Question

Let us consider a system of units in which mass and angular momentum are dimensionless. If the length has a 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

The correct option is D

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

The explanation for the correct options:

In the case of options A, C, and D.

Step 1. Explanation:

We know

$\left[M\right]=\left[Mass\right]=\left[M^1{L}^0{T}^0\right]$
$\left[J\right]=\left[Angularmomentum\right]=\left[M{L}^2{T}^{-1}\right]$
$\left[L\right]=\left[Length\right]=\left[M^0{L}^1{T}^0\right]$
Now; $\left[M{L}^2{T}^{-1}\right]=\left[M^0{L}^0{T}^0\right]$

Since mass is dimensionless, therefore now we are left with

$\left[L^2{T}^{-1}\right]=\left[L^0{T}^0\right]$

Comparing LHS with RHS, we get
∴$\left[L^2\right]=\left[T\right]$
Step 2. Dimensions:

Finding dimensions of Energy:

Energy = $m{c}^2$

$m$=mass, $c$=velocity

Velocity=$M^0{L}^1{T}^{-1}$
Therefore, Energy/work $\left[W\right]=\left[ML{T}^{-2}.L\right]$
=$\left[L^2{L}^{-4}\right]$
=$\left[L^{-2}\right]$

Finding dimensions of Force:

Force = Mass × Acceleration

Since, acceleration = velocity / time =$\left[L{T}^{-1}\right]\left[T^{-1}\right]$

Force $\left[F\right]=\left[ML{T}^{-2}\right]=\left[L.L^{-4}\right]=\left[L^{-3}\right]$

Finding dimensions of Linear momentum :

Linear Momentum = Mass × Velocity

Mass =$\left[M^1{L}^0{T}^0\right]$, Velocity =$\left[M^0{L}^1{T}^{-1}\right]$
So Linear momentum $\left[p\right]=\left[ML{T}^{-1}\right]=\left[L.L^{-2}\right]=\left[L^{-1}\right]$

Hence, the correct options are options A, C, and D.

The explanation for the incorrect option:

In the case of option B.

Finding dimensions of Power:

$$\begin{gathered} Power=\frac{work}{time} \\ Work=forceXdisplacement \\ Power=\frac{(forceXdisplacement)}{time} \end{gathered}$$

Writing the dimensional formula of all the above values, we get

Power $\left[P\right]=\left[ML{T}^{-2}.L{T}^{-1}\right]$
=$\left[M{L}^2{T}^{-3}\right]$
=$\left[L^2{L}^{-6}\right]$
$\left[P\right]=\left[L^{-4}\right]$

Hence, option B is the incorrect option.

Hence, the correct options are options A, C, and D.