Chapter 13 Gravitation

13.2 & 13.3 Newton and the Law of Universal Gravitation Newton was an English Scientist He wanted to explain why Kepler’s Laws work Newton is famous for his three laws of motion which are the basis of Mechanics which is a branch of Physics

Universal Gravitation Newton’s Law of Universal Gravitation states that any two objects with mass are attracted to each other The Force of Attraction is directly proportional to the product of the masses and inversely proportional to the square of the distance between them.

Universal Gravitation F = Gm 1 m 2 /r 2 G – the universal Gravitation constant. Experimentally determined to be G = 6.67 x N m 2 /kg 2 M 1 & M 2 – the masses of the objects R 2 – the distance between the objects This law explains why an object speeds up when it is closer to the sun and slows down when it is further away

13.4 Gravity Near the Earth’s Surface Since and F = ma Attraction between earth and another object is expressed as:  Since m p cancels on both sides you get: So this tells us that gravity caused by any object is the ratio of the mass to the distance between the centers of the objects multiplied by the universal gravitation constant

13.6 Gravitational Potential Energy In Chapter 8 we described Gravitational Potential Energy near the Earth’s surface. To generalize that equation in light of Universal Gravitation, it becomes: By Definition:

13.6 Gravitational Potential Energy Since Work done is equal to potential energy stored:

Escape Speed Usually when a projectile is fired upward it will slow, stop momentarily, and then return to earth. However there is a certain minimum initial speed which will cause it to move upward forever. This is the escape speed. At this point, where it has escaped gravity, its total energy is zero, so by conservation, its total energy at the surface must also have been zero. Hence: So:

Section 13.7 Planets and Satellites: Kepler’s Laws German Mathematician – he was not a scientist. He had no theories about the universe he simply wanted a mathematical model to fit the data collected by Brahe. Hired by Brahe in 1600 he worked for 29 years on the three laws of planetary motion

1 st Law of Planetary Motion 1 st Law Planetary Orbits are elliptical, not circular  Two focal points – sun is at one foci, every point on the ellipse is the sum of the distances from the foci  Major Axis - length of the long part of the ellipse  Semi-Major Axis - Half the length of the major axis

Eccentricity Eccentricity – how far off you are from a perfect circle E = distance between foci/ length of the major axis Mathematically E = d/2a

2 nd Law of Planetary Motion 2 nd Law – a planet sweeps equal areas in equal times  This requires planets to speed up and slow down  Perihelion – When the planet is closest to the sun and moving fastest  Aphelion – When the planet is furthest from the sun and moving slowest

3 rd Law of Planetary Motion The period of a celestial bodies orbit squared is equivalent to the length of its semi-major axis cubed Mathematically: p 2 = a 3

Sample problem If Earth is 1 AU from the sun and takes 365 days to go around the sun, how long does it take Mars to orbit the sun if it is 1.5 AUs from the sun?

Solution Using the formula: p 2 = a 3 Rearrange it for each object

Orbital Periods Since an orbit has a distance x = 2πr and V = x/T where T = the time for one orbit:

Section 13.8 Satellites: Orbits and Energy Since the weight of the satellite is given by the gravitational force: And since the weight of the satellite provides the centripetal force we get: Since m sat cancels you can rearrange the equation to get: Which is the minimum velocity required to keep a satellite in an orbit R (measured from the center of the earth to the satellite

Energy From our previous discussions: and It follows that: hence So for a satellite in orbit, with a semi-major axis, a, this becomes: