The new dimensional formula for the moment of inertia of a body is

Options

A. `[M^(1)L^(0)T^(-2)]`
B. `[M^(2)L^(2)T^(0)]`
C. `[M^(1)L^(1)T^(0)]`
D. `[M^(2)L^(3)T^(0)]`

Answer

To find the dimensional formula for the moment of inertia of a body, we can follow these steps:

### Step 1: Understand the Definition of Moment of Inertia
The moment of inertia (I) of a body is defined as the sum of the products of the mass of each particle (m_i) and the square of its distance (r_i) from the axis of rotation. Mathematically, it can be expressed as:
\[ I = \sum_{i=1}^{n} m_i r_i^2 \]
where \( n \) is the number of particles.

### Step 2: Identify the Dimensions of Mass and Distance
- The dimension of mass (m) is represented as [M].
- The dimension of distance (length, r) is represented as [L].

### Step 3: Write the Dimensional Formula for Moment of Inertia
Since the moment of inertia involves mass and the square of distance, we can express its dimensional formula as:
\[ I = m \cdot r^2 \]
Substituting the dimensions, we get:
\[ [I] = [M] \cdot [L^2] \]

### Step 4: Combine the Dimensions
Combining the dimensions, we have:
\[ [I] = [M] \cdot [L^2] = [M^1 L^2 T^0] \]
Here, T (time) does not appear in the formula, so its exponent is zero.

### Step 5: Final Result
Thus, the dimensional formula for the moment of inertia is:
\[ [I] = [M^1 L^2 T^0] \]