The rotaitonal kinetic energy `E=1/2I omega^(2)`. Use this equation to get the dimensional formula for `omega`, where I is the moment of inertia and `omega` is the angular velocity.

Explanation

To find the dimensional formula for angular velocity (ω) using the equation for rotational kinetic energy, we can follow these steps:

### Step-by-Step Solution:

1. **Write the Given Equation**:
The equation for rotational kinetic energy is given as:
\[ E = \frac{1}{2} I \omega^2 \]

2. **Identify Dimensions of Energy (E)**:
The dimension of energy (E) is known to be:
\[ [E] = ML^2T^{-2} \]
where M is mass, L is length, and T is time.

3. **Identify Dimensions of Moment of Inertia (I)**:
The moment of inertia (I) is defined as:
\[ I = MR^2 \]
Therefore, its dimensions are:
\[ [I] = ML^2 \]

4. **Substitute Dimensions into the Equation**:
Now, substituting the dimensions of E and I into the equation:
\[ [E] = \frac{1}{2} [I] [\omega]^2 \]
This simplifies to:
\[ ML^2T^{-2} = \frac{1}{2} (ML^2) [\omega]^2 \]

5. **Eliminate Constants and Rearrange**:
We can ignore the constant \( \frac{1}{2} \) for dimensional analysis. Thus, we have:
\[ ML^2T^{-2} = ML^2 [\omega]^2 \]

6. **Divide Both Sides by [I]**:
To isolate \( [\omega]^2 \), divide both sides by \( ML^2 \):
\[ \frac{ML^2T^{-2}}{ML^2} = [\omega]^2 \]
This simplifies to:
\[ T^{-2} = [\omega]^2 \]

7. **Take the Square Root**:
To find the dimension of \( \omega \), take the square root of both sides:
\[ [\omega] = T^{-1} \]

### Final Result:
The dimensional formula for angular velocity \( \omega \) is:
\[ [\omega] = T^{-1} \]