Dimension of which base quantity corresponds to that of `sqrt((Gh)/(c^(3))` = ?

Options

A. Time
B. Length
C. Mass
D. Temperature

Explanation

To find the dimension of the quantity corresponding to \(\sqrt{\frac{Gh}{c^3}}\), we will break down each component in the expression and derive their dimensions step by step.

### Step 1: Identify the quantities involved
The expression involves three quantities:
1. \(G\) - Gravitational constant
2. \(h\) - Planck constant
3. \(c\) - Speed of light

### Step 2: Determine the dimensions of each quantity

#### 2.1: Dimension of \(G\) (Gravitational constant)
The gravitational constant \(G\) is defined in the context of Newton's law of gravitation:
\[ F = \frac{G \cdot m_1 \cdot m_2}{r^2} \]
Where \(F\) is the gravitational force, \(m_1\) and \(m_2\) are masses, and \(r\) is the distance between the centers of the two masses.

Rearranging for \(G\):
\[ G = \frac{F \cdot r^2}{m_1 \cdot m_2} \]

The dimensions of force \(F\) are:
\[ [F] = MLT^{-2} \]

Thus, the dimensions of \(G\) become:
\[ [G] = \frac{MLT^{-2} \cdot L^2}{M^2} = M^{-1}L^3T^{-2} \]

#### 2.2: Dimension of \(h\) (Planck constant)
The Planck constant \(h\) can be derived from the relation of energy and frequency:
\[ E = h \cdot f \]
Where \(E\) is energy and \(f\) is frequency. The dimension of energy is:
\[ [E] = ML^2T^{-2} \]
And frequency has the dimension:
\[ [f] = T^{-1} \]

Thus, the dimensions of \(h\) are:
\[ [h] = \frac{ML^2T^{-2}}{T^{-1}} = ML^2T^{-1} \]

#### 2.3: Dimension of \(c\) (Speed of light)
The speed of light \(c\) has dimensions of:
\[ [c] = LT^{-1} \]

### Step 3: Substitute the dimensions into the expression
Now we substitute the dimensions into the expression \(\sqrt{\frac{Gh}{c^3}}\):
\[ \sqrt{\frac{Gh}{c^3}} = \sqrt{\frac{(M^{-1}L^3T^{-2})(ML^2T^{-1})}{(LT^{-1})^3}} \]

Calculating the denominator:
\[ c^3 = (LT^{-1})^3 = L^3T^{-3} \]

Now substituting back into the expression:
\[ \sqrt{\frac{(M^{-1}L^3T^{-2})(ML^2T^{-1})}{L^3T^{-3}}} \]

### Step 4: Simplify the expression
The numerator becomes:
\[ M^{-1}L^3T^{-2} \cdot ML^2T^{-1} = L^5T^{-3} \]

So we have:
\[ \sqrt{\frac{L^5T^{-3}}{L^3T^{-3}}} = \sqrt{L^{5-3}T^{-3+3}} = \sqrt{L^2} = L \]

### Final Result
Thus, the dimension of \(\sqrt{\frac{Gh}{c^3}}\) corresponds to the dimension of length \(L\).

### Conclusion
The base quantity corresponding to the dimension of \(\sqrt{\frac{Gh}{c^3}}\) is **Length**.

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