A plane mirror is placed at the centre of a concave mirror having radius of curvature 40\ m . The plane mirror rotates at the speed of 2600 revolutions per second. Calculate the angle between a ray…

Solution

Step 1: Understand the setup

• A plane mirror is placed at the center of curvature of a concave mirror with radius of curvature $R=40\mathrm{m}$.
• The plane mirror rotates at 2600 revolutions per second (rps).
• The speed of light is $c=3\times 10^{8}m/s$.

Step 2: Calculate the time for light to travel to the concave mirror and back

• The distance from the plane mirror to the concave mirror is $R=40\mathrm{m}$.
• Time for light to travel to the concave mirror: $$t_1=\frac{R}{c}=\frac{40}{3\times 10^8}\mathrm{s}$$
• Time for light to return to the plane mirror: $$t_2=\frac{R}{c}=\frac{40}{3\times 10^8}\mathrm{s}$$
• Total time for one round trip: $$t_{\text{total}}=t_1+t_2=\frac{80}{3\times 10^8}\mathrm{s}$$

Step 3: Calculate the angular displacement of the plane mirror during this time

• The plane mirror rotates at $2600\mathrm{rps}$.
• Angular speed $\omega$ in radians per second: $$\omega =2600\times 2\pi =5200\pi rad/s$$
• Angular displacement $\theta$ during $t_{\text{total}}$: $$\theta =\omega \cdot t_{\text{total}}=5200\pi \times \frac{80}{3\times 10^8}$$ $$\theta =\frac{416000\pi }{3\times 10^8}\mathrm{rad}$$ $$\theta \approx 4.36\times 10^{-3}\mathrm{rad}$$

Step 4: Relate angular displacement to the angle between incident and reflected rays

• The angle between the incident ray and the reflected ray after the round trip is twice the angular displacement of the plane mirror (since the ray reflects off the moving mirror).
• Therefore: $$\text{Angle between incident and reflected rays}=2\theta =2\times 4.36\times 10^{-3}\mathrm{rad}$$ $$\text{Angle}\approx 8.72\times 10^{-3}\mathrm{rad}$$
• Converting to degrees (since small angles are often expressed in degrees for clarity): $$\text{Angle}=8.72\times 10^{-3}\times \frac{180}{\pi }\approx 0.5^{\circ }$$

Final Answer

The angle between the incident ray and the reflected ray after the light has traveled to the concave mirror and back to the plane mirror is $8.72\times 10^{-3}\mathrm{rad}$ (or approximately $0.5^{\circ }$).