1) 29.4 N
The force of gravity between two objects is given by:
[tex]F=G\frac{Mm}{r^2}[/tex]
where
G is the gravitational constant
M and m are the masses of the two objects
r is the separation between the centres of mass of the two objects
In this problem, we have
[tex]M=5.97\cdot 10^{24} kg[/tex] (mass of the Earth)
[tex]m=3.0 kg[/tex] (mass of the box)
[tex]r=R=6.37\cdot 10^6 m[/tex] (Earth's radius, which is also the distance between the centres of mass of the two objects, since the box is located at Earth's surface)
Substituting into the equation, we find F:
[tex]F=\frac{(6.67\cdot 10^{-11})(5.97\cdot 10^{24})(3.0)}{(6.37\cdot 10^6)^2}=29.4 N[/tex]
2) [tex]g=9.8 m/s^2[/tex]
Let's now calculate the ratio F/m. We have:
F = 29.4 N
m = 3.0 kg
Subsituting, we find
[tex]\frac{F}{m}=\frac{29.4}{3.0}=9.8 N/kg = 9.8 m/s^2[/tex]
This is called acceleration of gravity, and it is the acceleration at which every object falls near the Earth's surface. It is indicated with the symbol [tex]g[/tex].
We can prove that this is the acceleration of the object: in fact, according to Newton's second law,
[tex]F=ma[/tex]
where a is the acceleration of the object. Re-arranging,
[tex]a=\frac{F}{m}[/tex]
which is exactly equal to the quantity we have calculated above.
