Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 6: Problem 728

Energy required to move a body of mass \(\mathrm{m}\) from from an orbit of radius \(2 \mathrm{R}\) to \(3 \mathrm{R}\) is \(\ldots \ldots \ldots \ldots\) (A) \(\left[(\mathrm{GMm}) /\left(12 \mathrm{R}^{2}\right)\right]\) (B) \(\left[(\mathrm{GMm}) /\left(3 \mathrm{R}^{2}\right)\right]\) (C) \([(\mathrm{GMm}) /(8 \mathrm{R})]\) (D) \([(\mathrm{GMm}) /(6 \mathrm{R})]\)

Short Answer

Expert verified
The energy required to move a body of mass m from an orbit of radius 2R to 3R is \(\Delta U = \frac{GMm}{6R}\). The correct answer is (D).

Step by step solution

01

Calculate Initial Potential Energy

In the initial state, the body is in an orbit of radius 2R. The gravitational potential energy at this radius can be calculated using the formula: \(U_1 = -\frac{GMm}{r_1}\) Substituting the given radius, \(r_1 = 2R\): \(U_1 = -\frac{GMm}{2R}\)
02

Calculate Final Potential Energy

In the final state, the body is in an orbit of radius 3R. The gravitational potential energy at this radius can be calculated using the same formula: \(U_2 = -\frac{GMm}{r_2}\) Substituting the given radius, \(r_2 = 3R\): \(U_2 = -\frac{GMm}{3R}\)
03

Find the Difference in Potential Energy

The energy required to move the body from radius 2R to 3R is the difference in potential energy between the two orbits: \(\Delta U = U_2 - U_1\) Substituting the values we calculated for \(U_1\) and \(U_2\): \(\Delta U = -\frac{GMm}{3R} - \left(-\frac{GMm}{2R}\right)\)
04

Simplify the Expression

Now we simplify the expression for the change in potential energy: \(\Delta U = \frac{GMm}{2R} - \frac{GMm}{3R} = \frac{3GMm - 2GMm}{6R} = \frac{GMm}{6R}\)
05

Match the Answer with the Given Options

Matching the expression we found for the change in potential energy with the answer options given in the exercise, we find that the correct answer is: \(\Delta U = [(GMm) /(6R)]\) So, the correct answer is (D).

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.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

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 acceleration of a body due to the attraction of the earth (radius R) at a distance \(2 \mathrm{R}\) from the surface of the earth is \(=\) (g \(=\overline{\text { acceleration due to gravity at the surface of earth })}\) (A) \(\mathrm{g} / 9\) (B) \(\mathrm{g} / 3\) (C) \(\mathrm{g} / 4\) (D) 9

If \(\mathrm{r}\) denotes the distance between the sun and the earth, then the angular momentum of the earth around the sun is proportional to (A) \(1^{3 / 2}\) (B) \(\mathrm{r}\) (C) \(r^{1 / 2}\) (D) \(r^{2}\)

Escape velocity on the surface of earth is \(11.2 \mathrm{kms}^{-1}\) Escape velocity from a planet whose masses the same as that of earth and radius $1 / 4\( that of earth is \)=\ldots \ldots \ldots \mathrm{kms}^{-1}$ (A) \(2.8\) (B) \(15.6\) (C) \(22.4\) (D) \(44.8\)

The additional K.E. to be provided to a satellite of mass \(\mathrm{m}\) revolving around a planet of mass \(\mathrm{M}\), to transfer it from a circular orbit of radius \(\mathrm{R}_{1}\) to another radius \(\mathrm{R}_{2}\left(\mathrm{R}_{2}>\mathrm{R}_{1}\right)\) is (A) $\operatorname{GMm}\left[\left(1 / R_{1}^{2}\right)-\left(1 / R_{2}^{2}\right)\right]$ \(\operatorname{GMm}\left[\left(1 / R_{1}\right)-\left(1 / R_{2}\right)\right]\) (C) $2 \mathrm{GMm}\left[\left(1 / \mathrm{R}_{1}\right)-\left(1 / \mathrm{R}_{2}\right)\right]$ (D) $(1 / 2) \mathrm{GMm}\left[\left(1 / \mathrm{R}_{1}\right)-\left(1 / \mathrm{R}_{2}\right)\right]$

The moon's radius is \(1 / 4\) that of earth and its mass is \(1 / 80\) times that of the earth. If g represents the acceleration due to gravity on the surface of earth, that on the surface of the moon is (A) \(g / 4\) (B) \(\mathrm{g} / 5\) (c) \(\mathrm{g} / 6\) (D) \(\mathrm{g} / 8\)
See all solutions

Recommended explanations on English Textbooks

Lexis and Semantics

Read Explanation

Linguistic Terms

Read Explanation

Essay Writing Skills

Read Explanation

Text Comparison

Read Explanation

Semiotics

Read Explanation

Creative Story

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.

Sign-up for free