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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).

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Most popular questions from this chapter

Two satellites \(\mathrm{A}\) and \(\mathrm{B}\) go round a planet in circular orbits having radii \(4 \mathrm{R}\) and \(\mathrm{R}\) respectively If the speed of satellite \(\mathrm{A}\) is \(3 \mathrm{v}\), then speed of satellite \(\mathrm{B}\) is (A) \((3 \mathrm{v} / 2)\) (B) \((4 \mathrm{v} / 2)\) (C) \(6 \mathrm{v}\) (D) \(12 \mathrm{v}\)

A particle of mass \(10 \mathrm{~g}\) is kept on the surface of a uniform sphere of mass \(100 \mathrm{~kg}\) and radius \(10 \mathrm{~cm}\). Find the work to be done against the gravitational force between them to take the particle is away from the sphere \(\left(\mathrm{G}=6.67 \times 10^{-11} \mathrm{SI}\right.\) unit \()\) (A) \(6.67 \times 10^{-9} \mathrm{~J}\) (B) \(6.67 \times 10^{-10} \mathrm{~J}\) (C) \(13.34 \times 10^{-10} \mathrm{~J}\) (D) \(3.33 \times 10^{-10} \mathrm{~J}\)

3 particle each of mass \(\mathrm{m}\) are kept at vertices of an equilateral triangle of side \(L\). The gravitational field at center due to these particles is (A) zero (B) \(\left[(3 \mathrm{GM}) / \mathrm{L}^{2}\right]\) (C) \(\left[(9 \mathrm{GM}) / \mathrm{L}^{2}\right]\) (D) \((12 / \sqrt{3})\left(\mathrm{Gm} / \mathrm{L}^{2}\right)\)

A satellite of mass \(\mathrm{m}\) is orbiting close to the surface of the earth (Radius \(\mathrm{R}=6400 \mathrm{~km}\) ) has a K.E. \(\mathrm{K}\). The corresponding \(\mathrm{K} . \mathrm{E}\). of satellite to escape from the earth's gravitational field is (A) \(\mathrm{K}\) (B) \(2 \mathrm{~K}\) (C) \(\mathrm{mg} \mathrm{R}\) (D) \(\mathrm{m} \mathrm{K}\)

Weight of a body is maximum at (A) moon (B) poles of earth (C) Equator of earth (D) Center of earth
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