Download Chapter 13 Gravitation

Chapter 13
Gravitation
PhysicsI 2048
Newton’s law of gravitation
Besides the three laws of motion,
Newton also discovered the
universal law of gravitation.
The force of gravity between two point object of mass m1
and m2 is attractive and of magnitude
where r is the distance between the masses and
G is a constant referred as the universal gravitation
constant.
Newton’s law of gravitation
Gravitation
The law of gravity
applies to all objects
small or large.
G is very small
(=6.67x10-11
N·m2kg-2)
The force is inverse
proportional to
distance.
Satisfies
superposition.
Gravitation and the Principle of
Superposition
Where F1,netis the net force on particle 1 due to n particles
Gravitation Near Earth's Surface
Newton was able to show that the net
force exerted by the sphere on a
point mass m is the same as if all the
mass of the sphere were
concentrated at its center.
For a mass is near the
surface of earth:
Variation of ag with Altitude
Gravitation Near Earth's Surface
Earth's mass is not uniformly
distributed.
Gravitation Near Earth's Surface
Earth is not a sphere. Earth is
approximately an ellipsoid, flattened
at the poles and bulging at the
equator.
Earth is rotating. The rotation axis
runs through the north and south
poles of Earth.
Gravitation Near
Earth's Surface
Gravitational Potential Energy
Proof of the gravitational potential
energy equation
Let us shoot a baseball directly away from
Earth along the path in Figure
Potential Energy and Force
This is Newton's law of gravitation.
The minus sign indicates that the
force on mass m points inward,
toward mass M
Escape Speed
When the projectile reaches infinity, it
stops and thus has no kinetic energy. It
also has no potential energy because an
infinite separation between two bodies is
zero potential energy
Some Escape Speeds
Kepler’s law of orbital motion
Kepler’s three laws
(1) Planets follow elliptical orbits, with
the Sun at one focus of the ellipse.
Kepler’s law of orbital motion
(2) As a planet moves in its orbit, it
sweeps out an equal amount of area in
an equal amount of time.
Kepler’s law of orbital motion
(3) The period of a planet increases
as its mean distance from the Sun, r
raised to the 3/2 power
Kepler’s law of orbital motion
Here we will show that
the Kepler’s third law can
be derived from the
definition of centripetal
acceleration and the
universal gravitation law.
Satellites: Orbits and Energy
The potential energy of the system is given by
Equation
we write Newton's second law (F = ma) as
Where a is the ellipsis semimajor axis
The mean diameters of planets M and
E are 6.9 × 103 km and 1.3 × 104 km,
respectively. The ratio of the mass of
planet M to that of planet E is 0.11. (a)
What is the ratio of the mean density
of M to that of E? (b) What is the ratio
of the gravitational acceleration on M to
that on E? (c) What is the ratio of
escape speed on M to that on E?
a-
b-
C-
Two neutron stars are separated by
a distance of 1.0 x 1010 m. They
each have a mass of 1.0 x 1030 kg
and a radius of 1.0 x 105 m. They
are initially at rest with respect to
each other. As measured from that
rest frame, how fast are they moving
when (a) their separation has
decreased to one-half its initial value
and (b) they are about to collide?
(a) Use the principle of conservation
of energy. The initial potential energy
is.
The initial kinetic energy is zero since
the stars are at rest.
The final potential energy is.
(b) Now the final separation of the
centers is