If mass is written as m = k c^p G^-1/2 h^1/2, then the value of p… — Solution | JEE Physics

MCQEasyJEE 2024Dimensions & Dimensional Analysis

If mass is written as $m = k c^p G^{-1/2} h^{1/2}$ , then the value of $p$ will be:

A
$\frac{1}{2}$
B
$\frac{1}{3}$
C
$2$
D
$-\frac{1}{3}$

Answer

Correct answer:B

Step-by-step solution

Standard Method

Given: $m = k c^P G^{-1/2} h^{1/2}$ , where $k$ is dimensionless.

Find: The value of $P$ .

The solution states that the correct option is B. However, the dimensional-analysis working shown there leads to:

[c^P] = [L]^P[T]^{-P}

[G^{-1/2}] = [M]^{1/2}[L]^{-3/2}[T]

[h^{1/2}] = [M]^{1/2}[L][T]^{-1/2}

So,

[M] = [L]^P[T]^{-P} \cdot [M]^{1/2}[L]^{-3/2}[T] \cdot [M]^{1/2}[L][T]^{-1/2}

Combining powers,

[M] = [M]^1 [L]^{P-1/2} [T]^{-P+1/2}

Equating powers of $L$ and $T$ gives:

P - \frac{1}{2} = 0

-P + \frac{1}{2} = 0

Hence,

P = \frac{1}{2}

This value matches option A, while the solution marks option B as correct. Since the source explicitly labels B as the correct option, the extracted answer is B, but the working shown on the page supports $P = \frac{1}{2}$ and indicates a source discrepancy.

Common mistakes

Using the listed correct option without checking the dimensional working. Here the shown derivation gives $P = \frac{1}{2}$ , so students should always verify exponents from dimensions instead of trusting the option label alone.
Writing the dimension of $G^{-1/2}$ incorrectly. Since $[G] = [M]^{-1}[L]^3[T]^{-2}$ , raising it to $-\frac{1}{2}$ gives $[M]^{1/2}[L]^{-3/2}[T]$ , not a negative power of $M$ .
Forgetting to apply the exponent $\frac{1}{2}$ to all dimensions of $h$ . From $[h] = [M][L]^2[T]^{-1}$ , we get $[h^{1/2}] = [M]^{1/2}[L][T]^{-1/2}$ .

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Answer

Step-by-step solution

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