A satellite of mass m, originally on the surface of the...

A satellite of mass $m,$ originally on the surface of the Earth, is placed into Earth orbit at an altitude $h .$ (a) Assuming a circular orbit, how long does the satellite take to complete one orbit? (b) What is the satellite's speed? (c) What is the minimum energy input necessary to place this satellite in orbit? Ignore air resistance but include the effect of the planet's daily rotation. Represent the mass and radius of the Earth as $M_{E}$ and $R_{E},$ respectively.

Key Concepts

Circular Orbital Motion

In a stable circular orbit, the gravitational force provides the necessary centripetal force to keep a satellite in its path. By equating the gravitational force to the centripetal force, one can derive important relationships such as the orbital speed and period. These relationships are critical for determining how long a satellite takes to complete one orbit and the speed required to maintain such an orbit at a given altitude.

Energy in Orbital Mechanics

Orbit insertion and maintenance involve changes in both gravitational potential energy and kinetic energy. The minimum energy input required to place a satellite into orbit is determined by the difference between the energy state on the surface and the orbital energy, which includes both the kinetic energy associated with its orbital speed and the change in gravitational potential energy. Understanding these energy considerations is key to calculating the efficiency and requirements of orbital insertion.

Newton's Law of Universal Gravitation

This law states that every point mass attracts every other point mass with a force along the line intersecting both points. The magnitude of the force is proportional to the product of the two masses and inversely proportional to the square of the distance between them. It is fundamental in calculating the gravitational force acting on satellites and forms the basis for understanding orbital dynamics.

Effect of Earth's Rotation

The rotation of the Earth imparts an initial tangential velocity to objects on its surface. When launching a satellite, this initial velocity can reduce the additional energy required to achieve orbital speed relative to an inertial frame. Accounting for Earth's rotation is therefore important for accurately determining the minimum energy input necessary to place a satellite into orbit.

Transcript

00:01 Our equation for orbital period is t equals, well i'm going to write t squared, equals 4 pi squared r cubed over gm.

00:22 If we rearrange this, we get t equals 2 pi square root of r cubed over gm.

00:44 So that would be the answer where.

00:53 We know what g is.

00:56 M is mass of the earth.

01:07 R is the radius of the earth plus h.

01:22 So i could rearrange this as, i mean i could substitute r plus h here, and that's a more complete answer.

01:45 B, what's the speed? well, the speed is 2 pi r over the period, although in this case r is re plus h.

02:35 Those cancel out.

02:48 This, i'm going to invert the denominator, and then the cube root of re plus h would be r .e plus h times, i mean, the square root of r .e.

03:12 Plus h cubed is going to be r .e.

03:14 Plus h times the square root of r .e.

03:19 Plus h.

03:21 And we see that those cancel out.

03:28 And so the speed is the square root of g .m.

03:38 Over r .e.

03:40 Plus h.

03:45 C.

04:03 Okay.

04:04 So we need the energy to put this satellite in orbit.

04:08 Well, first of all, ve is 2 pi re over the period of the earth, which is, i'm just going to raise.

04:35 T e for period of earth.

04:40 Now, we're asked to write m sub e as the mass of the earth, and we're asked to write r sub e as the radius of the earth.

04:54 So, change in energy equals kinetic energy final, minus, no, plus potential energy, final, minus kinetic energy initial, minus potential energy initial.

05:10 So change in energy is kinetic energy final.

05:15 One half, little m, and that's the mass of the satellite.

05:26 V squared final.

05:28 That's going to be the square root of gm over r .e.

05:32 Plus h.

05:39 One half m v squared, so we have to square that.

05:44 Minus g m -e -m over r.

05:52 This is r -final, and r -final will be r -e plus h.

06:02 Oops, r -e, wants us to use capital r.

06:11 And i'm going to do that over here too...