Lesson 5 of 11 · 6 min
Acceleration due to gravity
NCERT §7.5
Kavya weighs 50 kg. On the bathroom scale the earth pulls her with 490 N. If she stood on the Moon, the same 50 kg would press down with only about 81 N. Why does a smaller world pull less?
The lesson in notes
In short
Think of the earth as many concentric shells. A point outside is outside every shell, so the whole earth pulls as if its mass M_E were at the centre.
On the surface the force on a mass m is F = G M_E m / R_E², and since F = mg, g = G M_E / R_E².
g does not depend on the mass of the falling body; it depends only on the mass and radius of the earth.
Knowing g, R_E and G gives the earth's mass: M_E = g R_E² / G. With g = 9.81 m/s² and R_E = 6.37 × 10⁶ m this is 5.97 × 10²⁴ kg, hence the phrase 'Cavendish weighed the earth'.
The Moon's orbit gives the same mass another way, M_E = 4π²R³/(G T²) ≈ 6.02 × 10²⁴ kg, within 1% of the first value.
Inside a uniform earth, at distance r from the centre, only the mass within radius r pulls, and the force is F = G M_E m r / R_E³, proportional to r.