Flywheel Revolutions Calculation
The problem asks for the total number of revolutions a flywheel completes while its angular speed increases over a specific time period.
Initial Data Conversion
First, convert the initial and final angular speeds from revolutions per minute (rpm) to radians per second (rad/s), as physics calculations require SI units.
- Conversion factor: $1 \text{ rpm} = \frac{2\pi}{60} \text{ rad/s}$
- Initial angular speed ($\omega_0$): $600 \text{ rpm} = 600 \times \frac{2\pi}{60} \text{ rad/s} = 20\pi \text{ rad/s}$
- Final angular speed ($\omega_f$): $1200 \text{ rpm} = 1200 \times \frac{2\pi}{60} \text{ rad/s} = 40\pi \text{ rad/s}$
- Time ($t$): $10 \text{ s}$
Calculating Angular Acceleration
Determine the angular acceleration ($\alpha$) using the formula $\alpha = \frac{\omega_f - \omega_0}{t}$.
- $\alpha = \frac{40\pi \text{ rad/s} - 20\pi \text{ rad/s}}{10 \text{ s}}$
- $\alpha = \frac{20\pi \text{ rad/s}}{10 \text{ s}} = 2\pi \text{ rad/s}^2$
Calculating Angular Displacement
Use the kinematic equation for angular displacement: $\theta = \omega_0 t + \frac{1}{2} \alpha t^2$.
- $\theta = (20\pi \text{ rad/s})(10 \text{ s}) + \frac{1}{2} (2\pi \text{ rad/s}^2)(10 \text{ s})^2$
- $\theta = 200\pi \text{ rad} + \frac{1}{2} (2\pi \text{ rad/s}^2)(100 \text{ s}^2)$
- $\theta = 200\pi \text{ rad} + 100\pi \text{ rad}$
- $\theta = 300\pi \text{ radians}$
Determining Number of Revolutions
Convert the total angular displacement ($\theta$) from radians to revolutions. Since $1 \text{ revolution} = 2\pi \text{ radians}$.
- Number of revolutions = $\frac{\theta}{2\pi}$
- Number of revolutions = $\frac{300\pi \text{ radians}}{2\pi \text{ radians/revolution}}$
- Number of revolutions = $150$
Therefore, the flywheel completes 150 revolutions.
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