Newton’s Law of Universal Gravitation

By: Heather Britton

Newton’s Law of Universal Gravitation

Johannes Kepler (1571 - 1630) - a German scientist that researched and calculated the motions of the planets using the sun as the center of the system

He came up with three laws that describe his observations

Newton’s Law of Universal Gravitation

1. The paths of the planets are elipses with the center of the sun at one focus

2. Imaginary lines from the sun to a planets orbit sweep out. The lines have equal areas in equal time intervals. Thus planets move fastest when closest to the sun and slowest when farthest away

Newton’s Law of Universal Gravitation

3. The ratio of the squares of the periods of any two planets revolving around the sun is equal to the ratios of the cubes of their average distance from the sun

(Ta/Tb)2 = (ra/rb)3

The above equation may be used for any body revolving around another body in space

Newton’s Law of Universal Gravitation

T = the period of revolution

period - how much time an objects takes to complete one revolution

r = the distance between the two centers of the objects

Newton’s Law of Universal Gravitation

Example 1

Galileo discovered 4 moons of Jupiter. Io, which he measured to be 4.2 units from the center of Jupiter, has a period of 1.8 days. He measured the radius of Ganymede’s orbit as 10.7 units. Find the period of Ganymede.

Newton’s Law of Universal Gravitation

Gravity is a force that can act over distance

Aside from Newton’s three laws of motion, he also came up with the law of universal gravitation

The force of gravity is directly proportional to the product of the two masses and inversely proportional to the distance between the centers

Newton’s Law of Universal Gravitation

Fg = G(m1m2) / r2

Fg = gravitational force measured in Newtons

m = the masses measured in kilograms

r = the distance between centers measured in meters

Newton’s Law of Universal Gravitation

G = the constant of proportionality

The value of G never changes

G = 6.67 x 10-11 Nm2/kg2

Newton’s Law of Universal Gravitation

This in an inverse square law meaning that as the distance doubles the force is 1/4 as strong

If the distance triples the force is 1/9 as strong

Newton did not discover the value of G during his lifetime

He reasoned that it must be very small since we are not aware of the attraction to the objects around us

Newton’s Law of Universal Gravitation

The value of G was not discovered until Henry Cavendish (1731 - 1810) conducted his “weighing” the Earth experiment

He set up a balance, and then rolled 6 tons of lead under one side

After rebalancing the lever he knew the size of the force the 6 tons of lead exerted

Newton’s Law of Universal Gravitation

This gave him every variable but G

He then could calculate the value of G

With G now known the mass of the Earth could be determined

Earth’s mass is 5.98 x 1024 kg

Earth’s average radius is 6.37 x 106 m

Newton’s Law of Universal Gravitation

Example 2

The mass of the Hubble Space Telescope is 11,600 kg. Determine the weight of the telescope when it is on Earth and orbiting 598 km above the surface