physical quantities. Generally, M mass, T time and L length is represented.By following the quantity and their units we will derive the formula for the given quantity of the question.

Complete step by step solution:
Let’s first discuss the dimensional formula in more detail before proceeding for the calculations.
Dimensions of a derived unit are the powers to which the fundamental units of mass, length and time must be raised to represent that unit. Dimensional formula is an expression which shows how and which of the fundamental units are required to represent the unit of a physical quantity and dimensional equation is termed as the equation obtained by equating the physical quantity with its dimensional formula.
Let’s come to the calculation part now.
x = distance from a fixed origin.
$U = \dfrac{{A\sqrt x }}{{x + B}}$ (Given quantity)
Dimensions of B are same as of x because both are in addition
$x = L$
Therefore B is also equal to square of length.
We have:
$
   \Rightarrow U = \dfrac{{{x^{\dfrac{1}{2}}}A}}{{x + B}} \\
   \Rightarrow A = \dfrac{{U(x + B)}}{{{x^{\dfrac{1}{2}}}}} \\
 $(Wrote the equation to find the dimensions of A)
$
   \Rightarrow A = \dfrac{{M{L^2}(L)}}{{{T^2} \times {L^{\dfrac{1}{2}}}}} \\
   \Rightarrow A = M{L^{\dfrac{5}{2}}}{T^{ - 2}} \\
 $(Dimensions of A are being calculated0
We have dimensions of B;
So, AB is:
$
   \Rightarrow AB = M{L^{\dfrac{5}{2}}}{T^{ - 2}} \times L \\
   \Rightarrow AB = M{L^{\dfrac{7}{2}}}{T^{ - 2}} \\
 $ (We have multiplied the dimensions of both AB)

Option (D) is correct.

Note: The fundamental or primary units are mass, Length, time, electric current, temperature and luminous intensity which are unique and are not derived from any other units. In the similar manner secondary units are the one which are derived from primary units such as velocity, potential etc.