Download the App!

Get 24/7 study help with the Numerade app for iOS and Android! Enter your email for an invite.

Sent to:
Search glass icon
  • Login
  • Textbooks
  • Ask our Educators
  • Study Tools
    Study Groups Bootcamps Quizzes AI Tutor iOS Student App Android Student App StudyParty
  • For Educators
    Become an educator Educator app for iPad Our educators
  • For Schools

Problem

The force $ F $ acting on a body with mass $ m $ …

05:37

Question

Answered step-by-step

Problem 20 Hard Difficulty

Newton's Law of Gravitation says that the magnitude $ F $ of the force exerted by a body of mass $ m $ on a body of mass $ M $ is
$ F = \frac {GmM}{r^2} $
where $ G $ is the gravitational constant and $ r $ is the distance between the bodies.
(a) Find $ dF/dr $ and explain its meaning. What does the minus sign indicate?
(b) Suppose it is known that the earth attacks an objects an object with a force that decreases at the rate of 2 N/km when

$ r = 20,000 km. $ How fast does this force change when $ r = 10,000 km? $


Video Answer

Solved by verified expert

preview
Numerade Logo

This problem has been solved!

Try Numerade free for 7 days

Heather Zimmers
Numerade Educator

Like

Report

Textbook Answer

Official textbook answer

Video by Heather Zimmers

Numerade Educator

This textbook answer is only visible when subscribed! Please subscribe to view the answer

More Answers

01:04

Amrita Bhasin

Related Courses

Calculus 1 / AB

Calculus: Early Transcendentals

Chapter 3

Differentiation Rules

Section 7

Rates of Change in the Natural and Social Sciences

Related Topics

Derivatives

Differentiation

Discussion

You must be signed in to discuss.
Top Calculus 1 / AB Educators
Grace He
Kayleah Tsai

Harvey Mudd College

Caleb Elmore

Baylor University

Michael Jacobsen

Idaho State University

Calculus 1 / AB Courses

Lectures

Video Thumbnail

04:40

Derivatives - Intro

In mathematics, a derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a moving object with respect to time is the object's velocity. The concept of a derivative developed as a way to measure the steepness of a curve; the concept was ultimately generalized and now "derivative" is often used to refer to the relationship between two variables, independent and dependent, and to various related notions, such as the differential.

Video Thumbnail

44:57

Differentiation Rules - Overview

In mathematics, a differentiation rule is a rule for computing the derivative of a function in one variable. Many differentiation rules can be expressed as a product rule.

Join Course
Recommended Videos

04:05

Newton's Law of Gravi…

05:18

Newton' s Law of Grav…

03:53

20. Newton $ Law of Gravit…

Watch More Solved Questions in Chapter 3

Problem 1
Problem 2
Problem 3
Problem 4
Problem 5
Problem 6
Problem 7
Problem 8
Problem 9
Problem 10
Problem 11
Problem 12
Problem 13
Problem 14
Problem 15
Problem 16
Problem 17
Problem 18
Problem 19
Problem 20
Problem 21
Problem 22
Problem 23
Problem 24
Problem 25
Problem 26
Problem 27
Problem 28
Problem 29
Problem 30
Problem 31
Problem 32
Problem 33
Problem 34
Problem 35
Problem 36
Problem 37
Problem 38
Problem 39

Video Transcript

here we have an equation that represents the force as a function of the distance between the bodies and we want to find D FDR. So the FDR would be the rate of change of the force as a function of the distance. So that would tell us how the force is changing as a result of the change in the distance. So let's rewrite the function as G m M times are to the negative second power. And now we confined its derivative by bringing down the negative, too. G m and M are constants, and we have our to the negative. Third, we can write rewrite this as negative to G m m over r cubed. All right, So for Part B, we're interested in figuring out the rate of change for a 10,000 kilometer distance, given some information that we know about a 20,000 kilometer distance. So here's what we know that if r equals 20,000 kilometers, dfd are is negative two. It says it is decreasing at a rate of two. I believe it would be Newtons per kilometre. If that's the unit I'm interpreting correctly. Okay, so here's what we can do we can substitute these numbers into our derivative and that will give us the value of the constant. And then we can use that value as their constant and find the unknown value. So we have negative two for Dft R is equal to negative two g m m. Over 20,000 cubed. Okay, now we can divide both sides by negative too. And that will just be a one. And then we can multiply both sides by 20,000 cubed. So we have 20,000 cubed equals GMM. Now let's use that to solve for the rate of change of the force when the distance is 10,000. So now we have rate of change of force with respect to radius is negative two times that value. We just got 20,000 cubed. That's our GMM over are cute. We can substitute are 10,000 in for our we can reduce 20,000 cubed over 10,000 cubed, and that's going to leave us with two. Cute. We have negative two times, two cubed and that's negative 16. Now the problem doesn't really require negative because it just asks How fast is the force changing? So the forces changing at 16 Newtons per kilometre

Get More Help with this Textbook
James Stewart

Calculus: Early Transcendentals

View More Answers From This Book

Find Another Textbook

Study Groups
Study with other students and unlock Numerade solutions for free.
Math (Geometry, Algebra I and II) with Nancy
Arrow icon
Participants icon
67
Hosted by: Ay?Enur Çal???R
Math (Algebra 2 & AP Calculus AB) with Yovanny
Arrow icon
Participants icon
44
Hosted by: Alonso M
See More

Related Topics

Derivatives

Differentiation

Top Calculus 1 / AB Educators
Grace He

Numerade Educator

Kayleah Tsai

Harvey Mudd College

Caleb Elmore

Baylor University

Michael Jacobsen

Idaho State University

Calculus 1 / AB Courses

Lectures

Video Thumbnail

04:40

Derivatives - Intro

In mathematics, a derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a moving object with respect to time is the object's velocity. The concept of a derivative developed as a way to measure the steepness of a curve; the concept was ultimately generalized and now "derivative" is often used to refer to the relationship between two variables, independent and dependent, and to various related notions, such as the differential.

Video Thumbnail

44:57

Differentiation Rules - Overview

In mathematics, a differentiation rule is a rule for computing the derivative of a function in one variable. Many differentiation rules can be expressed as a product rule.

Join Course
Recommended Videos

04:05

Newton's Law of Gravitation says that the magnitude $F$ of the force exerted by…

05:18

Newton' s Law of Gravitation says that the magnitude F of the force exerted by …

03:53

20. Newton $ Law of Gravitation says that the magnitude F of the force exerted …
Additional Mathematics Questions

04:53

Cynthia Besch wants to buy a rug for a room that is 21 ft wide and 28 ft lon…

01:04

Graph the function h (x) =4x_7_

03:25

Find f (x) and find the equation of the line tangent to the graph of fat the…

01:23

(a) Newton $ Law of Gravitation states that two bodies with masses mI and mz…

00:53

Use function notation to write the equation of the line with the given slope…

00:30

13.
points
SPreCalc7 2.8.032.
table of values for one-to-one functi…

08:31

height zero wnen caught: feet of fencing rancher has 10,000 linear Ranchi…

01:36

Find an equation of the line passing through the given points. Use function …

03:38

Note; Triangle may not be drawn to scale;
Suppose a = 5 and A = 25 degree…

05:00

Note: Triangle may not be drawn to scale
Suppose € = 12 and A = 5 degrees…

Add To Playlist

Hmmm, doesn't seem like you have any playlists. Please add your first playlist.

Create a New Playlist

`

Share Question

Copy Link

OR

Enter Friends' Emails

Report Question

Get 24/7 study help with our app

 

Available on iOS and Android

About
  • Our Story
  • Careers
  • Our Educators
  • Numerade Blog
Browse
  • Bootcamps
  • Books
  • Topics
  • Test Prep
  • Ask Directory
  • Online Tutors
  • Tutors Near Me
Support
  • Help
  • Privacy Policy
  • Terms of Service
Get started