An airplane engine starts from rest, and 2 s later, it is rotating with an angular speed of 300 rev/min. If the angular acceleration is constant, how many revolutions does the propeller undergo during this time?

Options

A. 5
B. 50
C. 10
D. 300

Answer

To solve the problem step by step, we will follow the physics principles related to angular motion.

### Step 1: Convert Angular Speed to Radians per Second
The final angular speed (ω) is given as 300 revolutions per minute (rev/min). We need to convert this to radians per second (rad/s).

\[ \omega = 300 \, \text{rev/min} \times \frac{2\pi \, \text{rad}}{1 \, \text{rev}} \times \frac{1 \, \text{min}}{60 \, \text{s}} = 300 \times \frac{2\pi}{60} = 10\pi \, \text{rad/s} \]

### Step 2: Calculate Angular Acceleration
Since the engine starts from rest, the initial angular velocity (ω₀) is 0. We can use the formula for angular acceleration (α):

\[ \omega = \omega_0 + \alpha t \]

Substituting the known values:

\[ 10\pi = 0 + \alpha \cdot 2 \]

Solving for α:

\[ \alpha = \frac{10\pi}{2} = 5\pi \, \text{rad/s}^2 \]

### Step 3: Calculate Angular Displacement
Now, we will calculate the angular displacement (θ) using the formula:

\[ \theta = \omega_0 t + \frac{1}{2} \alpha t^2 \]

Since ω₀ is 0, this simplifies to:

\[ \theta = \frac{1}{2} \alpha t^2 \]

Substituting the values of α and t:

\[ \theta = \frac{1}{2} \cdot (5\pi) \cdot (2^2) = \frac{1}{2} \cdot (5\pi) \cdot 4 = 10\pi \, \text{rad} \]

### Step 4: Convert Angular Displacement to Revolutions
To find the number of revolutions, we convert the angular displacement from radians to revolutions:

\[ \text{Number of revolutions} = \frac{\theta}{2\pi} = \frac{10\pi}{2\pi} = 5 \]

### Final Answer
The propeller undergoes **5 revolutions** during the 2 seconds.

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