Universal Gravitation

Announcements We will have a short quiz on Monday 24th on: Momentum Collisions Rigid body rotations Moment of inertia Torque

Goal of the class To understand gravitational potential energy and the orbit of satellites. To understand the force of gravity Question of the day: How does the acceleration due to gravity vary as a function of height?

Gravity Gravity is an attractive force that pulls objects towards each other. Gravity is a force that affects all objects universally. Anything with mass feels the force of gravity.

Factors Affecting Gravity There are 2 factors that affect gravity Mass is a measure of how much matter an object has (how much stuff). - The force of gravity gets stronger with more mass Distance – the further apart 2 objects are, the lower the force of gravity. - For a spaceship travelling to Mars, the attraction to the Earth decreases and the attraction to Mars increases as it gets closer.

Factors Affecting Gravity

Gravity Newton discovered the laws of gravitation. G = universal gravitation constant = 6.67 x 10-11 Nm2/kg2 Newton did not discover gravity, he wrote the laws and equations for it. Sir. Isaac Newton

Variation of g with Altitude RE = 6.37x106 m ME = 5.97 x 1024 kg F= G Mm/(RE+h)2 = mg g(h) = G M/(RE+h)2 But as RE >> h On surface g= G M/RE2 = 9.81m/s2

Satellites F= G Mm/(RE+h)2 = mv2/r v=(GME/r)1/2 v=2πr/T Calculate the height of a geosynchronous (geostationary) orbit. h=3.6x107 m

Gravitational Potential Energy We say that 0 PE is on the surface Draw a small mass as say PE=mgh But this is only valid near Earth’s surface if g is constant We choose 0 PE when objects are at an infinite distance away as it’s more convenient. From infinity to a position r U= -G Mm/r = Int (F.dr) It’s negative the force is outward so the direction is different. Shoot an object v = 2x104m/s from the Earth. Total Energy before = 1/2mv02 -G Mm/RE Total Energy after = 1/2mv2 -G Mm/r If r is very large we can take the r is v. big (infinite) so U = 0 So at inf it will have 16.6km/s

Escape Velcoity What minimum speed should we shoot it so it doesn’t come back If you throw something it will come back, unless v is very large to escape earth’s gravity. The minimum velocity needed for this is called the escape velocity If we shoot a mass m up with velocity v it will reach some max height h. Rmax = RE + h Energy = 1/2mv02 -G MEm/RE At max height all the energy is potential = -G MEm/rmax We should make rmax = to infinity if we don’t want the object to come back. 1/2mv02 = G MEm/RE =>v0 = (G ME/RE)1/2 = 1.12x104m/s

Problem A 1000 kg satellite orbits the earth at an altitude of 200 km. Find the satellite’s speed and time period How much energy must be added to the system to move the satellite into a circular orbit with altitude 400 km.