Physics 151: Lecture 28 Today’s Agenda

1
Physics 151 Lecture 28 Todays Agenda

• Todays Topics
• Gravity and planetary motion (Chapter 13)

2
Gravitationaccording to Sir Isaac Newton
See text 13.1

• Newton found that amoon / g .000278
• and noticed that RE2 / R2 .000273
• This inspired him to propose the Universal Law
  of Gravitation
• FMm GMm / R2

amoon
g
R
RE
G 6.67 x 10 -11 m3 kg-1 s-2
3
Gravity...
See text 13.1

• The magnitude of the gravitational force F12
  exerted on an object having mass m1 by another
  object having mass m2 a distance R12 away is
• The direction of F12 is attractive, and lies
  along the line connecting the centers of the
  masses.

m1
m2
F12
F21
R12
4
Example

• What is the magnitude of the free-fall
  acceleration at a point that is a distance 2R
  above the surface of the Earth, where R is the
  radius of the Earth ?
• a. 4.8 m/s2
• b. 1.1 m/s2
• c. 3.3 m/s2
• d. 2.5 m/s2
• e. 6.5 m/s2

5
Keplers Laws
See text 13.4

• 1st All planets move in elliptical orbits
  with the sun at one focal point.
• 2nd The radius vector drawn from the sun to a
  planet sweeps out equal areas in equal times.
• 3rd The square of the orbital period of any
  planet is proportional to the cube of the
  semimajor axis of the elliptical orbit.
• It was later shown that all three of these laws
  are a result of Newtons laws of gravity and
  motion.

6
Example

• Which of the following quantities is conserved
  for a planet orbiting a star in a circular orbit?
  Only the planet itself is to be taken as the
  system the star is not included.
• a. Momentum and energy.
• b. Energy and angular momentum.
• c. Momentum and angular momentum.
• d. Momentum, angular momentum and energy.
• e. None of the above.

7
Example

• The figure below shows a planet traveling in a
  clockwise direction on an elliptical path around
  a star located at one focus of the ellipse. When
  the planet is at point A,
• a. its speed is constant.
• b. its speed is increasing.
• c. its speed is decreasing.
• d. its speed is a maximum.
• e. its speed is a maximum.

Animation
8
Example

• A satellite is in a circular orbit about the
  Earth at an altitude at which air resistance is
  negligible. Which of the following statements is
  true?
• a. There is only one force acting on the
  satellite.
• b. There are two forces acting on the satellite,
  and their resultant is zero.
• c. There are two forces acting on the satellite,
  and their resultant is not zero.
• d. There are three forces acting on the
  satellite.
• e. None of the preceding statements are correct.

9
Example

• A satellite is placed in a geosynchronous orbit.
  In this equatorial orbit with a period of 24
  hours, the satellite hovers over one point on the
  equator. Which statement is true for a satellite
  in such an orbit ?
• a. There is no gravitational force on the
  satellite.
• b. There is no acceleration toward the center of
  the Earth.
• c. The satellite is in a state of free fall
  toward the Earth.
• d. There is a tangential force that helps the
  satellite keep up with the rotation of the Earth.
• e. The force toward the center of the Earth is
  balanced by a force away from the center of the
  Earth.

10
Example

• Normally, if I throw a ball up in the air it will
  eventually come back down and hit the ground.
• What if I throw it REALLY hard so that I put it
  into an orbit !
• How HARD do I have to throw ?

11
Energy of Planetary Motion
See text 13.7

• A planet, or a satellite, in orbit has some
  energy associated with that motion.
• Lets consider the potential energy due to
  gravity in general.

Define ri as infinity
12
Energy of a Satellite
See text 13.7

• A planet, or a satellite, also has kinetic energy.

13
Energy of a Satellite
See text 13.7

• So, an orbiting satellite always has negative
  total energy.
• A satellite with more energy goes higher, so r
  gets larger, and E gets larger (less negative).
• Its interesting to go back to the solution for
  v.

v is smaller for higher orbits (most of the
energy goes into potential energy).
14
Lecture 28, Act 2Satellite Energies

• A satellite is in orbit about the earth a
  distance of 0.5R above the earths surface. To
  change orbit it fires its booster rockets to
  double its height above the Earths surface. By
  what factor did its total energy change ?
• (a) 1/2 (b) 3/4 (c) 4/3
• (d) 3/2
• (e) 2

15
Lecture 28, Act 2Satellite Energies
Note E2/E1 3/4 actually means that the energy
is larger (because it is negative)
(b)
16
Escape Velocity

• Normally, if I throw a ball up in the air it will
  eventually come back down and hit the ground.
• What if I throw it REALLY hard ?
• Two other options
• I put it into orbit.
• I throw it and it just moves away forever
• i.e. moves away to infinity

17
Orbiting

• How fast to make the ball orbit.
• I throw the ball horizontal to the ground.
• We had an expression for v above,

18
Escape Velocity

• What if I want to make the ball just go away from
  the earth and never come back ?
• (This is something like sending a space ship out
  into space.)
• We want to get to infinity, but dont need any
  velocity when we get there.
• This means ETOT 0 Why ??

19
Example

• A projectile is launched from the surface of a
  planet (mass M, radius R). What minimum
  launch speed is required if the projectile is to
  rise to a height of 2R above the surface of the
  planet? Disregard any dissipative effects of
  the atmosphere.

20
Example

• A satellite circles planet Roton every 2.8 h in
  an orbit having a radius of 1.2x107 m. If the
  radius of Roton is 5.0x106 m, what is the
  magnitude of the free-fall acceleration on the
  surface of Roton?
• a. 31 m/s2
• b. 27 m/s2
• c. 34 m/s2
• d. 40 m/s2
• e. 19 m/s2

21
Example

• A spacecraft (mass m) orbits a planet (mass
  M) in a circular orbit (radius R). What is the
  minimum energy required to send this spacecraft
  to a distant point in space where the
  gravitational force on the spacecraft by the
  planet is negligible?
• a. GmM/(4R)
• b. GmM/R
• c. GmM/(2R)
• d. GmM/(3R)
• e. 2GmM/(5R)

22
Recap of todays lecture

• Todays Topics
• Gravity and planetary motion (Chapter 13)