Disco Beam Revolution Calculation

The problem asks for the length covered by a revolving disco beam after a specific time, given its total revolution time and maximum coverage length.

Determining Angular Speed

First, find the angular speed of the beam.

• The beam completes a full $360^\circ$ revolution in 20 minutes.
• Angular speed ($\omega$) = Total Angle / Total Time
• $\omega = \frac{360^\circ}{20 \text{ min}} = 18^\circ \text{ per minute}$
• Convert the angular speed to radians per minute for arc length calculation:
• $\omega = 18^\circ/\text{min} \times \frac{\pi \text{ radians}}{180^\circ} = \frac{\pi}{10} \text{ radians per minute}$

Calculating Angle Covered

Next, calculate the angle the beam rotates through in 2 minutes.

• Angle ($\theta$) = Angular Speed ($\omega$) $\times$ Time ($t$)
• $\theta = \left(\frac{\pi}{10} \text{ radians/min}\right) \times (2 \text{ min})$
• $\theta = \frac{2\pi}{10} = \frac{\pi}{5} \text{ radians}$

Calculating Covered Length

Finally, calculate the arc length covered using the angle and the maximum coverage length (radius).

• The maximum coverage length is given as 20 m. Assume this is the radius ($r$) of the circle traced by the beam. So, $r = 20$ m.
• Arc Length ($L$) = Radius ($r$) $\times$ Angle in Radians ($\theta$)
• $L = 20 \text{ m} \times \frac{\pi}{5}$
• $L = 4\pi \text{ m}$

Therefore, after 2 minutes, the beam would have covered a length of $4\pi$ m.