Newton’s Universal Law of Gravity

 

Using his laws of motion, Newton set out to show that the gravitational attraction between an Earth-bound object (e.g. an apple) and the earth was fundamentally the same as that between Earth and Moon. He had at his disposal the work of Kepler  in particular Kepler’s third law which related a planet’s orbital period to its distance from the sun (  ) and, of course, his own equations of motion. Using his second law (  ), and using for a the centripetal acceleration , he proceeded in the following way: The velocity of an orbiting planet is the circumference of its orbit divided by the orbital period (  ), so

(1.1)                                                    

 

 (using Kepler’s third law) and so the force .

 

It was here that Newton surmised that the constant must also include the mass of the central object and so wrote his famous equation

(1.2)                                                         .

 

He then asked if this force exists between all the planets and the sun, why could not the moon and all other objects, even those close by (such as an apple) be attracted to the earth in a similar way? If this were true, the acceleration of an object on Earth should be related to the acceleration of the moon in its orbit by the inverse of the relative distances squared. The sidereal period of the moon (T) is a little over 27days (2.36x10⁶ seconds) and its mean distance (R) from Earth is 3.84x10⁸ meters, so its acceleration is (using 1.1) is 0.00272 m/s².

 

We know that an object on Earth accelerates at 9.81 m/s². Earth’s radius is 6.37x10⁶ meters, so we can calculate the ratio of the squares of the radii of Earth and Moon to compare the accelerations. We find

 

 , a comparison that Newton found to “answer pretty nearly “.

 

Newton was convinced his law of gravity was truly universal, holding for any two masses m and m’. Note that his comparison was from a ratio of the accelerations (and so a ratio of the forces), not from a calculation of the forces themselves. He needed G for that, and had no way of finding it. It took the work of Cavendish, some 100 years later, to show that the universal gravitational constant of Newton had the value (in SI units) G = 6.67x10^-11 N-m²/kg².

 

Knowing G opens many doors: It is a universal constant, dependent on only the units used. (Unlike Kepler’s k, whose value changes as the central object changes.) We can, for example, “weigh the sun”: Earth moves in its (almost) circular orbit because the required centripetal force is provided by gravity  

(1.3)                                                      .

 

Earth orbits at a radius of 1.5x10¹¹ meters (1 AU), and its velocity, calculated as above, is 29,865 m/s. Solving for m_s provides a value 2x10³⁰ kg. To weigh the earth requires only the earth-moon distance and the period of Moon’s orbit, both provided above. The result, as you should verify, is 6x10²⁴ kg. To weigh the moon using this method would require an orbiting satellite  only the central object’s mass can be calculated.