Physics HL · Chapter 17: Gravitation
17.1 Newton's Law of Gravitation and Gravitational Field Strength
Build inverse-square force intuition, convert to field language, and map radial versus approximately uniform gravitational fields.
Estimated time: 36 minutes
Inverse-Square Force Law and Scaling Consequences
Newton's law of gravitation states that any two point masses attract each other with force magnitude proportional to the product of masses and inversely proportional to the square of their separation. The force acts along the line joining the two masses and always points inward toward the other mass. This inward direction is what keeps orbital trajectories continuously curving.
F = G rac{M m}{r^2}
Doubling separation reduces force by factor 4; tripling separation reduces force by factor 9.
The inverse-square structure is the part students most often misread under time pressure. Many errors come from treating the dependence as 1/r instead of 1/r^2. A useful sanity check is geometric: spreading interaction over larger spherical surfaces gives area growth proportional to r^2, so field and force per unit probe mass naturally weaken with the same scaling.
Field Strength as Force Per Unit Mass
Gravitational field strength g at a location is defined as the force experienced per unit test mass placed there. This framing shifts attention from a particular object to the structure of space around the source mass. For spherical sources, the field is radial and points toward the center.
g = rac{F}{m} = G rac{M}{r^2}
Units N/kg are equivalent to m/s^2, so field strength is numerically equal to free-fall acceleration.
This equality between field strength and acceleration explains why satellite and projectile questions can be linked directly: once you know g at the orbital radius, you know the required centripetal acceleration scale there. Near a planet's surface g can be approximated as nearly constant over small height intervals, but globally it must decrease with r^2.
Radial Fields, Uniform Approximations, and Superposition
A single spherical mass produces a radial field: vectors point inward and their magnitude decreases with distance. Over a thin layer near a large planet's surface, field lines are approximately parallel, so a uniform-field model can be useful. In two-mass setups, total field is the vector sum of each contribution, while potential is the scalar sum. Keeping that vector-versus-scalar distinction clean prevents many sign errors.
Important
Field adds as vectors, potential adds as scalars. If you mix these rules, two-body gravitation questions become inconsistent immediately.
Simulation: Gravitational Field Mapper
Place one or two masses, move a probe point, and inspect field vectors, potential, and local escape speed from the same geometry.
Gravitation + Orbit Lab
|g| at probe
3.37e-3 N/kg
Field direction
-168.0 degrees
Potential at probe
-1.33e+6 J/kg
Escape speed at probe
1.634 km/s
Vector field map (arrows point in the direction of acceleration of a 1 kg test mass)
Test Yourself
A planet has mass 3.0 x 10^24 kg and radius 4.0 x 10^6 m. Enter g at its surface in N/kg.
Hint: Use g = GM/r^2 with G = 6.67 x 10^-11 N m^2 kg^-2.