# The value of d F d r and explain its meaning. What does the minus sign indicate? ### Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805 ### Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805

#### Solutions

Chapter 3.8, Problem 20E
To determine

## To find: The value of dFdr and explain its meaning. What does the minus sign indicate?

Expert Solution

Therefore, the value of dFdr is 2GmMr3_.

### Explanation of Solution

Given:

The equation for magnitude F is as given below.

F=GmMr2

Rewrite the equation as below.

F=(GmM)r2 (1)

Calculation:

Calculate the value of rate of change of force dFdr as below.

Differentiate the equation (1) as below.

dFdr=2(GmM)r3dFdr=2GmMr3

The expression 2GmMr3 is the rate of change of the force with respect to the distance between the bodies.

The minus sign before the dFdr indicates that as the distance r between the bodies increases, the magnitude of the force F exerted by the body of mass m on the another body of mass M is decreases.

Therefore, the value of dFdr is 2GmMr3_.

(b)

To determine

### To find: How fast does the force change when the distance between the bodies r is 10,000 km.

Expert Solution

Therefore, when distance between the earth and the object r is 10,000km, the force exerted by the earth on the object changes faster as 16N/km_.

### Explanation of Solution

Given:

The equation for magnitude F is as given below.

F=GmMr2

Rewrite the equation as below.

F=(GmM)r2 (1)

Calculation:

Calculate how fast the force change dFdr when the distance between the bodies r is 10,000km.

The force exerted by the earth on the object decreases at the rate of 2N/km, when the distance between the bodies is 20,000km.

Use the first derivative of force equation.

F'=2GmM(r3)2=2GmM(20,000)GmM=20,000

Substitute 10,000 for r and 20,000 for GmM.

F'(10,000)=2(20,000)10,0003F'(10,000)=16N/km

Therefore, when distance between the earth and the object r is 10,000km, the force exerted by the earth on the object changes faster as 16N/km_.

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