A flywheel is rotating at an angular speed of 150 rpm. If the moment of inertia of the flywheel somehow decreases from `10kgm^(2)` to `4kgm^(2)`, then what will be the new angular speed of the flywheel?

Answer

To solve the problem, we will use the principle of conservation of angular momentum. The angular momentum of a system is given by the product of its moment of inertia (I) and its angular speed (ω). According to the conservation of angular momentum, if no external torque acts on the system, the initial angular momentum will equal the final angular momentum.

### Step-by-Step Solution:

1. **Identify Initial Conditions**:
- Initial angular speed (ω₁) = 150 rpm
- Initial moment of inertia (I₁) = 10 kg·m²

2. **Identify Final Conditions**:
- Final moment of inertia (I₂) = 4 kg·m²
- Final angular speed (ω₂) = ? (This is what we need to find)

3. **Apply Conservation of Angular Momentum**:
- According to the conservation of angular momentum:
\[ I₁ \cdot ω₁ = I₂ \cdot ω₂ \]

4. **Substitute the Known Values**:
- Substitute I₁, ω₁, and I₂ into the equation:
\[ 10 \, \text{kg·m²} \cdot 150 \, \text{rpm} = 4 \, \text{kg·m²} \cdot ω₂ \]

5. **Calculate the Left Side**:
- Calculate \(10 \cdot 150\):
\[ 10 \cdot 150 = 1500 \, \text{kg·m²·rpm} \]

6. **Set Up the Equation**:
- Now we have:
\[ 1500 = 4 \cdot ω₂ \]

7. **Solve for ω₂**:
- Divide both sides by 4 to find ω₂:
\[ ω₂ = \frac{1500}{4} = 375 \, \text{rpm} \]

8. **Final Answer**:
- The new angular speed of the flywheel is **375 rpm**.