In this chapter we will explore the following topics: Gravitation In this chapter we will explore the following topics: -Newton’s law of gravitation, which describes the attractive force between two point masses and its application to extended objects -The acceleration of gravity on the surface of the Earth, above it as well as below it. -Gravitational potential energy -Kepler’s three laws of planetary motion -Satellites (orbits, energy, escape velocity) (13-1)

m1 m2

m1 m2 F12 F21 r

m1 m2 r F1 m2 m1

A particle is placed, in turn, outside of four objects, each of mass, m; 1)A large uniform solid sphere, 2) a large uniform spherical shell, 3) A small uniform solid sphere and 4) a small uniform shell. In each situation, the distance between the center of the object and the particle is a distance, d. Rank the objects according to the magnitude of the gravitational force, greatest first. Answer: all tie

m1 dm r

The picture below has four arrangements of three particles of equal mass. Rank the arrangements from greatest magnitude of gravitational force acting on particle m, greatest first. Answer: 1, 2=4, 3 b) In arrangement 2, is the direction of the net force closer to line D or d? Answer: d

g = free fall acceleration = -9.8 m/s2 ag = gravitational acceleration (just from the gravitational force.

3. The Earth's rotation causes the value of g to be less than ag 3. The Earth's rotation causes the value of g to be less than ag. At the equator where this effect is most pronounced it is a difference of only 0.034 m/s2

m1 m2 r F1

Consider a mass, m, inside the Earth at a distance, r, from the center of the Earth. If we consider the Earth as a series of nested shells, only the shells that are closer to the center of the Earth contribute to the acceleration due to gravity on the mass and they have an effective mass of the Earth, Meff.



 Find the acceleration due to gravity within the Earth. Plot the acceleration due to gravity for the Earth for any distance from r = 0 to infinity. m2 m1 r

r m

m B v = 0 v A

S N Star (13-11) Polaris Earth Celestial sphere Rotation Axis of the Celestial Sphere Earth N S Star (13-11)

The smaller the eccentricity the more circular the orbit,

angular

Fc = FG - N; FG = Newton's Law of G N = W = mg (reading on scale) mac = maG - N ; N = W = mg b/c @ rest on scale mac = maG - mg mg = maG - mac; ac g = aG - ac y-axis Fc = FG - N; FG = Newton's Law of G N = W = mg (reading on scale) mac = maG - mg; ac = v2/r =  r2 / r =  r  r = aG - g g = aG -  r ; aG = GM/r2