The dimensional formula of `sqrt((hc^5)/G)` is

Options

A. `[ML^(2)T^(-3)]`
B. `[ML^(2)T^(-2)]`
C. `[ML^(-2)T^(2)]`
D. `[MLT^(-2)]`

Explanation

To find the dimensional formula of \(\sqrt{\frac{hc^5}{G}}\), we will break it down step by step.

### Step 1: Identify the dimensional formulas of the constants involved

1. **Planck's constant (h)**:
- The unit of Planck's constant is Joule-seconds (Js).
- Joule is defined as \(1 \, \text{J} = 1 \, \text{kg} \cdot \text{m}^2/\text{s}^2\).
- Therefore, the dimensional formula of \(h\) is:
\[ [h] = [\text{J}] \cdot [\text{s}] = [M L^2 T^{-2}] \cdot [T] = [M L^2 T^{-1}] \]

2. **Speed of light (c)**:
- The dimensional formula of speed is distance/time.
- Therefore, the dimensional formula of \(c\) is:
\[ [c] = [L T^{-1}] \]

3. **Gravitational constant (G)**:
- The gravitational constant can be derived from Newton's law of gravitation, where \(F = \frac{G m_1 m_2}{r^2}\).
- Rearranging gives \(G = \frac{F r^2}{m_1 m_2}\).
- The dimensional formula of force \(F\) is \(M L T^{-2}\) and \(r\) is \(L\).
- Thus, the dimensional formula of \(G\) is:
\[ [G] = \frac{[M L T^{-2}] \cdot [L^2]}{[M^2]} = [M^{-1} L^3 T^{-2}] \]

### Step 2: Substitute the dimensional formulas into the expression

Now, we substitute the dimensional formulas into the expression \(\sqrt{\frac{hc^5}{G}}\):

\[ \frac{hc^5}{G} = \frac{[M L^2 T^{-1}] \cdot ([L T^{-1}]^5)}{[M^{-1} L^3 T^{-2}]} \]

Calculating \(c^5\):
\[ c^5 = (L T^{-1})^5 = L^5 T^{-5} \]

Now substituting back:
\[ \frac{hc^5}{G} = \frac{[M L^2 T^{-1}] \cdot [L^5 T^{-5}]}{[M^{-1} L^3 T^{-2}]} \]

### Step 3: Simplify the expression

Now we simplify the numerator:
\[ [M L^2 T^{-1}] \cdot [L^5 T^{-5}] = [M L^{2+5} T^{-1-5}] = [M L^7 T^{-6}] \]

Now substituting this back into the expression:
\[ \frac{[M L^7 T^{-6}]}{[M^{-1} L^3 T^{-2}]} = [M L^7 T^{-6}] \cdot [M^1 L^{-3} T^{2}] = [M^{1+1} L^{7-3} T^{-6+2}] = [M^2 L^4 T^{-4}] \]

### Step 4: Take the square root

Now we take the square root of the entire expression:
\[ \sqrt{[M^2 L^4 T^{-4}]} = [M^{2/2} L^{4/2} T^{-4/2}] = [M^1 L^2 T^{-2}] \]

### Final Answer

Thus, the dimensional formula of \(\sqrt{\frac{hc^5}{G}}\) is:
\[ [M L^2 T^{-2}] \]