A huge circular arc of length 4.4 ly subtends an angle 4s at the centre of the circle. How long it w

Q: Why do we need to convert arcseconds to radians?

The standard SI unit for angle in mathematical formulas like $l=R\theta$ is radians. Using degrees or arcseconds directly will yield an incorrect radius.

Q: How do I convert light-years to meters?

Multiply the given value in light-years by the conversion factor provided in the question: $1 \text{ ly} = 9.46 \times 10^{15} \text{ m}$.

Q: Why is the total distance $8\pi R$?

The distance covered in one full revolution is the circumference of the circle, which is $2\pi R$. Since the body completes 4 revolutions, the total distance is $4 \times 2\pi R = 8\pi R$.

Q: Can I solve this without converting to meters?

While theoretically possible if you maintain consistent custom units, it's highly error-prone. Converting everything to a common base SI unit like meters and seconds is the safest and most standard approach.

Q: What does 4 arcseconds mean?

An arcsecond is a unit of angular measurement equal to 1/3600th of a degree. It's a very small angle commonly used in astronomy to measure tiny apparent sizes or distances in the sky.

Question

A huge circular arc of length $4.4 	ext{ ly}$ subtends an angle $4	ext{s}$ at the centre of the circle. How long it would take for a body to complete $4 	ext{ rev}$ if its speed is $8 	ext{ AU per sec}$ ?
[Given, $1 	ext{ ly} = 9.46 	imes 10^{15} 	ext{ m}, 1 	ext{ AU} = 1.5 	imes 10^{11} 	ext{ m}$]

Options:

(A) $4.1 	imes 10^8 	ext{ s}$

(B) $4.5 	imes 10^{10} 	ext{ s}$

(C) $3.5 	imes 10^6 	ext{ s}$

(D) $7.2 	imes 10^8 	ext{ s}$

Sigma Percentile · Accepted Answer

## A Cosmic Journey: Calculating Time on a Galactic Scale

Imagine a circle so vast that just a tiny piece of its arc is $4.4 	ext{ light-years}$ long! This arc subtends a minuscule angle of $4 	ext{ arcseconds}$ at the center. We have a body zooming around this circle at $8 	ext{ Astronomical Units per second}$. Our goal is to find out how long it takes to complete $4$ full revolutions.

### The Geometry of the Cosmos (Finding R)

To find the time, we first need the total distance, which means we need the radius of this massive circle. Remember the fundamental relation from geometry? Angle equals arc length divided by radius.

$$l = R	heta \implies R = \frac{l}{	heta}$$

Before we plug in the numbers, we must ensure all units are in the standard SI system. We convert the arc length from light-years to meters by multiplying by $9.46 	imes 10^{15}$.

$$l = 4.4 	imes 9.46 	imes 10^{15} 	ext{ m} = 4.1624 	imes 10^{16} 	ext{ m}$$

For the angle, we convert $4 	ext{ arcseconds}$ to degrees by dividing by $3600$, and then to radians by multiplying by $\frac{\pi}{180}$.

$$	heta = 4 	imes \left(\frac{1}{3600}ight) 	imes \left(\frac{\pi}{180}ight) 	ext{ rad}$$

Now, let's crunch these astronomical numbers. Dividing the arc length by our tiny angle in radians gives us a staggering radius:

$$R = \frac{4.1624 	imes 10^{16}}{4 	imes \frac{1}{3600} 	imes \frac{\pi}{180}} \approx 2.146 	imes 10^{21} 	ext{ m}$$

That's the scale we are dealing with!

### The Speed of the Journey

With the radius in hand, we can find the total distance. One full revolution is the circumference, $2\pi R$. Since the body completes $4$ revolutions, the total distance is $4 	imes 2\pi R$, or $8\pi R$. The time taken is simply this total distance divided by the speed $v$.

$$t = \frac{d}{v} = \frac{8\pi R}{v}$$

Next, let's look at the speed. It's given as $8 	ext{ Astronomical Units per second}$. We convert this to meters per second by multiplying by $1.5 	imes 10^{11}$.

$$v = 8 	imes 1.5 	imes 10^{11} 	ext{ m/s} = 1.2 	imes 10^{12} 	ext{ m/s}$$

### The Final Countdown

Now, we substitute our values for $R$ and $v$ into the time equation.

$$t = \frac{8\pi 	imes 2.146 	imes 10^{21}}{1.2 	imes 10^{12}}$$

Finally, we evaluate this expression. Carefully calculating the numerator and dividing by the denominator, we find that the time taken is approximately:

**$$t \approx 4.5 	imes 10^{10} 	ext{ s}$$**

To put this in perspective, $4.5 	imes 10^{10} 	ext{ seconds}$ is over $1400 	ext{ years}$! This problem beautifully illustrates how simple kinematic equations apply even at galactic scales, provided we are meticulous with our unit conversions.

Animated Solution for Physics - Kinematics: A huge circular arc of length $4.4 ly$ subtends an angle $4 s$ at the centre of the circle. How long it would take for a body to complete $4 rev$ if its speed is $8 AU per sec$ ? [Given, $1 ly = 9.46 \times 1 0^{15} m, 1 AU = 1.5 \times 1 0^{11} m$ ]

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Visualized Solution

$The Astronomical Setup$

$Finding the Radius R$

$\text{Unit Conversions & Substitution}$

$Calculating R$

$Time for 4 Revolutions$

$\text{Speed Conversion & Substitution}$

$Final Calculation$

$Cosmic Perspectives$

The Sigma Insight: Kinematics of Circular Motion

The Geometry of the Cosmos (Finding R)

The Speed of the Journey

The Final Countdown

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Visualized Solution

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The Astronomical Setup

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Finding the Radius R

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } \text{Unit Conversions & Substitution}

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Calculating R

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Time for 4 Revolutions

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } \text{Speed Conversion & Substitution}

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Final Calculation

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Cosmic Perspectives