/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 4 Our sun rotates in a circular or... [FREE SOLUTION] | 91Ó°ÊÓ

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Our sun rotates in a circular orbit about the center of the Milky Way galaxy. The radius of the orbit is \(2.2 \times 10^{20} \mathrm{m},\) and the angular speed of the sun is \(1.1 \times 10^{-15} \mathrm{rad} / \mathrm{s} .\) How long (in years) does it take for the sun to make one revolution around the center?

Short Answer

Expert verified
It takes approximately 181 million years for the sun to make one revolution around the Milky Way's center.

Step by step solution

01

Understand the Problem

We need to find the time it takes for the sun to complete one full revolution around the center of the Milky Way. The given values are:- Radius of the orbit: \(2.2 \times 10^{20} \text{ meters}\)- Angular speed of the sun: \(1.1 \times 10^{-15} \text{ rad/s}\)
02

Use the Angular Speed Formula

The time taken for one complete revolution can be found using the formula for angular speed \(\omega: \ \omega = \frac{\theta}{t}\), where \(\theta\) is the angle in radians and \(t\) is the time in seconds. For one full revolution, \(\theta = 2\pi\) radians.
03

Rearrange the Formula to Find Time

Rearrange the formula to solve for time \(t\):\[t = \frac{\theta}{\omega}\ = \frac{2\pi}{1.1 \times 10^{-15}}\ \ = \2\pi \times \frac{1}{1.1 \times 10^{-15}}\text{ seconds}\\].
04

Calculate the Time in Seconds

Substitute the given values into the formula:\[t = 2\pi \times \frac{1}{1.1 \times 10^{-15}} \approx 5.71 \times 10^{15} \text{ seconds}\\].
05

Convert Time to Years

To convert the time from seconds to years, use the conversion factor: \(1\text{ year} = 365.25 \times 24 \times 60 \times 60 \text{ seconds}\).Calculate the time in years:\[t = \frac{5.71 \times 10^{15}}{365.25 \times 24 \times 60 \times 60} \approx 1.81 \times 10^8 \text{ years}\\].

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Circular Orbit
When an object travels in a circular path around a point or another object, it is said to be in a circular orbit. Imagine a merry-go-round; every horse on it follows a circular orbit. In the context of astronomy, our sun travels in a massive circular orbit around the center of the Milky Way Galaxy.
The path the sun follows is due to gravitational forces pulling it toward the galaxy's center, and due to its velocity, it continues in a curved path. The key aspects of a circular orbit include:
  • The constant radius of the path.
  • The continuous application of centripetal force toward the center of the orbit.
  • A consistent angular speed as long as the forces involved are in equilibrium.
Understanding this concept helps us comprehend the vast scales and dynamics involved in astronomical movements.
Revolution Time
Revolution time refers to the period it takes for an object to complete one full cycle along its orbit. In our exercise, it specifically relates to how long it takes the sun to make one full rotation around the Milky Way Galaxy's center. Measuring this time allows astronomers to understand the dynamics and scaling of the galaxy.
To calculate the revolution time of the sun, we use the formula for angular speed: \( \omega = \frac{\theta}{t} \), where \( \omega \) is the angular speed, \( \theta \) is the angular displacement (\( 2\pi \) radians for a complete cycle), and \( t \) is the time period. This formula connects angular displacement with time, allowing for a direct calculation of revolution time. By understanding revolution time, we can gauge how different celestial objects function within their orbits and predict future positions.
Milky Way Galaxy
The Milky Way Galaxy is a barred spiral galaxy that hosts our solar system. Imagine it as a swirling city of stars, dust, and dark matter. Within this vast galaxy, billions of stars move along their orbits, creating an incredible cosmic dance.
The Milky Way is characterized by:
  • A bulging center often composed of dense stars and a supermassive black hole.
  • Arms that spiral outward, where our sun is located, approximately 25,000 light years from the center.
  • An estimated 100,000 light years in diameter overall, making it a colossal feature of the cosmos.
Understanding the Milky Way's structure and dynamics helps us position our solar system within the universe and provides context for the gravitational forces affecting our cosmic neighborhood.
Convert Seconds to Years
Converting seconds to years is a common step in dealing with astronomical calculations given the vast time scales. It helps humans comprehend astronomical data in more relatable terms.
To convert seconds into years, use the conversion factor: 1 year = 365.25 days * 24 hours/day * 60 minutes/hour * 60 seconds/minute. Here's the breakdown:
  • 365.25 accounts for the inclusion of leap years every four years.
  • Multiply by 24 for hours in a day, then by 60 for minutes, and finally by another 60 for seconds.
Therefore, the total seconds in a year is approximately 31,557,600. Using this, we can convert any value from seconds to years, simplifying the representation of large timespans in astronomy. This conversion ensures we interpret the data effectively and gauge celestial events across correctly understood timescales.

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Most popular questions from this chapter

The table that follows lists four pairs of initial and fi nal angles of a wheel on a moving car. The elapsed time for each pair of angles is 2.0 s. For each of the four pairs, determine the average angular velocity (magnitude and direction as given by the algebraic sign of your answer). $$ \begin{array}{lcc} & \text { Initial angle } \theta_{0} & \text { Final angle } \theta \\ \hline \text { (a) } & 0.45 \mathrm{rad} & 0.75 \mathrm{rad} \\ \hline \text { (b) } & 0.94 \mathrm{rad} & 0.54 \mathrm{rad} \\ \hline \text { (c) } & 5.4 \mathrm{rad} & 4.2 \mathrm{rad} \\ \hline \text { (d) } & 3.0 \mathrm{rad} & 3.8 \mathrm{rad} \\ \hline \end{array} $$

Review Multiple-Concept Example 7 in this chapter as an aid in solving this problem. In a fast-pitch softball game the pitcher is impressive to watch, as she delivers a pitch by rapidly whirling her arm around so that the ball in her hand moves on a circle. In one instance, the radius of the circle is \(0.670 \mathrm{m} .\) At one point on this circle, the ball has an angular acceleration of \(64.0 \mathrm{rad} / \mathrm{s}^{2}\) and an angular speed of \(16.0 \mathrm{rad} / \mathrm{s} .\) (a) Find the magnitude of the total acceleration (centripetal plus tangential) of the ball. (b) Determine the angle of the total acceleration relative to the radial direction.

A pitcher throws a curveball that reaches the catcher in 0.60 s. The ball curves because it is spinning at an average angular velocity of 330 rev/min (assumed constant) on its way to the catcher’s mitt. What is the angular displacement of the baseball (in radians) as it travels from the pitcher to the catcher?

The initial angular velocity and the angular acceleration of four rotating objects at the same instant in time are listed in the table that follows. For each of the objects (a), (b), (c), and (d), determine the fi nal angular speed after an elapsed time of 2.0 s. $$ \begin{array}{lcc} & \begin{array}{c} \text { Initial angular } \\ \text { velocity } \omega_{0} \end{array} & \begin{array}{c} \text { Angular } \\ \text { acceleration } \alpha \end{array} \\ \text { (a) } & +12 \mathrm{rad} / \mathrm{s} & +3.0 \mathrm{rad} / \mathrm{s}^{2} \\ \hline \text { (b) } & +12 \mathrm{rad} / \mathrm{s} & -3.0 \mathrm{rad} / \mathrm{s}^{2} \\ \hline \text { (c) } & -12 \mathrm{rad} / \mathrm{s} & +3.0 \mathrm{rad} / \mathrm{s}^{2} \\ \hline \text { (d) } & -12 \mathrm{rad} / \mathrm{s} & -3.0 \mathrm{rad} / \mathrm{s}^{2} \\ \hline \end{array} $$

At the local swimming hole, a favorite trick is to run horizontally off a cliff that is \(8.3 \mathrm{m}\) above the water. One diver runs off the edge of the cliff, tucks into a "ball," and rotates on the way down with an average angular speed of 1.6 rev/s. Ignore air resistance and determine the number of revolutions she makes while on the way down.

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