Newton’s Universal Law of Gravitation

The Gravitational Force Newton’s Universal Law of Gravitation states that every particle in the universe exerts an attractive force on every other particle. Where “G” is the “universal gravitational constant” G = 6.67 x 10-11 N∙m2/kg2

What happens to the Force if one of the masses is doubled? 1. It will now be F x ? What happens to the Force if both of the masses were doubled? 2. It will now be F x ? What happens to the Force if one of the masses is doubled and then other one is halved? 3. It will now be F x ?

This is an “inverse square” law, since the Force is proportional to the inverse of the distance squared. Example: At twice the distance, the gravitational force between two objects would be less. How much less?

What if the distance was 4d? 6. At ½ d, the force would by F x ?? Two objects are separated by some distance, d. How would the gravitational force differ if the distance was tripled? 1/9 the original force What if the distance was 4d? 1/16 the original force 4. the distance was 5d? 5. The distance was 10d? ½ d? 6. At ½ d, the force would by F x ??

Enter 1.3 (one decimal place) Example: Two masses of 5 kg and 9 kg are separated by 1.5 m. What is the gravitational force they exert on each other? How do you enter all those numbers in your calculator? Use your exponent button (EE) for “G”!! Do NOT type in “ x 10^ ” 6.67E-11*5*9÷1.52 = F = 1.3 x 10-9 N Enter 1.3 (one decimal place) Enter the exponent, -9 G = 6.67 x 10-11

Enter the number with 1 decimal place What is the gravitational force between a 600 kg mass and a 850 kg mass if they are 0.4 meters apart? Enter the number with 1 decimal place Enter the exponent G = 6.67 x 10-11

Enter the number with 1 decimal place. Example: Two masses of 3 x 103 kg and 1.8 x 1015 kg are separated by d = 1.4 x 1021m. What is the gravitational force they exert on each other? How do you enter all those numbers in your calculator? Use your exponent button!! 6.67E-11*3E3*1.8E15÷1.4E21 2 = Enter the number with 1 decimal place. Enter the exponent G = 6.67 x 10-11

13. Enter the value of “d” using 2 decimal places If the gravitational force between a 95 kg mass and a 120 kg mass is 4 x 10-4N, how far apart are they? What’s the shortcut to get d2 out of the denominator? Trade places with F!! And don’t forget to take the square root! 13. Enter the value of “d” using 2 decimal places G = 6.67 x 10-11

NET Gravitational Force Two masses pull on the central mass. How would you get the NET gravitational force? Subtract the two forces.

NET Gravitational Force Two masses pull on the left mass. How would you get the NET gravitational force? Add the two forces. (Be careful about your distances!)

NET Gravitational Force Two masses pull on the mass at the origin. How would you get the NET gravitational force? Calculate the 2 different forces, then pythagorize the two forces. .

Cavendish and “G”, the gravitational constant Henry Cavendish, a British scientist, first devised an experiment to determine “G” in 1797. He suspended two small known masses from a “torsion wire” of which he knew the strength. These two small masses were gravitationally attracted to two large known masses, which caused the wire to twist until the torsion force was balanced by the gravitational force. Because he knew the strength of the torsion force, he also knew the strength of the gravitational force. With known masses, known Force, and known distance, the only “unknown” left was G! * You need to know who first determined “G”,

The acceleration due to gravity, “g” What is the acceleration due to gravity at places other than the Earth? We don’t have the ability to drop objects on those planets and measure the acceleration as they fall. How does a different value of “g” affect the weight of objects on other planets?

Finding “g” The weight of an object is the gravitational force a planet exerts on the object. Weight (mg) = Gravitational Force (G 𝒎 𝟏 𝒎 𝟐 𝒅 𝟐 ) 𝒎 𝒐𝒃𝒋𝒆𝒄𝒕 𝒈 = 𝑮 𝒎 𝒑𝒍𝒂𝒏𝒆𝒕 𝒎 𝒐𝒃𝒋𝒆𝒄𝒕 𝒅 𝟐 “g”, the acceleration due to gravity can be found by canceling the mobject . The distance, d, is measured from the center of the planet to the location of interest. On the surface, that distance would be the RADIUS of the planet. The acceleration due to gravity, “g”, is also called the “gravitational field strength”.

How large is “g” on the planet Venus, which has a mass of 4 How large is “g” on the planet Venus, which has a mass of 4.87 x 1024 kg and has a radius of 6,050,000 meters? 6.67E-11 x 4.87 E24 ÷ 6,050,0002 = g = 8.87 m/s2 Would you weigh more or less on Venus than you do on Earth?

Example: An asteroid of radius 500 m has a mass of 6. 5 x 1013 kg Example: An asteroid of radius 500 m has a mass of 6.5 x 1013 kg. What is the gravitational field strength at its surface? 6.67E-11 x 6.5 E13 ÷ 5002 = g = 0.0173 m/s2 How much would a 60 kg astronaut WEIGH on this asteroid? W = mg W = 60 kg x 0.0173 m/s2 W = 1.04 N

Proportional Reasoning… another “inverse square” law

Weight = mg If the mass of the Earth were to double with no change in its radius, by what factor would your weight change? If the radius of the Earth were to double with no change in its mass, by what factor would your weight change? (answer as a fraction, not decimal) If the radius of the Earth were half as much with no change in its mass, by what factor would your weight change? If you were on a planet whose mass with 8 times Earth’s mass and whose radius was twice Earth’s radius, by what factor would your weight change?

G = 6.67E-11 The Moon, mass = 7.36 x 1022 kg, radius = 1.74 x 106 m Find the acceleration due to gravity, “g”, for the following locations. Record answer with ONE decimal place! The Moon, mass = 7.36 x 1022 kg, radius = 1.74 x 106 m Mars, mass = 6.4 x 1023 kg, radius = 3.395 x 106 m 7. Jupiter, mass = 1.9 x 1027 kg, radius = 7.15 x 107 m

Aristotle Geocentric universe 384 BC “geocentric” – Earth centered universe…… WRONG!

Ptolemy, 83 AD Ptolemy (also geocentric universe) presented his astronomical models in convenient tables, which could be used to compute the future or past position of the planets, the Sun, and Moon, the rising and setting of the stars, and eclipses of the Sun and Moon. His model showed the planets turning in small circles as they orbited the Earth! The tables actually produced fairly good predictions, but his model and his geocentric universe was….. WRONG! Ptolemy was also the first to use latitude and longitude lines.

Copernicus 1473 heliocentric universe “sun-centered” universe Although others before him had proposed that the planets orbit the sun rather than the Earth, Copernicus was the first to publish mathematical evidence

Tycho Brahe 1546 Built “The Castle of the Stars” Had an accident in a duel Died an unusual death…

A mathematician hired as Brahe’s assistant Johannes Kepler 1571 A mathematician hired as Brahe’s assistant Wrote Three Laws of Planetary Motion

Galileo 1564 Made the first telescope (not really, but he gets credited with it) Observed the rings of Saturn and some of the moons of Jupiter

Sir Isaac Newton 1642 Said that gravity was not just an Earthly force, but a Universal force. Wrote the Universal Law of Gravitation

Albert Einstein 1879 The effects of gravity are just the results of the distortion of the “fabric of space-time” or the “space-time continuum” Gravity is not really a “force” at all!