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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

More subtly....

More subtly....

There are also more subtle similarities (both forces are conservative) and differences too (electric charge is invariant with respect to speed but mass is not). But these can wait until S3.5 and S4.7
 
 

[A] About the force

Key Point 2.4

The gravitational force of interaction between two point masses m1 and m2 separated by distance r:
  • has magnitude
    \[ F_G = \frac{Gm_1m_2}{r^2} \]
    with G the universal constant of gravitation.
Graphic - No title - gravforcezero
IA (nearly) perfect law

[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.

IPlanetary orbits are not circles...quite

Solution

Solution

  • We can regard sun as ‘fixed’ (inertial) if Mm.

    Why?

    Why?

    Because the accelerations the masses induce in oneanother are inversely proportional to their masses. So in a first approximation we can forget about what the earth ‘does’ to the sun in much the same way as we forget about what a falling stone ‘does’ to the earth (S2.5).
     
  • Then choose coordinates fixed w.r.t. the sun.
  • Newton’s 2nd Law (full vector form):
    \[ \xmlInlineElement[\xmlAttr{}{target}{slides}]{http://www.ph.ed.ac.uk/aardvark/NS/aardvark-latex}{aardvark:reveal}{m\frac{d^2 \vec{r}}{dt^2} = - \frac {GMm}{r^2} \hat{r}} \]
Graphic - No title - keplercoords
  • The general solution is an ellipse with the sun at one focus.
    Graphic - No title - ellipsecircle
  • 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):

    \[ \xmlInlineElement[\xmlAttr{}{target}{slides}]{http://www.ph.ed.ac.uk/aardvark/NS/aardvark-latex}{aardvark:reveal}{m \times \frac{v^2}{r} = \frac{GMm}{r^2}} \]

  • Identify

    \[ \xmlInlineElement[\xmlAttr{}{target}{slides}]{http://www.ph.ed.ac.uk/aardvark/NS/aardvark-latex}{aardvark:reveal}{v = \frac{2\pi r}{T}} \]

    Then

    \[ \frac{v^2}{r} = \xmlInlineElement[\xmlAttr{}{target}{slides}]{http://www.ph.ed.ac.uk/aardvark/NS/aardvark-latex}{aardvark:reveal}{\frac{4\pi^2 r}{T^2} = \frac{GM}{r^2} } \]
    or
    \[ T^2 = \xmlInlineElement[\xmlAttr{}{target}{slides}]{http://www.ph.ed.ac.uk/aardvark/NS/aardvark-latex}{aardvark:reveal}{\frac {4 \pi^2}{GM} r^3} \]

 

Results

In words: the square of the period is proportional to the cube of the orbit radius.

This is Kepler’s Law of Periods.

IKepler's law of periods

Explicitly:

$$T^2 = \frac {4 \pi^2}{GM} r^3$$

Commentary

Commentary

  • The result is independent of the planet mass m
  • So all planets in the solar system satisfy this equation. …and comets too!
  • But ‘orbits’ are not always simple …
 

Learning Resources

Textbook: 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.
Course Questions:
Self-Test Questions: