Consider an object of mass m, not necessarily small compared...

Consider an object of mass $m$, not necessarily small compared with the mass of the Earth, released at a distance of $1.20 \times 10^{7} \mathrm{m}$ from the center of the Earth. Assume the Earth and the object behave as a pair of particles, isolated from the rest of the Universe. (a) Find the magnitude of the acceleration $a_{\mathrm{rel}}$ with which each starts to move relative to the other as a function of $m$. Evaluate the acceleration (b) for $m=5.00 \mathrm{kg},(\mathrm{c})$ for $m=2000 \mathrm{kg},$ and (d) for $m=2.00 \times 10^{24} \mathrm{kg} .(\mathrm{e})$ Describe the pattern of variation of $a_{\text {rel with }} m$.

Key Concepts

Newton's Law of Gravitation

Newton's law of universal gravitation states that every pair of masses attracts each other with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers. This principle is fundamental for analyzing the gravitational interaction between two bodies, regardless of their sizes, and forms the basis for calculating forces and subsequent accelerations in a two-body system.

Two-Body Problem

The two-body problem in classical mechanics involves predicting the motion of two interacting bodies, each exerting a gravitational force on the other. When both masses are taken into account, their mutual influence must be considered rather than assuming one body to be fixed. This analysis uses concepts like relative acceleration and reduced mass to describe how each body moves in response to the gravitational force, making it essential for understanding systems where both masses are significant.

Center-of-Mass Frame

In analyzing a two-body system, the center-of-mass frame is a viewpoint in which the overall motion of the system is simplified by separating the motion of the center of mass from the relative motions of the bodies around it. This concept is critical when neither mass is negligible, as it allows the acceleration of each mass to be considered relative to one another and highlights how the gravitational force affects both bodies symmetrically.

Transcript

00:01 So we have the earth, which i'm going to make blue, and we have another object that i'm going to make red.

00:11 The earth is going to have a mass m sub e, and the other object is just going to have a mass m, and it's going to have a distance between the two.

00:26 Distance is 1 .20 times 10 to the 7th power meters.

00:34 So we're looking for the acceleration between the two.

00:54 So i'm going to write that the sum of the forces equals mass times acceleration.

01:08 Well, the force on each of them is going to be the same.

01:13 It's going to be g, m, e, m over r squared.

01:21 Which is going to be, let's just focus on the red one for now.

01:28 Mass times acceleration toward the earth.

01:35 If we cancel that out, then acceleration of the red one is gme over r squared.

01:57 And we know those.

01:59 However, if we focus on the earth, g, m, e, m over r squared, equals m, a.

02:14 Well, now we get a is g, lowercase, m over r squared.

02:32 If we add these two accelerations together, that will give us the relative acceleration.

02:42 So when we add the two together, relative acceleration is going to be g over r squared times m -e plus m.

03:04 All right, let's put that into a calculator.

03:09 6 .67 times 10 to the, whoops, to the negative 11th power.

03:19 Over r squared.

03:22 I used r and d interchangeably here.

03:27 1 .2 times 10 to the seventh power.

03:36 And then i have to add the two masses together times mass of the earth, which is 5 .97 times 10 to the 24th power.

03:53 Okay, mass of the earth, 5 .97 times 10 of the 24th power, plus m, but i don't think we're given the mass of the object.

04:08 So i can't really put this in a calculator.

04:27 All right.

04:27 So i can't really put this in a calculator.

04:30 So i'm just going to leave it like this.

04:36 A, what if m was five kilograms? m .e.

04:48 Plus five.

04:55 It doesn't like something that i did here.

04:57 Okay.

05:07 I get the...

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