Question

Find the dimensions of $A\times (B+C)D$ where $A$ is mass, dimensions of $B$ is unknown, $C$ is energy and $D$ is dimensionless constant:

• A. $M^2{L}^2{T}^{-1}$
• B. $M^2{L}^2{T}^{-2}$
• C. $M^2{L}^1{T}^{-2}$
• D. $M^1{L}^2{T}^{-2}$

Answer 1

The correct option is B

$M^2{L}^2{T}^{-2}$

Step 1. Given data

$A$ is mass,

$B$ is unknown,

$C$ is energy and

$D$ is dimensionless constant:

Step 2. Calculating dimensional formula

If $B$ and $C$ are addable that means the unit of both the quantities are the same

Hence, the dimensions of $B$ and $C$ are equal

Dimension of $A\times (B+C)D$ is equal to dimension of $A\times C\times D$ because $B+C$ will have the same dimension after adding as the dimension of $B$ is unknown,

Where, $A$ is mass, the dimensional formula is $\left[M\right]$

$C$ is energy, dimensional formula $\left[M^1{L}^2{T}^{-2}\right]$

$D$ is dimensionless constant

Therefore, by substituting dimensional formulas in a given relation, we get

$$\begin{gathered} A\times C\times D=M\times M{L}^2{T}^{-2} \\ =M^2{L}^2{T}^{-2} \end{gathered}$$

Hence, the correct option is (B).