Download the App!

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

Question

Answered step-by-step

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

This problem has been solved!

Try Numerade free for 7 days

Like

Report

Official textbook answer

Video by Heather Zimmers

Numerade Educator

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

01:04

Amrita Bhasin

Calculus 1 / AB

Chapter 3

Differentiation Rules

Section 7

Rates of Change in the Natural and Social Sciences

Derivatives

Differentiation

Harvey Mudd College

Baylor University

Idaho State University

Lectures

04:40

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.

44:57

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.

04:05

Newton's Law of Gravi…

05:18

Newton' s Law of Grav…

03:53

20. Newton $ Law of Gravit…

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

View More Answers From This Book

Find Another Textbook

04:53

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

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.pointsSPreCalc7 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 scaleSuppose € = 12 and A = 5 degrees…