2.3 - Universal gravitation
Uniform gravitational fields (field due to a surface)
● ***Gravitational field: a region where a mass experiences a gravitational force, ie
everything with a mass has a gravitational field
Fg
○ *** g( gravitational field strength)= (direction of g = direction of Fg)
m
○ When drawing a diagram of Earth’s gravitational field:
Lines of gravitational force are parallel [for Fig 2(c)], equally spaced (showing
uniform field [assumption])
Arrows (of force) are downwards, ⟂ (tangential) surface of Earth
Newton’s law of gravitation
G m1 m2
● *** F g= 2 where G = 6.67 x 10-11 N m2 kg-2 (base units m3 kg-1 s-2)
r
● Because of Newton’s 3rd law of motion, the gravitational force between any 2 objects
is of equal magnitude BUT opposite direction
Orbital motion
● For satellites, the only force acting on them is the gravitational attraction of the Earth
towards the Earth’s centre (ie they are in a state of free fall)
● Applying Newton’s 2nd law, his law of gravitation & equation for centripetal force:
G mE m m v2
F=ma= = (where mE = mass of Earth), v=√ ❑
r2 r
2 πr Gm E 3 G mE T 2
And ∵ v= for orbits, ∴ =¿, r =
T r 4π2
Gm E 3 2
∵ 2 is constant, r ∝T (Kepler’s 3rd law), ∴ the further out the satellite is,
4π
2 3
4π r
the longer it takes to orbit the Earth; m E = 2
GT
○ Check: replace mE with mS (mass of Sun), let T be Earth’s orbiting period
2
mS =4 π ׿ ¿
True value of mS: 1.989 ×1 030 kg (% difference: ≈0.847%)
1
Uniform gravitational fields (field due to a surface)
● ***Gravitational field: a region where a mass experiences a gravitational force, ie
everything with a mass has a gravitational field
Fg
○ *** g( gravitational field strength)= (direction of g = direction of Fg)
m
○ When drawing a diagram of Earth’s gravitational field:
Lines of gravitational force are parallel [for Fig 2(c)], equally spaced (showing
uniform field [assumption])
Arrows (of force) are downwards, ⟂ (tangential) surface of Earth
Newton’s law of gravitation
G m1 m2
● *** F g= 2 where G = 6.67 x 10-11 N m2 kg-2 (base units m3 kg-1 s-2)
r
● Because of Newton’s 3rd law of motion, the gravitational force between any 2 objects
is of equal magnitude BUT opposite direction
Orbital motion
● For satellites, the only force acting on them is the gravitational attraction of the Earth
towards the Earth’s centre (ie they are in a state of free fall)
● Applying Newton’s 2nd law, his law of gravitation & equation for centripetal force:
G mE m m v2
F=ma= = (where mE = mass of Earth), v=√ ❑
r2 r
2 πr Gm E 3 G mE T 2
And ∵ v= for orbits, ∴ =¿, r =
T r 4π2
Gm E 3 2
∵ 2 is constant, r ∝T (Kepler’s 3rd law), ∴ the further out the satellite is,
4π
2 3
4π r
the longer it takes to orbit the Earth; m E = 2
GT
○ Check: replace mE with mS (mass of Sun), let T be Earth’s orbiting period
2
mS =4 π ׿ ¿
True value of mS: 1.989 ×1 030 kg (% difference: ≈0.847%)
1
