The dimensional formula of intensity of gravitational field

Options

A. `MLT^(-2)`
B. `ML^(-1)T^(-2)`
C. `MLT^(-3)`
D. `M^(0)LT^(-2)`

Answer

To find the dimensional formula of the intensity of the gravitational field, we can follow these steps:

### Step 1: Understand the Concept of Gravitational Field Intensity
The intensity of the gravitational field (often referred to as gravitational field strength) at a point in space is defined as the force experienced by a unit mass placed at that point. Mathematically, it is expressed as:
\[ E = \frac{F}{m} \]
where \( E \) is the intensity of the gravitational field, \( F \) is the gravitational force, and \( m \) is the mass of the object experiencing the force.

### Step 2: Identify the Units of Gravitational Force
The gravitational force \( F \) can be expressed using Newton's second law:
\[ F = m \cdot a \]
where \( a \) is the acceleration due to gravity. The unit of force is Newton (N), which can be expressed in terms of base units:
\[ 1 \text{ N} = 1 \text{ kg} \cdot \text{m/s}^2 \]

### Step 3: Substitute the Units into the Formula for Gravitational Field Intensity
Since the intensity of the gravitational field \( E \) is the force per unit mass, we can write:
\[ E = \frac{F}{m} = \frac{\text{N}}{\text{kg}} \]
Substituting the units of force:
\[ E = \frac{1 \text{ kg} \cdot \text{m/s}^2}{1 \text{ kg}} = \frac{\text{m}}{\text{s}^2} \]

### Step 4: Determine the Dimensional Formula
Now, we can express the units \( \frac{\text{m}}{\text{s}^2} \) in terms of their dimensional formula:
- The dimension of length (m) is represented as \( [L] \).
- The dimension of time squared (s²) is represented as \( [T^2] \).

Thus, the dimensional formula for the intensity of the gravitational field is:
\[ [E] = [L][T^{-2}] \]

### Step 5: Final Answer
The dimensional formula of the intensity of the gravitational field is:
\[ [E] = [M^0 L^1 T^{-2}] \]