/*! 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 9 Two satellites \(S_{1}\) and \(S... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 7: Problem 9

Two satellites \(S_{1}\) and \(S_{2}\) describe circular orbits of radii \(r\) and \(2 r\) respectively around a planet. If the orbital angular velocity of \(S_{1}\) is \(\omega\), the orbital angular velocity of \(S_{2}\) is (A) \(\frac{\omega}{2 \sqrt{2}}\) (B) \(\frac{\omega \sqrt{2}}{3}\) (C) \(\frac{\omega}{\sqrt{2}}\) (D) \(\omega \sqrt{2}\)

Short Answer

Expert verified
The orbital angular velocity of S2 is \(\frac{\omega}{2\sqrt{2}}\).

Step by step solution

01

Write down the formula for gravitational force

The gravitational force acting on the satellites is given by Newton's law of gravitation: \[F = G\frac{Mm}{r^2}\] where F is the gravitational force, G is the gravitational constant, M is the mass of the planet, m is the mass of the satellite, and r is the distance between the planet and the satellite.
02

Write down the formula for centripetal force

The centripetal force acting on a satellite in a circular orbit is given by the formula: \[F = m\omega^2r\] where F is the centripetal force, m is the mass of the satellite, 蠅 is the angular velocity, and r is the radius of the orbit.
03

Set the gravitational force equal to the centripetal force for S1 and S2

For S1, we have: \[G\frac{Mm}{r^2} = m\omega^2r\] For S2, we have: \[G\frac{Mm}{(2r)^2} = m\omega_2^2(2r)\] where 蠅 is the orbital angular velocity of S1, and 蠅_2 is the orbital angular velocity of S2.
04

Solve the equations for 蠅_2

Divide the equation for S2 by the equation for S1 to eliminate M and m: \[\frac{G\frac{Mm}{(2r)^2}}{G\frac{Mm}{r^2}} = \frac{\omega_2^2(2r)}{\omega^2r}\] Simplify the equation: \[\frac{1}{4} = \frac{2\omega_2^2r}{\omega^2r}\] Solve for 蠅_2: \[\omega_2^2 = \frac{\omega^2}{8}\] Take the square root of both sides: \[\omega_2 = \frac{\omega}{\sqrt{8}}\] Simplify: \[\omega_2 = \frac{\omega}{2\sqrt{2}}\] This corresponds to option (A). So, the orbital angular velocity of S2 is \(\frac{\omega}{2\sqrt{2}}\).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

The height at which the acceleration due to gravity becomes \(\frac{g}{9}\) (where \(g=\) the acceleration due to gravity on the surface of the earth) in terms of \(R\), the radius of the earth, is (A) \(2 R\) (B) \(\frac{R}{\sqrt{3}}\) (C) \(\frac{R}{2}\) (D) \(\sqrt{2} R\)

The period of a satellite in a circular orbit around a planet is independent of, (A) the mass of the planet. (B) the radius of the planet. (C) the mass of the satellite. (D) all of three parameters \(a, b\) and \(c\).

This question contains Statement 1 and Statement 2 of the four choices given after the statements, choose the one that best describes the two statements. [2008] Statement 1: For a mass \(M\) kept at the centre of a cube of side \(a\) the flux of gravitational field passing through its sides \(4 \pi G M\). Statement 2: If the direction of a field due to a point source is radial and its dependence on the distance \(r\) from the source is given as \(\frac{1}{r^{2}}\), its flux through a closed surface depends only on the strength of the source enclosed by the surface and not on the size or shape of the surface. (A) Statement 1 is false, Statement 2 is true. (B) Statement 1 is true, Statement 2 is true; Statement 2 is a correct explanation for Statement 1 . (C) Statement 1 is false, Statement 2 is true; Statement 2 is not a correct explanation for Statement \(1 .\) (D) Statement 1 is true, Statement 2 is false.

The change in the value of \(g\) at a height \(h\) above the surface of the earth is the same as that of a depth \(d\) below the surface of earth. When both \(d\) and \(h\) are much smaller than the radius of earth, then which one of the following is correct? (A) \(d=\frac{h}{2}\) (B) \(d=\frac{3 h}{2}\) (C) \(d=2 h\) (D) \(d=h\)

The acceleration due to gravity on the surface of the moon is \(\frac{1}{6}\) that of the surface of earth and the diameter of the moon is \(\frac{1}{4}\) that of earth. The ratio of escape velocities on earth and moon will be (A) \(\frac{\sqrt{6}}{2}\) (B) \(\sqrt{24}\) (C) 3 (D) \(\frac{\sqrt{3}}{2}\)
See all solutions

Recommended explanations on Physics Textbooks

Electricity

Read Explanation

Classical Mechanics

Read Explanation

Electricity and Magnetism

Read Explanation

Engineering Physics

Read Explanation

Geometrical and Physical Optics

Read Explanation

Atoms and Radioactivity

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.