Lecture 21: Gravitation

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Lecture 21Gravitation

• Phys 2101
• Gabriela González

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The Law of Gravitation

• Newtons gravitation law (not Fma!!)

G 6.67 x 10 -11 N m2/kg2
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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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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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Free fall
If energy is negative, we can calculate the
maximum height
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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