Question

The potential energy of a particle varies with distance $x$ from a fixed origin as $\vee =\left(\frac{A\sqrt{x}}{x+B}\right),$ where $A$ and $B$ are constants. The dimensions of AB are

• (A) [ML⁵/2T^-2]
• (B) [ML²T^-2]
• (C) [M3/2L3/2T-2]
• (D) [ML^7/2T^-2]

Answer 1

The Correct Option is D

Explanation:
Given, $\mathrm{V}=\frac{A\sqrt{x}}{x+B}$Dimensions of $\mathrm{V}=$ dimensions of potential energy $=\left[{\mathrm{ML}}^2{\mathrm{T}}^{-2}\right]$From Eq. (i), Dimensions of $B=$ dimensions of $x=\left[M^{0}L{T}^0\right]$ Dimensions of $A=\frac{\text{ dimensions of }\mathrm{V}\times \text{ dimensions of }(x+B)}{\text{ dimensions of }\sqrt{x}}$ $=\frac{\left[{\mathrm{ML}}^2{\mathrm{T}}^{-2}\right]\left[{\mathrm{M}}^0{\mathrm{LT}}^0\right]}{\left[{\mathrm{M}}^0{\mathrm{L}}^{1/2}{\mathrm{T}}^0\right]}$$=\left[{\mathrm{ML}}^{5/2}{\mathrm{T}}^{-2}\right]$Hence, dimensions of $AB$ $=\left[{\mathrm{ML}}^{5/2}{\mathrm{T}}^{-2}\right]\left[{\mathrm{M}}^0{\mathrm{LT}}^0\right]$$=\left[{\mathrm{ML}}^{7/2}{\mathrm{T}}^{-2}\right]$

Answer 2

Dimensions of Physical Quantities

The number of times a fundamental quantity is contained in the given derived physical quantity is known as the dimension of that physical quantity.

• The expression shows which of the fundamental units and with what powers enter into the derived unit of a physical quantity is known as the dimensional formula of the physical quantity.
• For example, the dimensional formula of force is [MLT^-2].
• The equation obtained by equating the symbol of a physical quantity with its dimensional formula is called a dimensional equation.
• For example, the dimensional equation of force is F = [MLT^-2].

Principle of Homogeneity

 

According to the principle of homogeneity, the dimensions of the fundamental quantities of two sides of a physical relation must be the same.

For example, if [M^a L^b T^c] = [M^x L^y T^z]

Then, a = x, b = y, and c = z

Uses of Dimensional Analysis 

The following are the applications of dimensional analysis

• It is used to convert one system of units into the other.
• It is used to check the correctness of the given physical relation.
• It is used to derive the relationship between various physical quantities.