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Calculus (MindTap Course List)

8th Edition
James Stewart
ISBN: 9781285740621

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Calculus (MindTap Course List)

8th Edition
James Stewart
ISBN: 9781285740621
Textbook Problem

(a) Newton’s Law of Gravitation states that two bodies with masses mi and m2 attract each other with a force

F = G m 1 m 2 r 2

where r is the distance between the bodies and G is the gravitational constant. If one of the bodies is fixed, find the work needed to move the other from r = a  and  r = b

(b) Compute the work required to launch a 1000-kg satellite vertically to a height of 1000 km. You may assume that the earth’s mass is 5.98 × 10 24  kg and is concentrated at its center. Take the radius of the earth to be 6.37 × 10 6  m and G = 6.67 × 10 11  N .m 2 / kg 2 .

To determine

a)

To find:

The work to move the other body from r=a to r=b if one of the bodies is fixed

Explanation

1) Concept:

The work is calculated by using the formula W=abf(x)dx

2) Given:

F=Gm1m2r2

m1, m2 are masses, G  is the gravitational constant and r is the distance between bodies.

3) Calculation:

The work is calculated by using the formula:

W=abf(x)dx

The gravitational force is given by F(r)=Gm1m2r2 We need to find the work done when the position changes from r=a to r=b

Therefore,

W=abGm1m2<

To determine

b)

To compute:

The work required to launch the satellite

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