Question

atellite is one that So the first part to this question is as follows:

A geosynchronus satellite is one that stays above the same point on the Earth’s equator such satellites are useful for tasks such as weather forecasting and transmission relay. Determine the height above the Earth’s surface such as a satellite must orbit. The Earth’s mean radius is 6.38 ✕ 106m, mass is 5.98 ✕ 1024kg. (The first thing you need to know is the orbital period of the geosynchronus satellite)

(A) 3.59 X 10^4km

(B) 5.08 X 10^3km

(C) 4.23 X 10^4km

(D) 5.55 X 10^4km

And the second part to this Question is then:

How far from the Earth’s center, on a straight line between the Earth and the moon, will an astronaut feel the same forces of attraction from the Earth and the moon? with...

Earth’s mass: 5.98 ✕ 1024kg

Moon’s mass: 7.35 ✕ 1022kg

Mean Earth-moon distance: 3.84 ✕ 105km

(A) 4.32 X 10^8m

(B) 3.83 X 10^7m

(C) 3.79 X 10^8m

(D) 3.46 X 10^8m

If someone could please provide clarity to this entire question I would greatly appreciate it as I've been stumped on the full answer for both parts for a while now.

Answer 1

Question 1

The height of a geo synchronus satellite can be found using the expression

where, distance of the satellite from the center of Earth

Mass of Earth kg

Time period of  geosynchronous satellite

N m² kg^-2

The time period of  geosynchronous satellite

  

  

Hence, the height of the satellite = satellite distance from the Earth's centerEarth's Radius

Therefore, the height of the geosynchronous satellite above Earth's surface is A.) .

Question 2

Let the distance between the Earth and Moon be and be the distance from the Earth's centre on a straight line between the Earth and an astronaut where he/she feels the same forces of attraction from the Earth and the moon.

If the astronaut feel the same forces of attraction from the Earth,then

is the gravitational force acting on the astronaut due to the Earth

is the gravitational force acting on the astronaut due to the Moon

Now, by Newton's law of gravitation, the gravitational force between due to bodies of mass and which are at a distance apart is given by

  

  

Taking square root on both sides, we get

Given: Earth’s mass: 5.98 ✕ 10²⁴ kg

Moon’s mass: 7.35 ✕ 10²² kg

Mean Earth-moon distance: 3.84 ✕ 10⁵ km =3.84 ✕ 10⁸ m

Therefore, an astronaut at a distance of (D) 3.46 10⁸ m from the Earth’s center, on a straight line between the Earth and the moon, feels the same forces of attraction from the Earth and the moon.