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Step-by-Step Solution
Step 1: Understand the Problem
We are given:
1) The arc length of a circular path: 4.4 ly (light years).
2) The arc subtends an angle of 4 arcseconds (denoted as '4s') at the center of the circle.
3) We want the time taken for an object to complete 4 revolutions around this circle if its linear speed is 8 AU/s.
4) Conversion factors:
• 1 ly = 9.46 × 1015 m
• 1 AU = 1.5 × 1011 m
Step 2: Express the Given Quantities in Standard Units
1) Convert the arc length l from light years to meters:
$l = 4.4 \times 9.46 \times 10^{15}\text{ m}.$
2) Convert the speed v from AU/s to m/s:
$v = 8 \times 1.5 \times 10^{11} \text{ m/s} = 12 \times 10^{11} \text{ m/s} = 1.2 \times 10^{12}\text{ m/s}.$
Step 3: Convert the Angle from Arcseconds to Radians
The angle subtended is 4 arcseconds. Recall:
1° = 3600 arcseconds, and
1° in radians = $ \frac{\pi}{180}.$
Thus,
$ \theta = 4 \,\text{arcseconds} = 4 \times \frac{1}{3600}\,^\circ = \frac{4}{3600}\,^\circ.$
In radians, this becomes
$ \theta = \left(\frac{4}{3600}\right) \times \frac{\pi}{180}
= \frac{4 \pi}{3600 \times 180}
= \frac{4 \pi}{648000}
= \frac{\pi}{162000}.$
Step 4: Determine the Radius of the Circular Path
If an arc length $l$ on a circle subtends an angle $\theta$ radians at the center, the radius $R$ is given by
$ R = \frac{l}{\theta}.$
Here,
$ R = \frac{4.4 \times 9.46 \times 10^{15}}{\frac{\pi}{162000}}
= \frac{4.4 \times 9.46 \times 10^{15} \times 162000}{\pi}.$
(We do not necessarily need to calculate this number explicitly to see the relationship for time.)
Step 5: Calculate Total Distance for 4 Revolutions
One complete revolution around a circle of radius $R$ is the circumference $2\pi R$.
Hence, for 4 revolutions, the total distance $D$ traveled is
$ D = 4 \times 2\pi R = 8\pi R.$
Step 6: Time Taken to Complete 4 Revolutions
Time = (Total distance traveled) / (Speed). Therefore,
$ T = \frac{D}{v} = \frac{8\pi R}{v}.$
Substitute $R = \frac{l}{\theta}$ into the expression:
$ T = \frac{8\pi}{v} \times \frac{l}{\theta}.$
Step 7: Substitute Numerical Values
• $l = 4.4 \times 9.46 \times 10^{15}\text{ m},$
• $\theta = \frac{\pi}{162000},$
• $v = 1.2 \times 10^{12}\text{ m/s}.$
Putting these into the time formula gives (after simplification and numerical calculation):
$ T \approx 4.5 \times 10^{10} \text{ s}.$
Final Answer
4.5 × 1010 s
