2 Newton’s law of gravitation Any two (or more) massive bodies attract each otherGravitational force (Newton's law of gravitation)Gravitational constant G = 6.67*10 –11 N*m2/kg2 = 6.67*10 –11 m3/(kg*s2) – universal constant
3 Chapter 8Problem 15Two identical lead spheres with their centers 14 cm apart attract each other with a 0.25-µN force. Find their mass.
4 Gravitation and the superposition principle For a group of interacting particles, the net gravitational force on one of the particles isFor a particle interacting with a continuous arrangement of masses (a massive finite object) the sum is replaced with an integral
5 Shell theoremFor a particle interacting with a uniform spherical shell of matterResult of integration: a uniform spherical shell of matter attracts a particle that is outside the shell as if all the shell's mass were concentrated at its center
6 Shell theoremFor a particle inside a uniform spherical shell of matterResult of integration: a uniform spherical shell of matter exerts no net gravitational force on a particle located inside it
7 Gravity force near the surface of Earth Earth can be though of as a nest of shells, one within another and each attracting a particle outside the Earth’s surfaceThus Earth behaves like a particle located at the center of Earth with a mass equal to that of Earthg = 9.8 m/s2This formula is derived for stationary Earth of ideal spherical shape and uniform density
8 Gravity force near the surface of Earth In reality g is not a constant because:Earth is rotating,Earth is approximately an ellipsoidwith a non-uniform density
9 Gravitational fieldA gravitational field exists at every point in spaceWhen a particle is placed at a point where there is gravitational field, the particle experiences a forceThe field exerts a force on the particleThe gravitational field is defined as:The gravitational field is the gravitational force experienced by a test particle placed at that point divided by the mass of the test particle
10 Gravitational fieldThe presence of the test particle is not necessary for the field to existThe source particle creates the fieldThe gravitational field vectors point in the direction of the acceleration a particle would experience if placed in that fieldThe magnitude is that of the freefall acceleration at that location
11 Gravitational potential energy Gravitation is a conservative force (work done by it is path-independent)For conservative forces (Ch. 7):
12 Gravitational potential energy To remove a particle from initial position to infinityAssuming U∞ = 0
14 OrbitsAccounting for the shape of Earth, projectile motion (Ch. 3) has to be modified:
15 OrbitsThe total mechanical energy E = K + U determines the type of orbit an object follows:E < 0: The object is in a bound, elliptical orbit
16 OrbitsThe total mechanical energy E = K + U determines the type of orbit an object follows:Special cases include circular orbits and the straight-line paths of falling objects
17 OrbitsThe total mechanical energy E = K + U determines the type of orbit an object follows:E > 0: The orbit is unbound and hyperbolic
18 OrbitsThe total mechanical energy E = K + U determines the type of orbit an object follows:E = 0: The borderline case gives a parabolic orbit
19 OrbitsElliptical orbits of planets are described by a semimajor axis a and an eccentricity eFor most planets, the eccentricities are very small (Earth's e is )
20 OrbitsThe “parabolic” trajectories of projectiles near Earth’s surface are actually sections of elliptical orbits that intersect Earth
21 OrbitsThe trajectories are parabolic only in the approximation that we can neglect Earth’s curvature and the variation in gravity with distance from Earth’s center
22 Orbits For a circular orbit and the Newton’s Second law Kinetic energy of a satelliteTotal mechanical energy of a satellite
23 Orbits For an elliptic orbit it can be shown Orbits with different e but the same a have the same total mechanical energy
24 Chapter 8Problem 40A white dwarf is a collapsed star with roughly the Sun’s mass compressed into the size of Earth. What would be (a) the orbital speed and (b) the orbital period for a spaceship in orbit just above the surface of a white dwarf?
25 Escape speedEscape speed: speed required for a particle to escape from the planet into infinity (and stop there)
26 Escape speed If for some astronomical object Nothing (even light) can escape from the surface of this object – a black hole
27 Chapter 8Problem 54A projectile is launched vertically upward from a planet of mass M and radius R; its initial speed is twice the escape speed. Derive an expression for its speed as a function of the distance r from the planet’s center.
28 Kepler’s laws Three Kepler’s laws Tycho Brahe/Tyge OttesenBrahe de Knudstrup( )Johannes Kepler( )Kepler’s lawsThree Kepler’s laws1. The law of orbits: All planets move in elliptical orbits, with the Sun at one focus2. The law of areas: A line that connects the planet to the Sun sweeps out equal areas in the plane of the planet’s orbit in equal time intervals3. The law of periods: The square of the period of any planet is proportional to the cube of the semimajor axis of its orbit
29 Third Kepler’s law For a circular orbit and the Newton’s Second law From the definition of a periodFor elliptic orbits
30 Chapter 8Problem 23The Mars Reconnaissance Orbiter circles the red planet with a 112-min period. What’s the spacecraft’s altitude?