Question

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 = \(9.46 \times 10^{15}\) m
1 AU = \(1.5 \times 10^{11}\) m

• A. \(4.5 \times 10^{10}\) s
• B. \(4.1 \times 10^8\) s
• C. \(3.5 \times 10^6\) s
• D. \(7.2 \times 10^8\) s

Hint:

In astronomical calculations, unit conversion is the most crucial first step. Pay close attention to units like light-years (ly), astronomical units (AU), and angles in degrees, minutes, or seconds. Always convert to a consistent system (like SI) before applying formulas.

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
We are given the length of a circular arc and the angle it subtends at the center. From this, we can find the radius of the circle. We are also given the speed of a body moving along this circle. We need to find the time taken for this body to complete 4 full revolutions.
Step 2: Key Formula or Approach:
1. Convert all units to SI units (meters, radians, seconds).
2. Use the arc length formula: \(L = r\theta\), where \(L\) is arc length, \(r\) is radius, and \(\theta\) is the angle in radians.
3. Calculate the time period for one revolution: \(T = \frac{\text{Circumference}}{\text{Speed}} = \frac{2\pi r}{v}\).
4. Calculate the total time for 4 revolutions: \(t = 4T\).
Step 3: Detailed Explanation:
Unit Conversion:
Arc Length, \(L = 4.4 \text{ ly} = 4.4 \times 9.46 \times 10^{15} \text{ m} = 41.624 \times 10^{15} \text{ m}\).
Angle, \(\theta = 4s = 4 \text{ arcseconds}\). We convert this to radians. \[ 1^\circ = 3600 \text{ arcseconds} \] \[ \theta = 4'' \times \frac{1^\circ}{3600''} \times \frac{\pi \text{ rad}}{180^\circ} = \frac{4\pi}{3600 \times 180} \text{ rad} = \frac{\pi}{162000} \text{ rad} \] \[ \theta \approx 1.939 \times 10^{-5} \text{ rad} \] Speed, \(v = 8 \text{ AU/s} = 8 \times 1.5 \times 10^{11} \text{ m/s} = 12 \times 10^{11} \text{ m/s}\).
Calculate Radius (\(r\)):
\[ r = \frac{L}{\theta} = \frac{41.624 \times 10^{15}}{1.939 \times 10^{-5}} \approx 2.146 \times 10^{21} \text{ m} \] Calculate Time for One Revolution (\(T\)):
\[ T = \frac{2\pi r}{v} = \frac{2 \times \pi \times (2.146 \times 10^{21})}{12 \times 10^{11}} \] \[ T \approx \frac{13.48 \times 10^{21}}{12 \times 10^{11}} \approx 1.123 \times 10^{10} \text{ s} \] Calculate Time for 4 Revolutions (\(t\)):
\[ t = 4 \times T = 4 \times 1.123 \times 10^{10} \approx 4.492 \times 10^{10} \text{ s} \] Step 4: Final Answer:
The total time taken for 4 revolutions is approximately \(4.5 \times 10^{10}\) seconds. This matches option (A).