The rotor's velocity of a helicopter engine changes from 330 rev/min to 110 rev/min in 2 minutes. How long does the rotor blades take to stop

Options

A. 3 min
B. 4 min
C. 5 min
D. 6 min

Explanation

To solve the problem of how long it takes for the rotor blades of a helicopter to stop, we can follow these steps:

### Step 1: Determine Initial and Final Angular Velocities
The initial velocity of the rotor is given as 330 revolutions per minute (rev/min) and the final velocity is 110 rev/min after 2 minutes.

1. **Initial Angular Velocity (ω₀)**:
\[ ω₀ = 2\pi \times f₀ = 2\pi \times 330 \text{ rev/min} = 660\pi \text{ rad/min} \]

2. **Final Angular Velocity (ω)**:
\[ ω = 2\pi \times f = 2\pi \times 110 \text{ rev/min} = 220\pi \text{ rad/min} \]

### Step 2: Calculate Angular Acceleration (α)
Angular acceleration is defined as the change in angular velocity over time. We can use the formula:
\[ α = \frac{ω - ω₀}{t} \]
Where \(t\) is the time interval (2 minutes = 2 min).

Substituting the values:
\[ α = \frac{220\pi - 660\pi}{2} = \frac{-440\pi}{2} = -220\pi \text{ rad/min}^2 \]

### Step 3: Use Angular Motion Equation to Find Time to Stop
To find the time it takes for the rotor blades to stop, we can use the equation:
\[ ω = ω₀ + αt \]
Setting the final angular velocity (ω) to 0 (since we want to find the time to stop):
\[ 0 = 660\pi + (-220\pi)t \]

### Step 4: Solve for Time (t)
Rearranging the equation gives:
\[ 220\pi t = 660\pi \]
Dividing both sides by \(220\pi\):
\[ t = \frac{660\pi}{220\pi} = 3 \text{ minutes} \]

### Conclusion
The rotor blades take **3 minutes** to stop completely.

---