Question

The angular speed of a flywheel is increased from 600 rpm to 1200 rpm in 10 s. The number of revolutions completed by the flywheel during this time is: ____.

• A. 300
• B. 150
• C. 900
• D. 600

Hint:

Alternatively, use the formula $N = \frac{\theta}{2\pi}$. Calculate $\alpha = (\omega_2 - \omega_1)/t$ and then $\theta = \omega_1 t + \frac{1}{2}\alpha t^2$. However, the "average frequency" method used above is much faster for simple acceleration.

Answer 1

The Correct Option is B

Step 1: Understanding the Topic:
This problem falls under "Rotational Mechanics," specifically rotational kinematics. It involves calculating the angular displacement (total revolutions) of a body experiencing constant angular acceleration, similar to how we calculate linear displacement in 1D kinematics.
Step 2: Key Formulas and Approach:
The most efficient way to find total revolutions ($N$) under constant acceleration is using the average angular frequency:
Frequency in revolutions per second ($f$ or $n$) = $\frac{\text{rpm}}{60}$.
Total Revolutions ($N$) = $\text{Average frequency } (n_{avg}) \times \text{Time } (t)$.
$n_{avg} = \frac{n_1 + n_2}{2}$.

Step 3: Detailed Explanation:

Identify given values: Initial speed $n_1 = 600 \text{ rpm}$, Final speed $n_2 = 1200 \text{ rpm}$, and Time $t = 10 \text{ s}$.
Convert units to seconds: Since the time is in seconds, we convert rotations per minute (rpm) to rotations per second (rps).
$n_1 = 600 / 60 = 10 \text{ rps}$.
$n_2 = 1200 / 60 = 20 \text{ rps}$.

Calculate average frequency: Assuming the angular acceleration is uniform, the average speed during the 10-second interval is: \[ n_{avg} = \frac{10 + 20}{2} = 15 \text{ rps} \]
Calculate total revolutions: The total number of completed turns is the average rate multiplied by the duration: \[ N = 15 \text{ rps} \times 10 \text{ s} = 150 \text{ revolutions} \]
Alternatively, one could find angular acceleration $\alpha = (\omega_2 - \omega_1)/t$ and then use $\theta = \omega_1 t + 0.5\alpha t^2$, then divide by $2\pi$. However, the frequency method is significantly faster and less prone to $\pi$-related errors.
Step 4: Final Answer:
The flywheel completes 150 revolutions in the given time.