Physics 2211a Kulp Class 35: How do planets orbit the sun

1
Physics 2211a (Kulp)Class 35 How do planets
orbit the sun?

• W. D. Kulp
• william.kulp at physics.gatech.edu
• Office W501b
• Office Hours MF 11-12 AM, Tu 9-10 AM, by appt.
• www.physics.gatech.edu/academics/Classes/fall200
  6/2211/a/

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Where are we?

• Last class (universal gravity)

• Today Class 35
• Use conservation of energy to analyze
  gravitational problems.
• Determine the escape velocity from a massive
  body.
• Apply Keplers Laws to analyze the orbit of an
  object or calculate a change of orbits.
• Determine the height of a geosynchronous orbit.

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Stars struck

• Two neutron stars are separated by a distance of
  1010 m. They each have a mass of 1030 kg and a
  radius of 105 m. They are initially at rest with
  respect to each other. How fast are they moving
  just before they collide?

m
R
ri

• Can we approach this with Newtons Law of
  Gravity?
• Is there a better way to solve this problem?
• What is the final geometry of the collision?

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Gravitational Potential Energy

• We could approach the colliding neutron star
  problem by calculating the work done by the
  gravitational force.
• However, there are two bodies moving!
• Note that the force depends only on the position
  of the bodies gravity is a conservative force.

• This becomes larger as the objects get closer
  together.

5
Stars struck

• Two neutron stars are separated by a distance of
  1010 m. They each have a mass of 1030 kg and a
  radius of 105 m. They are initially at rest with
  respect to each other. How fast are they moving
  just before they collide?

m
R
ri

• The initial gravitational potential energy is

6
Stars struck

• Two neutron stars are separated by a distance of
  1010 m. They each have a mass of 1030 kg and a
  radius of 105 m. They are initially at rest with
  respect to each other. How fast are they moving
  just before they collide?

• The final energy just before the collision and
  speed are

1.8 x 107 m/s
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Escape velocity

• What is the escape speed from the sun for an
  object in Earths orbit (orbital radius R), but
  far from Earth?

42.1 km/s

• Note that we set the final energies to zero, and
  this does not consider Earths gravitational
  potential, or its orbital speed.

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Keplers Third Law

• The square of the planets period T is
  proportional to the cube of the orbits
  semimajor-axis length.

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Keplers Third Law

• The square of the planets period T is
  proportional to the cube of the orbits
  semimajor-axis length.

• This result shows that the period is independent
  of the mass.

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Keplers Second Law

• A line between the sun and a planet sweeps out
  equal areas during equal intervals of time.

• This is an empirical observation dating from
  1609.
• If we look more closely, we find a new kind of
  conservation law angular momentum is conserved.

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Keplers Second Law

• A line between the sun and a planet sweeps out
  equal areas during equal intervals of time.

• The only force present is a radial force,
  perpendicular by definition, thus there is no
  tangential acceleration. In this case, L is a
  constant - angular momentum is conserved.

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Orbit forming

• How much work must be done to boost a 1000 kg
  satellite from low Earth orbit (h 300 km) to a
  geosynchronous orbit?

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Orbital Energetics

• An object in orbit is a bound system (the total
  mechanical energy is negative). If we consider
  an object of mass m in a circular orbit of radius
  r

• This is not a requirement.
• (but specifies a circular orbit)

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Geosynchronous orbit

• An orbit with a period equal to one day. The
  orbital motion is synchronous with the planet,
  and appears motionless over the planets surface.

• rgeo 35,900 km

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Orbit forming

• How much work must be done to boost a 1000 kg
  satellite from low Earth orbit (h 300 km) to a
  geosynchronous orbit?

• ?E W 2.52 x 1010 J

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Whats next?

• Wednesday, 11/15 Class 36 What is the motion of
  a spring?
• Determine the amplitude, period, phase, and
  frequency of an object in oscillatory motion.
• Calculate acceleration, velocity, and
  displacement for an object executing SHM, and
  determine where these quantities are maximum,
  minimum, or zero.
• Calculate the kinetic and potential energies of
  an oscillating system as functions of time.
• Analyze an oscillating system using conservation
  of energy.

• To-do list
• Homework 33 before class on Wednesday
• Read assignment (14.1 - 14.3)

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Its whats inside that counts

• Consider a spherical shell of matter. What is
  the force on a mass placed inside the shell?

• By symmetry, the forces cancel out at every
  point.
• The net force is zero.

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Its whats inside that counts

• Consider a sphere of density ?. What is the
  acceleration on a mass placed outside the sphere?

• We need the mass of the sphere,

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• Which of the following depends on the inertial
  mass of an object (as opposed to its
  gravitational mass)?

• The time the object takes to fall from a certain
  height.
• The weight of the object as measured on a
  bathroom-type spring scale.
• The acceleration given to the object by a the
  release of a compressed spring.
• The weight of the object on an ordinary balance.

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• Two satellites, A and B, of the the same mass are
  going around Earth in concentric orbits. The
  distance of satellite B from Earths center is
  twice that of satellite A. What is the ratio of
  the centripetal force acting on B to that acting
  on A?

• 1. 2. 3. 4. 5.

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• What is the direction of the net gravitational
  force on a mass m placed in the center of the
  four masses arranged in a square pictured below?

• 1. 2. 3. 4.
  5.

300 kg
500 kg
directed along a straight line connecting the
objects
m
100 kg
500 kg
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Planet Shuffle
Consider the planet Jupiter compared with Earth

• If the earth could be moved to the distance of
  Jupiter and placed in a circular orbit around the
  sun, its orbital period would be

• 1 year
• between 1 and 11.9 years
• 11.9 years
• more than 11.9 years
• it is impossible for a planet of Earths mass to
  orbit at the distance of Jupiter

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Look out below!

• A rock, initially at rest with respect to Earth
  and located an infinite distance away is released
  and accelerates toward Earth. An observation
  platform 3 Earth-radii above the surface is built
  to observe the rock as it plummets to the ground.
  Neglecting friction, the rocks speed when it
  hits the ground is

• twice
• three times
• four times
• six times
• eight times
• nine times
• sixteen times
• its speed at the observation platform.