Newtons Law of Universal Gravitation SP1. Students will analyze the relationships between force, mass, gravity, and the motion of objects. e. Measure and calculate the magnitude of gravitational forces. SP6. The student will describe the corrections to Newtonian physics given by quantum mechanics and relativity when matter is very small, moving fast compared to the speed of light, or very large. d. Describe the gravitational field surrounding a large mass and its effect on a ray of light. Gravitation Every object with mass attracts

every other object with mass. Newton realized that the force of attraction between two massive objects: Increases as the mass of the objects increases. Decreases as the distance between the objects increases. Law of Universal Gravitation M1M2

FG = G G = Gravitational Constant G r2 = 6.67x10-11 N*m2/kg2 M1 and M2 = the mass of two bodies

r = the distance between them Law of Universal Gravitation The LoUG is an inverse-square law: If the distance doubles, the force drops to 1/4. If the distance triples, the force drops to 1/9. Distance x 10 = F / 100.

G Law of Universal Gravitation Law of Universal Gravitation Jimmy is attracted to Betty. Jimmys mass is 90.0 kg and Bettys mass is 57.0 kg. If Jim is standing 10.0 meters away from Betty, what is the gravitational force between them? F = GM M / r2 G 1 2

FG = (6.67x10-11 Nm2/kg2)(90.0 kg)(57.0 kg) / (10.0 m)2 FG = (3.42x10-7 Nm2) / (100. m2) FG = 3.42x10-9 N = 3.42 nN In standard terms, thats 7.6 ten-billionths of a pound of force. Law of Universal Gravitation The Moon is attracted to the Earth.

The mass of the Earth is 6.0x1024 kg and the mass of the Moon is 7.4x1022 kg. If the Earth and Moon are 345,000 km apart, what is the gravitational force between them? Gravitational Field Gravitational field an area of influence surrounding a massive body. Field strength = acceleration due to gravity (g).

g = GM / r2 Notice that field strength does not depend on the mass of a second object. GM M /r2 = M g = F = F 1 2 2 G w So gravity causes mass to have weight.

Gravitational Field Strength The mass of the Earth is 6.0x1024 kg and its radius is 6378 km. What is the gravitational field strength at Earths surface? g = GM/r2 g = (6.67x10-11 Nm2/kg2)(6.0x1024 kg) / (6.378x106 m)2 g = 9.8 m/s2 A planet has a radius of 3500 km and a surface gravity of 3.8 m/s2. What is the mass of the planet? (3.8 m/s2) = (6.67x10-11 Nm2/kg2)(M) / (3.5x106 m)2 (3.8 m/s2) = (6.67x10-11 Nm2/kg2)(M) / (1.2x1013

m2 ) (4.6x1013 m3/s2) = (6.67x10-11 Nm2/kg2)(M) Variations in Gravitational Field Strength Things Newton Didnt Know Newton didnt know what caused gravity, although he knew that all objects with mass have gravity and respond to gravity. To Newton, gravity was simply a property of objects with mass. Newton also couldnt explain how gravity was able to span between objects that werent touching.

He didnt like the idea of action-at-adistance. Discovery of Neptune Newtons Law of Universal Gravitation did a very good job of predicting the orbits of planets.

In fact, the LoUG was used to predict the existence of Neptune. The planet Uranus was not moving as expected. The gravity of the known planets wasnt sufficient to explain the disturbance. Urbain LeVerrier (and others) predicted the existence of an eighth planet and worked out the details of its orbit. Neptune was discovered on September 23, 1846 by Johann Gottfried Galle, only 1 away from where LeVerrier predicted it would be. Precession of Perihelion

Newtons Law has some flaws, however: It does not predict the precession of Mercurys perihelion, nor does it explain it. All planets orbit the Sun in slightly elliptical orbits. Mercury has the most elliptical orbit of any planet in the solar system (if you dont count Pluto). The closest point in Mercurys orbit to the

Sun is called the perihelion. Over long periods of time, Mercurys Precession of Mercurys Perihelion Other Things Newton Didnt Know Newton didnt know that gravity bends light. This was verified by a solar eclipse in 1919.

He also didnt know that gravity slows down time. Clocks near the surface of Earth run slightly slower than clocks higher up. This effect must be accounted for by GPS satellites, which rely on accurate time measurements to calculate your position. Einstein and Relativity Einsteins theory of relativity explains many of the things that Newtonian

mechanics cannot explain. According to Einstein, massive bodies cause a curvature in space-time. Objects moving through this curvature move in locally straight paths through curved space-time. To any observer inside this curved space-time, the objects motion would appear to be curved by gravity.

Curvature of Space-Time Curvature of Space-Time Gravity: Not So Simple Anymore According to Einstein, gravity isnt technically a force. Its an effect caused by the curvature of spacetime by massive bodies.

Why treat it as a force if it isnt one? Because in normal situations, Newtons LoUG provides an excellent approximation of the behavior of massive bodies. And besides, using the LoUG is a lot simpler than using the theory of relativity, and provides results that are almost as good in most cases. So there.