At what altitude above Earth's surface is the acceleration due to gravity equal to g / 2 ?

Name: Problem 17, Chapter 9: Pearson Physics
Uploaded: 2023-03-30T19:37:08-08:00
Duration: 5 min 53 s
Channel: H M
Description: At what altitude above Earth's surface is the acceleration due to gravity equal to g / 2 ?

At what altitude above Earth's surface is the acceleration...

Key Concepts

Newton's Law of Universal Gravitation

This fundamental law states that every mass attracts every other mass with a force which is proportional to the product of their masses and inversely proportional to the square of the distance between their centers. It forms the basis for understanding how gravitational forces act over distances and is quantified by the formula F = GMm/r².

Inverse-Square Law

The inverse-square law explains that the strength of the gravitational force decreases with the square of the distance from the source. This means that as one moves further from the center of a mass like Earth, the gravitational acceleration reduces rapidly, a critical concept for understanding variations in gravity with altitude.

Variation of Gravitational Acceleration with Altitude

Gravitational acceleration is dependent on the distance from the center of the Earth, expressed as g = GM/(R+h)². This relationship shows how gravity decreases with increased altitude (h), and by equating this expression to a fraction of the surface gravity, one can solve for the altitude at which a specified acceleration is achieved.

Transcript

00:01 The force due to gravity is f equals g, which is the gravitational constant, times the mass of one of the objects, in this case, m of the mass of the earth, times the mass of the other object that's being attracted by the gravitational force, divided by the distance between them squared, which is r squared.

00:31 And when i say the distance between them, i mean the distance between their centers of mass.

00:37 So, for example, if you have the earth with its center of mass here, and you're looking at some object here with center of mass, then r would be this distance here.

00:56 The acceleration due to gravity is given by newton's second law, f equals m .a, where m is the mass of m.

01:08 This object here.

01:10 So if we set the forces equal to each other, we have m .a is equal to g times m times m divided by r squared.

01:22 I can divide by little m on those sides to get that the acceleration due to gravity is equal to g times m over r squared.

01:37 We want to find the altitude above earth's surface where acceleration is equal to g over 2.

01:45 So the altitude above earth's surface is only a section of r.

01:53 It's only this part.

01:54 So this i'll call h for height.

01:59 And then we still have this section here, which is the radius of earth.

02:06 So we see that a little r is equal to 8.

02:14 Plus the radius of earth so i can plug that in to the equation so we have a equals g times m e divided by h plus r e and this is squared okay so we're trying to find the altitude h where a equals g over two so when i plug that in we have g over two is equal to big g times m e divided by h plus r .e all squared.

02:53 And we're solving for h.

02:55 So i'll multiply by h plus r e squared on both sides.

02:59 So i have g over two times h plus re squared is equal to g m .e...

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