Lecture 22 Gravitation

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Lecture 22Gravitation

• Phys 2101
• Gabriela González

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Gravitational acceleration and weight

• The force of a spherical planet of mass M and
  radius R on a particle of mass m at a height h
  from the surface is FGMm/r2GMm/(Rh)2.
• Using the other Newtons law (Fma), then bodies
  on the planets surface fall with acceleration
  agGM/(Rh)2.
• If Rgtgth, aGM/R2 9.8 m/s2 on Earth at sea
  level (and in BR).
• The higher the object (larger h), the smaller
  the acceleration not constant g!
• If planet is not spherical, we need integrals or
  approximations.
• The normal force of the planets surface on the
  particle is its weight Nmg.
• If planet is rotating, there is a centripetal
  acceleration that contributes to the weight gag
  - ?2Rcos?GM/r2 - ?2Rcos?.

A persons weight is smallest at the equator, due
to Earths flattening at the poles, and due to
Earths rotation. The effect is only 0.5, though
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Example

• Neutron stars are extremely dense stars, that can
  have rotation with large angular velocities. A
  neutron star is rotating at 1200 rpm (20 rev/s
  126 Hz) has a radius of 15km, and a mass of 1.4
  solar masses.
• What is the gravitational acceleration on the
  stars surface?
• How fast a rotation would make a particle
  weightless at the surface?

Fastest spinning pulsar The scientists
discovered the pulsar, named PSR J1748-2446ad, in
a globular cluster of stars called Terzan 5,
located some 28,000 light-years from Earth in the
constellation Sagittarius. The newly-discovered
pulsar is spinning 716 times per second, or at
716 Hertz (Hz), readily beating the previous
record of 642 Hz from a pulsar discovered in
1982. For reference, the fastest speeds of common
kitchen blenders are 250-500 Hz.
http//www.nrao.edu/pr/2006/mspulsar/
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Potential Energy

• Potential energy -work done by gravitational
  forces.
• Work required to move a mass m from R to ?

Potential energy of a system of masses
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Gravitational Potential Energy

• We have a new formula for gravitational
  potential energy, U-GMm/R, but we had already a
  formula for change in gravitational potential
  energy, ?Umgh. The old formula is an
  approximation of the new, more precise formula

h
R
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Escape Speed

• If no other forces are acting, we know that
  mechanical energy is conserved
• ?KE?U0. When an object is launched from Earth
  with some kinetic energy, the kinetic energy is
  transferred to potential energy as it goes up

If initial energy is negative (v20lt2GM/R), then
there is a maximum height hmax, when the
projectile momentarily stops, and comes back to
the surface. If initial energy is positive, the
projectile can reach infinity with some speed.
The minimum velocity for the energy to be
positive is the escape speed
In Earth, ve11.2 km/s25,000 mph
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Keplers laws I

• Three laws, but all consequences of just one
  Newtons law of gravitation!
• LAW OF ORBITS All planets move in elliptical
  orbits, with the Sun at one focus.

a semimajor axis b semiminor axis Suns
distance from center ea e eccentricity If e0,
orbit is circular.
Animations by by Bill Drennon, Physics
TeacherCentral Valley Christian High
SchoolVisalia, CA USA , http//home.cvc.org/scie
nce/kepler.htm
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Solar system orbits not circular, but close
1 AU 149 598 000 000 meters
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Keplers laws II

• LAW OF AREAS A line that connects the planet to
  the Sun sweeps out equal areas in the plane of
  the planets orbit in equal times.

Areal velocity dA/dt is constant
Angular momentum
Conservation of angular momentum!
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Keplers laws III

• LAW OF PERIODS The square of the period of any
  planet is proportional to the cube of the
  semimajor axis of the orbit.

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A very real example

• In spring 2002 S2 was passing with the
  extraordinary velocity of more than 5000 km/s at
  a mere 17 light hours distance -- about three
  times the size of our solar system -- through the
  perinigricon, the point of closest approach to
  the black hole. By combining all measurements of
  the position of S2 made between spring 1992 and
  summer 2002, we have obtained enough data in
  order to determine a unique keplerian orbit for
  this star, presented in Figure 1.

• It is highly elliptical (eccentricity 0.87), has
  a semimajor axis of 5.5 light days, a period of
  15.2 years and an inclination of 46 degrees with
  respect to the plane of the sky. From Kepler's
  3rd law we can determine the enclosed mass in a
  straightforward manner to be 3.71.5 million
  solar masses. Therefore at least 2.2 million
  solar masses have to be enclosed in a region with
  a radius of 17 light hours.

http//www.mpe.mpg.de/www_ir/GC/gc.html Schödel,
R. et al. A star in a 15.2-year orbit around the
supermassive black hole at the centre of the
Milky Way. Nature, 419, 694 - 696, (2002).
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Examples

• (a) What is the gravitational acceleration on
  Earth, on the equator?
• (Use g ag-?2R GM/R2 - ?2R)
• (b) How fast would the Earth had to rotate to
  produce weightlessness at the Equator?
• (c) What is the escape speed in Earth? (Use Ve2
  2GM/R)
• (d) What would be the escape velocity is the
  Earth had the same mass concentrated in only a
  radius of 5mm?
• (e) How far from the surface will a particle
  thrown at 112m/s go?EKU (1/2) mv2 -GMm/r
  constant
• (f) With what speed will an object hit the Earth
  if it is dropped from the space shuttle orbiting
  at 400km above the Earth?
• (g) How long does the space shuttle take to orbit
  once around the Earth? (law of periods)
• (h) How high should a satellite be above the
  Earth to have a circular geosynchronous orbit?