A huge circular arc of length 4.4 ly subtends an angle '4s' at the centre of the circle. How long it would take for a body to complete 4 revolution if its speed is 8 AU per second?
Given : 1 ly = 9.46 × 10¹⁵ m
1 AU = 1.5 × 10¹¹ m

Solution

Physics sir · 4 steps

That '4s' means 4 arc-seconds. Turn it into radians, find the radius, and the π cancels neatly.

1. Angle in radians. One arc-second is 1/3600 of a degree, so 1'' = π/(180 × 3600) = π/648000 rad. That makes θ = 4'' = 4π/648000 rad.
2. Radius from arc. Arc length is L = rθ, so r = L/θ. Here L = 4.4 × 9.46 × 10¹⁵ = 4.16 × 10¹⁶ m.

   t = (4 × 2π r)/v = 8π/θ·L/v, 8π/θ = (8π × 648000)/4π = 1.296 × 10⁶

3. Put in numbers. Distance for 4 revolutions is 1.296 × 10⁶ × 4.16 × 10¹⁶ = 5.39 × 10²² m. The speed is v = 8 × 1.5 × 10¹¹ = 1.2 × 10¹² m s⁻¹.
4. So. t = (5.39 × 10²²)/(1.2 × 10¹²) ≈ 4.5 × 10¹⁰ s, which is option (b).

The idea to keep. Use L = rθ only with θ in radians. Convert arc-seconds by 1'' = π/648000 rad, and keep everything in SI units before dividing.

Answer from the NTA final answer key. Solution by PrepWiser.