Knowing that mass of Earth #m=5.98 imes 10^24# kg and mean radius #R=6.37 imes 10^6# meters, at which height or altitude gravity of Earth will drop by 5 times, or drop to zero ?

Question

Knowing that mass of Earth #m=5.98	imes 10^24# kg and mean radius #R=6.37	imes 10^6# meters, at which height or altitude gravity of Earth will drop by 5 times, or drop to zero ?

William Audy · Accepted Answer

Drop by 5 times: #7870# km
Drop to zero: Never

Two objects' masses act as a gravitational force on each other, and this force is #F=G(m_1m_2)/r^2# where #F=#force in newtons #m_1=#mass of first object in kilograms #m_2=#mass of second object in kilograms #r=#distance between centres of the objects in metres #G=#the "gravitational constant", approximately equal to #6.674xx10^(-11)Nkg^(-2)m^2# Isaac Newton came up with this concept initially and published it in 1686. So the gravitational force "goes as the inverse of #r^2#", which is to say that if you travel twice as far away, it will drop in size by factor #2^2=4#. To drop the force by a factor 5, we must make the distance from the centre of the earth #sqrt(5)~=2.24# times what the distance is to the surface, i.e. a height of #(sqrt(5)-1)R# above the surface. Using the supplied value for #R#, this is to three significant figures a height of #7.87xx10^6# m#=7870# km. Note that this is outside earth's atmosphere (up to 100 km) and beyond "Low Earth Orbit" (up to 2000 km); it is within "Medium Earth Orbit". See https://tutor.hix.ai To drop the force to zero, the distance has to become infinite - the graph of #1/r^2# becomes arbitrarily small as #r# becomes arbitrarily large, but it does not go to zero. Wherever you are in the universe, you will experience some contribution of earth's gravity - but in most places it will be such a tiny effect as to be undetectable, swamped by more local gravitational fields. So the force never becomes zero, although for practical purposes it becomes negligible when you move out far enough from earth that the much larger Sun's gravitational field makes it small in comparison.

Isabella Alvarez · Answer

undefined The gravity of Earth drops by 5 times at approximately 1,397 kilometers above the surface. Gravity becomes zero at an infinite distance from the Earth.

Knowing that mass of Earth #m=5.98\times 10^24# kg and mean radius #R=6.37\times 10^6# meters, at which height or altitude gravity of Earth will drop by 5 times, or drop to zero ?