How did Newton fumble his way towards the Universal Law of Gravitation, and then satisfy himself that it was correct. Aim was to provide a theory which explained Kepler’s empirical formula relating a planet’s radius of orbit to its orbital time. ‘Standing on the shoulders of giants’
Newton started with formula for centripetal force And the fact that, for a circular orbit, So the centripetal force needed to make a planet orbit becomes: Then ‘guessed’ that the gravitational force on a planet would be related to its mass and the inverse square of its orbital radius.
The formulahad come from his guess that So now he needed to prove his formula correct, and in order to do this he needed to find the value of ‘k’. He used the Moons known orbital Time (27.3 days) and distance from the Earth ( km) to find k. He then used this value of k to find the force exerted on 1 kg of mass at the Earth’s surface. (radius 6353 km, mass of Earth ) Finding the correct value for ‘g’ on Earth meant that his ‘guessed’ inverse square law was correct. It was obvious that part of the ‘k’ would be the mass of the Sun. He calculated this using simultaneous equations from the orbits of the known planets to arrive at: He could then calculate the value of ‘G’ (6.673 Nm 2 kg -2 ) using both the Moon and the ‘g’ methods. With this knowledge, and working back using F=mv 2 /r and v=2 r/T he was able to predict the motion of everything in the Solar System extremely accurately. (Brian Cox Clip)
‘G’ was later measured directly by Cavendish using the apparatus shown below: Link to YouTube clip on Cavendish
(Page 61)
Slow way… use given data to find Mass of Sun then use this to find the force Quick Way. R has decreased by factor 3. Therefore force increases by factor 3 2 R has increased by factor 5,so F has decreased by factor (1/5) 2 NB this method for multiple choice questions
At half way point, d from both = 1.9 × 10 8 m
How did Newton fumble his way towards the Universal Law of Gravitation, and then satisfy himself that it was correct. Aim was to provide a theory which explained Kepler’s empirical formula relating a planet’s radius of orbit to its orbital time. ‘Standing on the shoulders of giants’
Newton started with formula for centripetal force And the fact that, for a circular orbit, So the centripetal force needed to make a planet orbit becomes: Then ‘guessed’ that the gravitational force on a planet would be related to its mass and the inverse square of its orbital radius.
The formulahad come from his guess that So now he needed to prove his formula correct, and in order to do this he needed to find the value of ‘k’. He used the Moons known orbital Time (27.3 days) and distance from the Earth ( km) to find k. He then used this value of k to find the force exerted on 1 kg of mass at the Earth’s surface. (radius 6353 km, mass of Earth ) Finding the correct value for ‘g’ on Earth meant that his ‘guessed’ inverse square law was correct. It was obvious that part of the ‘k’ would be the mass of the Sun. He calculated this using simultaneous equations from the orbits of the known planets to arrive at: He could then calculate the value of ‘G’ (6.673 Nm 2 kg -2 ) using both the Moon and the ‘g’ methods. With this knowledge, and working back using F=mv 2 /r and v=2 r/T he was able to predict the motion of everything in the Solar System extremely accurately. (Brian Cox Clip)
‘G’ was later measured directly by Cavendish using the apparatus shown below:
(Page 61)