S2.11 The gravitational force
Preamble
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Preamble
In the next two sections, we deal with two of the fundamental forces, which share similar characteristics, but are important on very different lengthscales. We will see that:- Both forces are inverse square: they depend on 1/r2
- Both forces are central: they act along the line of centres
- Gravity dominates solar-system-scale physics
- Coulomb dominates atomic-scale physics
[A] About the force
Key Point 2.4
The gravitational force of interaction between two point masses m1 and m2 separated by distance r:
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- is attractive
[B] Example problem
A planet of mass m moves in a circular orbit of radius r about the sun, of mass M. Establish the relationship between the period of the orbit and its radius. Treat the sun and planet as point masses.

Solution
- Sun and planet can be treated as point masses
- as long as separation is large, or
- as long as bodies are spherical
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- The general solution is an ellipse with the sun at one focus.
- But orbit can be circular centred on sun and at constant speed since, then,
force FG
- has constant magnitude
- acts directly towards sun
Newton’s 2nd law (centripetal component):
Identify
Then
or
Results
In words: the square of the period is proportional to the cube of the orbit radius.
This is Kepler’s Law of Periods.
Explicitly:

Learning Resources
![]() | HRW Chapters 13.2 and 13.7 deal with Kepler’s Laws, including the Law of Periods. They rest on concepts, notably angular momentum conservation, to be discussed in S5. |
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