A satellite circles a spherical planet of unknown mass in a...

A satellite circles a spherical planet of unknown mass in a circular orbit of radius $2.0 \times 10^{7} \mathrm{m} .$ The magnitude of the gravitational force exerted on the satellite by the planet is 120 $\mathrm{N} .(a)$ What would be the magnitude of the gravitational force exerted on the satellite by the planet if the radius of the orbit were increased to $3.0 \times 10^{7} \mathrm{m} ?$ (b) If the satellite circles the planet once every 2.0 $\mathrm{h}$ in the larger orbit, what is the mass of the planet?

A satellite circles a spherical planet of unknown mass in a
circular orbit of radius $2.0 \times 10^{7} \mathrm{m} .$ The magnitude of the
gravitational force exerted on the satellite by the planet is
120 $\mathrm{N} .(a)$ What would be the magnitude of the gravitational
force exerted on the satellite by the planet if the radius of
the orbit were increased to $3.0 \times 10^{7} \mathrm{m} ?$ (b) If the satellite
circles the planet once every 2.0 $\mathrm{h}$ in the larger orbit, what is
the mass of the planet?

Key Concepts

Orbital Period and Planet Mass Determination

The orbital period is the time a satellite takes to complete one full orbit. By using the relationship between the orbital period, the orbital radius, and the gravitational force (or centripetal acceleration), one can derive the mass of the central body. This is often done by combining Newton’s law of gravitation with the equations of circular motion, leading to an expression that relates the planet’s mass to the satellite's orbital period and radius.

Centripetal Force in Circular Motion

When an object moves in a circular path, it experiences an inward force called the centripetal force, which is necessary to keep the object in its curved trajectory. In orbital mechanics, the gravitational force acting on a satellite provides the required centripetal force to maintain its circular orbit. Understanding this balance is key to analyzing the motion of satellites and other orbiting bodies.

Inverse Square Law

The inverse square law indicates that a specified physical quantity or strength is inversely proportional to the square of the distance from the source of that physical quantity. In the context of gravitation, as the distance between two bodies increases, the gravitational force decreases by the square of that increase. This concept is crucial when determining how changes in the distance between a satellite and a planet affect the gravitational force.

Newton's Law of Universal Gravitation

This law states that every point mass attracts every other point mass in the universe with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between them. It is commonly expressed as F = G(m?m?)/r², where G is the gravitational constant, m? and m? are the masses involved, and r is the distance between their centers. This principle is fundamental in calculating gravitational forces in orbital mechanics.

Transcript

00:01 Okay, in this problem we have a planet and a satellite that is circling this planet.

00:06 And the distance from the center of the planet to the center of the satellite is the orbital radius r.

00:13 And we are told in this problem that at the first radius that they give us, the force, the gravitational force, is 120 newtons.

00:25 And we have r1, which is 2 .0 times 10 to the 7 meters.

00:34 So this is the distance that the satellite starts off at.

00:38 And then in part a of this problem, they ask what the force would be if the radius was then changed to a further out radius of 3 .0 times 10 to the 7 meters.

00:52 So now we have two different distances for this satellite.

00:56 And we want to know how the force changes.

00:59 So we'll call this first force that they give us f, the gravitational force, 1.

01:05 And we want to find the gravitational force, 2, at the second radius.

01:11 So what we can do is take a ratio of these two forces.

01:15 And first we'll just say that the gravitational force in general is the gravitational constant times the mass of the satellite times the mass of the planet divided by the distance between them squared.

01:31 So we can take a ratio of the two.

01:34 So fg1 over fg2 is going to be equal to gravitational times mass of the satellite times mass of the planet.

01:50 These are the same, the two radii that they give us change.

01:53 The masses aren't changing.

01:56 Divided by r1 squared.

01:59 Take that whole thing and divide that by the equation for the second force.

02:06 So gmsmp over r2 squared.

02:11 Now a lot of this will cancel out.

02:13 So the gravitational constant isn't changing, the masses aren't changing.

02:17 So we're actually just going to go ahead and cancel these out and simplify this.

02:23 So we have something, we have r2 squared in the denominator of the denominator.

02:28 So that's actually just going to go to the top.

02:32 Divided by r1 squared.

02:37 And what we want to do is just solve for fg2.

02:41 So just kind of rearranging things here, we can get that our second force is going to be equal to the first force times r1 squared over r2 squared.

03:03 So we actually just had to kind of flip these around.

03:06 So we can go ahead and plug in what we were given here...

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