4Newton’s law of gravity Always attractiveIn vector form, the gravitational force exerted by m2 on m1 isThe negative sign means that it is an attractive force.
5Gravity for spherically symmetric bodies For an object which has spherically symmetric mass distribution: concentrate all the mass of the object at its center.Earth of mass mE
6Acceleration due to gravity Ex 12.2Acceleration due to gravity
7Superposition of gravitational forces Ex 12.3Superposition of gravitational forcesGravitational force is a vector.The gravitation force exerted on m= vector sum of two forces
8At the surface of the Earth, we can neglect other stellar objects. WeightWeight of a body: the total gravitational force exerted on the body by ALL other bodies in the universeAt the surface of the Earth, we can neglect other stellar objects.Mass of the EarthRadius of the Earth
9Use this information to know the mass of the Mars lander Ex 12.4Gravity on MarsUse this information to know the mass of the Mars landerAt d = m above the surface of Mars
10Gravitational potential energy When the gravitational acceleration is constantIn general, the gravitational acceleration depends on rGravitational forcedisplacement
11Gravitational potential energy II Gravitational force is conservativeAt the surface of the Earth= constant, so can be dropped
12From the earth to the moon Ex 12.5From the earth to the moonMuzzle speed needed to shoot the shell from RE to 2RETo obtain the speed, we use energy conservation.
13From the earth to the infinity Ex 12.5bFrom the earth to the infinityMuzzle speed needed to shoot the shell from RE to infinityIndependent of the mass of the objectThis is called the escape speed
15Satellites: circular orbits The radius of the circular orbit of the satellite is determined by its speed.Independent of the satellite mass
16Satellites: circular orbits For a given radius, satellite speed is determined, so is its energy
17From the earth to the infinity Ex 12.6From the earth to the infinitySpeed, period, acceleration
18The work needed to place this satellite in orbit Ex 12.6Cont’dThe work needed to place this satellite in orbitThe additional work to make this satellite escape the earth
19Kepler’s lawsKepler’s First Law: each planet moves in an elliptical orbit, with the sun at one focus of the ellipseThis can be shown by solving the equation of motion based on Newton’s theory on gravity and Newton’s second law of motion. (but needs higher level of math)e: eccentricity in most cases, e is very small and the orbit is close to a circleAphelion: distance between P and S is maximum.Perihelion: distance between P and S is minimum.
20A result of angular momentum conservation Kepler’s Second Law: A line from the sun to a given planet sweeps out equal area in equal timesA result of angular momentum conservationThe line SP sweeps out equal areas in equal timesSee the textbook for the proof.Kepler’s Third Law: The period of the planets are proportional to the 3/2 powers of the major axis lengths of their orbitsWe have seen this for the case of circular orbit. But this is true even for elliptic orbits.