Question

Learning Goal:To teach you how to find the parameters characterizing an object in a circular orbit around a much heavier body like the earth.The motivation for Isaac Newton to discover his laws of motion was to explain the properties of planetary orbits that were observed by Tycho Brahe and analyzed by Johannes Kepler. A good starting point for understanding this (as well as the speed of the space shuttle and the height of geostationary satellites) is the simplest orbit--a circular one. This problem concerns the properties of circular orbits for a satellite orbiting a planet of mass M.For all parts of this problem, where appropriate, use G for the universal gravitational constant.Part A)Find the orbital speed v for a satellite in a circular orbit of radius R. (Express the orbital speed in terms of G, M, and R),Part B)Find the kinetric energy K of a satellite with mass m in a circular orbit with radius R. (Express your answer in terms of m, M, G, and R).Part D)Find the orbital period T. (Express your answer in terms of G, M, R, and pi).Part F)Find L, the magnitude of the angular momentum of the satellite with respect to the center of the planet. (Express your answer in terms of m, M, G, and R.Part G)The quantities v, K, U, and L all represent physical quantities characterizeing the orbit that depend on radius R. Indicate the exponent (power) of the radial dependence of the absolute value of each. (Express your answer as a comma-separated list of exponents corresponding to v, K, U, and L, in that order.

Answer 1

A) v = √ G M / r , B) K = ½ G m M / r , D) T = 2π / √ G M r , F) L = m r² √ (G M r)

Step-by-step explanation:

Part A

For this part we use the law of universal gravitation

F = G m M / r²

And Newton's second law

F = m a

Where the acceleration is centripetal

a = v² / r

We replace

G m M / r² = m v² / r

v = √ G M / r

Part B

Kinetic energy

K = ½ m v²

K = ½ m G M / r

K = ½ G m M / r

Part D

The period can be searched with angular velocity

w = v r

In addition, angular velocity is related to frequency and period.

w = 2π f = 2π / T

2π f = v R

f = v r / 2π

f = 1 /2π r √ GM / r = 1 / 2π √ G M r

.f = 1 / T

T = 2π / √ G M r

Part F

The angular momentum is

L = I w

We can consider the satellite as a particle, so its moment of inertia is

I = m r²

L = m r² √ (G M r)

Part G

Find the components of the quantities

Assume a radius of the orbit, height of the spatial space h = 400 103 m

r = Re + h

r = 6.37 10⁶ + 4 10⁵

r = 10⁶ m

v = √ G M / r

v = √ 6.67 10⁻¹¹ 5.98 10²⁴ / 6.37 10⁶

v = 10⁷7 m / s

Kinetic energy

The mass of the space station is of the order of

m = 10⁷ kg

K = ½ 10⁷ (10⁷)²2

K = 10¹⁰ J

Angular momentum

L = m r² √ (G M r)

L = 10⁷ (10⁶)² √ 10⁻¹¹ 10²⁴ 10⁶

L = 10¹⁹ 10¹⁴

L = 10³³