Ch. 8 Universal Gravitation

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Ch. 8 Universal Gravitation. Milbank High School. Sec. 8.1 Motion in the Heavens and on Earth. Objectives Relate Kepler’s laws of planetary motion to Newton’s law of universal gravitation. Calculate the periods and speeds of orbiting objects.

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### Ch. 8 Universal Gravitation

Milbank High School

Sec. 8.1Motion in the Heavens and on Earth
• Objectives
• Relate Kepler’s laws of planetary motion to Newton’s law of universal gravitation.
• Calculate the periods and speeds of orbiting objects.
• Describe the method Cavendish used to measure G and the results of knowing G.
Kepler
• Johannes Kepler (1571-1630)
• Was convinced that geometry and mathematics could be used to explain the motion of the planets.
Kepler’s laws

1. The orbits of the planets are ellipses, with the sun at one focus.

(Law of Ellipses)

The closer the planets

are to one another,

the more circular the

orbit.

Kepler’s Laws

2. An imaginary line drawn from the center of the sun to the center of the planet will sweep out equal areas in equal intervals of time.

(Law of Equal Areas)

Deals with speed…

faster when closer

to the sun.

Kepler’s Laws

3. The ratio of the squares of the periods of any two planets is equal to the ratio of the cubes of their average distances from the sun.

(Law of Harmonies)

(TA/TB)2 = (rA/rB)3

Universal Gravitation
• Isaac Newton
• 24yrs old…
• Watching an apple fall to the ground made him wonder if gravity extended beyond Earth
• Came up with universal gravitation
• Attractive force between two objects
• The apple was also attracting the Earth
• Proposed Law of Universal Gravitation
Law of Universal Gravitation
• The force of attraction between any two masses is constant throughout the universe
• F = G(mAmB/d2)
• G is a universal gravitational constant between two masses

6.67 x 10-11 N·m2/kg2

Sec. 8.2Using the Law of Universal Gravitation
• Objectives
• Solve problems involving orbital speed and period
• Relate weightlessness to objects in free fall
• Distinguish between inertia mass and gravitational mass
• Contrast Newton’s and Einstein’s views about gravitation
Satellite Motion
• If a projectile moves fast enough, it falls at the same rate that the Earth curves
How fast are satellites moving?
• F = ma or F = mv2/r

(ac = v2/r)

• F = G(mAmB/d2)
• Solve for velocity? Set them equal to each other
• G(mAmB/d2) = mv2/r

which gives you….

Period of a Satellite Circling Earth

T = 2π√r3/GmE

or if we know the velocity…

T = 2πr/v

Weightlessness
• What is gravity in outer space?
• Where space shuttle orbits…g = 8.7m/s2
• How come astronauts

are “floating” then?

• g = F/m