Dimensional Analysis in System with Dimensionless Mass and Angular Momentum

To determine the correct statements, we analyze the dimensions of the given physical quantities under the new system of units.

1. Given Conditions:

• Mass $[M]$ is dimensionless: $[M] = 1$
• Angular momentum $[J]$ is dimensionless: $[J] = [M L^2 T^{-1}] = 1$
• Length has the dimension $L$: $[\text{Length}] = L$

2. Dimension of Time ($T$): Using the relation for angular momentum: $[J] = [M][L]^2[T]^{-1} = 1$ Since $[M] = 1$, we have: $1 \cdot L^2 \cdot [T]^{-1} = 1 \implies [T] = L^2$

Now, we calculate the dimensions of the physical quantities mentioned in the options:

• Linear Momentum ($p$): $[p] = [M][L][T]^{-1}$ Substituting $[M] = 1$ and $[T] = L^2$: $[p] = 1 \cdot L \cdot (L^2)^{-1} = L \cdot L^{-2} = L^{-1}$ Thus, Statement (A) is correct.

• Energy ($E$): $[E] = [M][L]^2[T]^{-2}$ Substituting $[M] = 1$ and $[T] = L^2$: $[E] = 1 \cdot L^2 \cdot (L^2)^{-2} = L^2 \cdot L^{-4} = L^{-2}$ Thus, Statement (B) is correct.

• Force ($F$): $[F] = [M][L][T]^{-2}$ Substituting $[M] = 1$ and $[T] = L^2$: $[F] = 1 \cdot L \cdot (L^2)^{-2} = L \cdot L^{-4} = L^{-3}$ Thus, Statement (C) is correct.

• Power ($P$): $[P] = \left[\frac{E}{T}\right] = \frac{[E]}{[T]}$ Substituting $[E] = L^{-2}$ and $[T] = L^2$: $[P] = \frac{L^{-2}}{L^2} = L^{-4}$ Since Statement (D) gives the dimension as $L^{-5}$, Statement (D) is incorrect.

Correct Answer: The correct statements are A, B, and C.