How many kilometers from the Earth's surface will a mass of 65.0 kg be just 611.8 N in weight?
The weight of an object is determined by the force of gravity acting on the object. Near the Earth's surface, the force of gravity acting on the object of mass m is F = mg, where g is the gravitational acceleration, which near the Earth's surface is 9.8 m/s^2. This value does not vary significantly when the object is not far from the Earth. However, when the object is further away, the gravitational force vary with the distance as
`F = (GMm)/R^2` . Here, G is the gravitational constant
`G = 6.67*10^(-11) m^3/(kg*s^2)` , M is the mass of the Earth `M = 5.97*10^24 kg` and R is the distance between the center of the Earth and the object in question.
To calculate the distance R such that F = 611.8 N for the object of mass 65 kg,
plug in these values in the above formula:
`611.8 = (6.67*10^(-11)*5.97*10^24*65)/R^2`
From here `R^2 = (2.59*10^16)/611.8 =4.23*10^13 `
and `R = 6.5*10^6 m = 6500*10^3 km` .
Since this is the distance between the center of the Earth and the object, we need to subtract the radius of the Earth in order to find the distance from the object to the Earth surface. The average radius of the Earth is r = 6,371 km (average, because the Earth is not a perfect sphere.)
The distance to the Earth's surface is R - r = 6500 km - 6371 km = 129 km.
So, the mass of 65 kg will weigh only 611.8 N if it is 129 km away from the Earth's surface.