Question

A quantity $x$ is given by $x=\frac{IF{v}^2}{W{L}^4}$, in terms of moment of inertia $I$, force $F$, velocity $v$, work $W$ and length $L$. The dimensional formula for $x$ is the same as that of :

• A. planck’s constant
• B. force constant
• C. energy density
• D. coefficient of viscosity

Answer 1

The correct option is C energy density

Here,
Dimension of Force, $F=M^1{L}^1{T}^{–2}$
Dimension of velocity, $V=L^1{T}^{–1}$
Dimension of work, $=M^1{L}^2{T}^{–2}$
Dimension of length, $=L$
Moment of inertia, $=M{L}^2$

Now, $x=\frac{IF{v}^2}{W{L}^4}$

$=\frac{\left(M^1{L}^2\right)\left(M^1{L}^1{T}^{-2}\right)(L^1{T}^{-1}{)}^2}{\left(M^1{L}^2{T}^{-2}\right)\left(L^4\right)}$

$=\frac{M^1{L}^{-2}{T}^{-2}}{L^3}=M^1{L}^{-1}{T}^{-2}$

So, the dimensions of $x$ are $M^1{L}^{-1}{T}^{-2}$, which are dimensions of energy density.

Hence, $\left(C\right)$ is the correct answer.