Question

Let us consider a system of units in which mass and angular momentum are dimensionless. If length has dimension of $L$, which of the following statement(s) is/are correct?

• A. The dimension of force is $L^{-3}$
• B. The dimension of power is $L^{-5}$
• C. The dimension of linear momentum is $L^{-1}$
• D. The dimension of energy is $L^{-2}$

Answer 1

Answer: (A), (C), (D)

The dimension of energy is $L^{-2}$

$\begin{array}{cc}\text{Angular momentum}\left(\ell \right)=& mvr\\ \Rightarrow \left[\ell \right]=\frac{{ML}^2}{T}& \because M\to dimensionless\end{array}$

$\therefore T=L^2\ell$ also dimensionless
Now $p=mv=\frac{ML}{T}$
$\therefore \left[p\right]=L^{-1}$

$\left[\text{Energy}\right]=\frac{M{L}^2}{T^2}=\frac{L^2}{L^2{L}^2}=L^{-2}$

[Power $]=\frac{M{L}^2}{T^{2}T}=\frac{L^2}{L^2{L}^2{L}^2}=L^{-4}$

[Force] $=\frac{ML}{T^2}=\frac{L}{L^2{L}^2}=L^{-3}$