If `g` is the acceleration due to gravity and `R` is the radius of earth, then the dimensional formula for g `R` is

Options

A. (a)`M^(2)L^(2)T^(2)`
B. (b)`M^(0)L^(0)T^(0)`
C. (c)`ML^(2)T^(-2)`
D. (d)`M^(0)L^(2)T^(-2)`

Answer

To find the dimensional formula for the product \( gR \), where \( g \) is the acceleration due to gravity and \( R \) is the radius of the Earth, we can follow these steps:

### Step 1: Determine the dimensional formula for \( g \)
Acceleration due to gravity \( g \) can be defined as the change in velocity per unit time. The dimensional formula for velocity is given by:
\[ \text{Velocity} = \frac{\text{Distance}}{\text{Time}} = \frac{L}{T} \]
Thus, the dimensional formula for acceleration (which is the change in velocity per unit time) is:
\[ \text{Acceleration} = \frac{\text{Velocity}}{\text{Time}} = \frac{L}{T^2} \]
This means the dimensional formula for \( g \) is:
\[ [g] = M^0 L^1 T^{-2} \]

### Step 2: Determine the dimensional formula for \( R \)
The radius \( R \) is a measure of length, so its dimensional formula is:
\[ [R] = M^0 L^1 T^0 \]

### Step 3: Calculate the dimensional formula for \( gR \)
Now, we can find the dimensional formula for the product \( gR \):
\[ [gR] = [g] \times [R] = (M^0 L^1 T^{-2}) \times (M^0 L^1 T^0) \]
When we multiply these, we combine the dimensions:
\[ [gR] = M^{0+0} L^{1+1} T^{-2+0} = M^0 L^2 T^{-2} \]

### Conclusion
The dimensional formula for \( gR \) is:
\[ gR = M^0 L^2 T^{-2} \]

### Final Answer
Thus, the dimensional formula for \( gR \) is \( M^0 L^2 T^{-2} \).

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