Transcript Universal Gravitationx

Universal
Gravitation
Motion in the Heavens and
on Earth
Objectives
 Relate
Kepler's Law to Newton's Law.
 Calculate the periods & speeds of orbiting
objects.
 Describe the method Cavendish used to
measure G.
Motion in the Heavens and on
Earth
 Because
of the work
of early scientists
(Galileo, Kepler,
Newton, etc..) we
know that planets,
stars, comets and
other bodies follow
the same laws as
objects do on Earth.
Observed Motion
 Kepler,
only an assistant to
Tycho Brahe in the 1600's,
was convinced that our
universe was sun centered
and that mathematics
could explain the number,
distance and motion of
the planets.
 Kepler discovered laws
that describe the motion
of astronomical bodies
after careful analysis of
Brahe’s data
Observed Motion
 These
1.
2.
3.
are Keppler’s laws:
The paths of the planets are ellipses with the
sun at one focus.
Planets move faster when they are closer to
the sun.
The square of the ratio of the periods of any
two planets revolving about the sun ( TA/TB )2,
is equal to the cube of the ratio of their
average distances from the sun ( RA/RB)3.
Universal Gravitation
 Newton
used Kepler's 1st
law, (paths of planets
are ellipses) to determine
that the magnitude of
the force (F) on the
planet resulting from the
sun must vary inversely
with the square of the
distances between the
center of the planet and
the center of the sun.
Universal Gravitation
(F is proportional to 1/d2)
(∝)=means proportional to
d = distance between the centers of the
two bodies.
Universal Gravitation
 Newton
later wrote that the apple
falling straight down made him
wonder if the same force
extended beyond to the clouds,
moon and even beyond.
 The force of attraction is
proportional to their masses, (the
apple to the Earth, and the Earth
to the apple).
 This attractive force between all
objects is Gravitational Force.
Universal Gravitation
 Newton
was confident that the laws
governing Earth would work anywhere in
the universe on any two masses, (Ma and
Mb).
 This is the Law of Universal Gravitation.
 G is a universal constant.
F= G (MaMb/d2)
Universal Gravitation
 If
the mass of a planet near
the sun were doubled, the
force of the attraction
would be doubled.
 If a planet were near a star
having twice the mass of
the sun, the force between
the two bodies would be
twice as great.
 If a planet were twice the
distance from the sun, the
gravitational force would
be only one quarter as
strong.
Using Newton's Law of
Universal Gravity
 Mp
= Mass of planet
 ac = Centripetal acceleration
 Ms = Mass of sun
 Fg = Gravitational Force
 r = radius of the planet's orbit
F = ma
F = Mp ac
Using Newton's Law of
Universal Gravity
 Assume
circular orbits and use
ac = 4π2r/T2 and substitute for ac, so......
F
= Mpac now becomes F= Mp(4π2r/T2)
Using Newton's Law of
Universal Gravity

Now, set this equal to Newton's Law of
Universal Gravitation and arrive at:
G(MSMP/r2) = MP4π2r/T2
or
T2=(4π2/GMS)r3
This is Kepler's 3rd law of planetary
motion, or the period of an planet
orbiting the Sun!!
Weighing the Earth
 Cavendish
(1798)
invented the equipment
to measure the
gravitational force
between two objects,
G.
 He used a rod, 4 lead
spheres, wire, a mirror
and a light source.
Weighing the Earth
 He
substituted values for force, mass and
distance into Newton's law and found a
value for G.
G = 6.67 x 10-11 Nm2/Kg2
Weighing the Earth
 Ex:
Find the gravitational force between two
objects where mass is 7.26 Kg, and their centers
are separated by .30 m.
Fg = G(MAMB/r2 )
Fg = 6.67×10-11Nm2/kg2 (7.26kg × 7.26kg/(.30m)2)
Fg = 3.9×10-8 N
Weighing the Earth
 Determine
the force of gravitational attraction
between the Earth (m=5.98 x 1024kg) and a 70 kg
physics student if the student is standing at sea level,
a distance of 6.37 x 106 m from Earth's center.
Fg = G(MAMB/r2 )
Fg = 6.67×10-11Nm2/kg2 (5.98 x 1024kg × 70kg/(6.37x106 m)2)
Fg = 688.09 N
Weighing the Earth
 Ex:
use
To find the weight (force) of the Earth,
 Fg=GME/r2
 We
or ME=gr2/G
know that the Earth's radius is 6.38 x
106m and that ag or g is 9.8 m/s2, and that
G is 6.67 x 10-11Nm2/Kg2