Gravitation - physics.louisville.edu 298 summer 18/lectures/0… · Summer 2018 Prof. Sergio B....

1Prof. Sergio B. MendesSummer 2018

Chapter 8 of Essential University Physics, Richard Wolfson, 3rd Edition

Gravitation


• Newton's law of universal gravitation

• About motion in circular and other orbits

• How to calculate gravitational potential energy

• How to describe orbital types in terms of total mechanical energy

• The concept of escape speed

• The concept of gravitational field

What you are about to learn:


Kepler’s Observational Laws(1609 – 1619)

1. The orbit of a planet is an ellipse with the Sun at one of the two foci.

2. A line segment joining a planet and the Sun sweeps out equal areas during equal intervals of time.

3. The square of the orbital period of a planet is proportional to the cube of the semi-major axis of its orbit.


• This law strictly holds only for point masses. However Newton showed that it applies to spherically symmetric masses. Also, it is a good approximation for any objects that are small compared with their separation.

𝐹𝐹 = 𝐹𝐹1 = 𝐹𝐹2 = 𝐺𝐺𝑚𝑚1 𝑚𝑚2

𝑟𝑟2

Universal Law of Gravitation

• Newton's law of universal gravitation states that any two point particles attract each other with a force that is proportional to the product of their masses and inversely proportional to the square of their separation.

• G is the constant of universal gravitation.

(1686)


Cavendish’s Experiment

G = 6.67408(31) × 10–11 N·m2/kg2

(1797–1798)

current value:


𝐹𝐹 = 𝐺𝐺𝑀𝑀𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 𝑚𝑚𝑅𝑅𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸2

= 𝑚𝑚 𝑔𝑔

𝑀𝑀𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 =𝑔𝑔 𝑅𝑅𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸2

𝐺𝐺

𝜌𝜌𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 =3 𝑔𝑔

4 𝜋𝜋 𝑅𝑅𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 𝐺𝐺

Earth’s mass

Earth’s density

Once G is known, then:

= (5.9722 ± 0.0006) × 1024 kg

= 5.5153 g/cm3

𝑅𝑅𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 = 6.3781 × 106 m

Earth’s radius

close to Earth’s surface


Example 8.1𝑔𝑔 =

𝐺𝐺 𝑀𝑀𝑟𝑟2

𝑟𝑟 = 𝑅𝑅𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝑔𝑔𝐸𝐸 =

𝐺𝐺𝑀𝑀𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸

𝑅𝑅𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸2

𝑟𝑟 = 𝑅𝑅𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 + ℎℎ = 380 𝑘𝑘𝑚𝑚

𝑔𝑔𝐸 =𝐺𝐺𝑀𝑀𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸

𝑅𝑅𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 + ℎ 2

𝑀𝑀 = 𝑀𝑀𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸


𝑟𝑟 = 𝑅𝑅𝑀𝑀𝐸𝐸𝐸𝐸𝑀𝑀𝑀𝑀 = 𝑀𝑀𝑀𝑀𝐸𝐸𝐸𝐸𝑀𝑀 𝑔𝑔𝑀𝑀 =

𝐺𝐺 𝑀𝑀𝑀𝑀𝐸𝐸𝐸𝐸𝑀𝑀

𝑅𝑅𝑀𝑀𝐸𝐸𝐸𝐸𝑀𝑀2

𝐺𝐺 = 6.67 × 10–11 N·m2/kg2

= 9.81 𝑚𝑚/𝑠𝑠2

= 8.74 𝑚𝑚/𝑠𝑠2

= 3.75 𝑚𝑚/𝑠𝑠2


The Dependence on the Center-to-Center Distance of

Spherical Objects


Closed Orbits are actually Ellipses

1𝑟𝑟

= 𝑏𝑏 + 𝑎𝑎 𝑐𝑐𝑐𝑐𝑠𝑠 𝜃𝜃


PhET

https://phet.colorado.edu/en/simulation/legacy/my-solar-system


𝐹𝐹 = 𝐺𝐺𝑀𝑀 𝑚𝑚𝑟𝑟2

The Circular Orbit

= 𝑚𝑚𝑣𝑣2

𝑟𝑟

𝑣𝑣 =𝐺𝐺 𝑀𝑀𝑟𝑟

𝑇𝑇2 =4 𝜋𝜋2𝑟𝑟3

𝐺𝐺 𝑀𝑀

𝑣𝑣 =2 𝜋𝜋 𝑟𝑟𝑇𝑇

K3: The square of the orbital period of a planet is proportional to the cube of the semi-major axis of its orbit.

when the ellipse becomes a circle


Example 8.2

𝑣𝑣 =𝐺𝐺 𝑀𝑀𝑟𝑟

=𝐺𝐺 𝑀𝑀𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸

𝑅𝑅𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 + ℎ= 7.68 𝑘𝑘𝑚𝑚/𝑠𝑠

𝑇𝑇 =4 𝜋𝜋2 𝑟𝑟3

𝐺𝐺 𝑀𝑀=

4 𝜋𝜋2 𝑅𝑅𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 + ℎ 3

𝐺𝐺 𝑀𝑀𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸= 5.52 × 103 𝑠𝑠

= 90 𝑚𝑚𝑚𝑚𝑚𝑚

ℎ = 380 𝑘𝑘𝑚𝑚


PhET

https://phet.colorado.edu/en/simulation/gravity-and-orbits


Example: Period of the Moon

𝑇𝑇 =4 𝜋𝜋2 𝑟𝑟3

𝐺𝐺 𝑀𝑀𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸

𝑟𝑟 = 384,402 km

= 27.3 days


PhET

https://phet.colorado.edu/en/simulation/gravity-and-orbits


Example 8.3

𝑟𝑟 =𝐺𝐺 𝑀𝑀𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 𝑇𝑇2

4 𝜋𝜋2

1/3

𝑇𝑇2 =4 𝜋𝜋2𝑟𝑟3

𝐺𝐺 𝑀𝑀

= 4.22 × 107 𝑚𝑚

= 42.2 × 103 𝑘𝑘𝑚𝑚

ℎ = 𝑟𝑟 − 𝑅𝑅𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 = 36.0 × 103 𝑘𝑘𝑚𝑚

𝑅𝑅𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 = 6.4 × 103 km

Geostationary Orbit: 𝑇𝑇 = 24 ℎ

Geostationary Operational Environmental Satellite


Gravitational Potential Energy

𝑊𝑊𝑔𝑔𝐸𝐸𝐸𝐸𝑔𝑔 = �𝐸𝐸1

𝐸𝐸2𝑭𝑭𝑔𝑔𝐸𝐸𝐸𝐸𝑔𝑔 𝒓𝒓 .𝑑𝑑𝒓𝒓 = − 𝑈𝑈𝐸𝐸2 − 𝑈𝑈𝐸𝐸1

𝐹𝐹𝑔𝑔𝐸𝐸𝐸𝐸𝑔𝑔 = 𝐺𝐺𝑀𝑀 𝑚𝑚𝑟𝑟2

𝑈𝑈𝐸𝐸2 − 𝑈𝑈𝐸𝐸1 = − 𝐺𝐺𝑀𝑀𝑚𝑚1𝑟𝑟2−

1𝑟𝑟1

𝑭𝑭𝑔𝑔𝐸𝐸𝐸𝐸𝑔𝑔 𝒓𝒓 .𝑑𝑑𝒓𝒓 = −𝐹𝐹𝑔𝑔𝐸𝐸𝐸𝐸𝑔𝑔 𝑑𝑑𝑟𝑟



Example 8.4

−𝑊𝑊𝑔𝑔𝐸𝐸𝐸𝐸𝑔𝑔 = 𝑈𝑈𝐸𝐸2 − 𝑈𝑈𝐸𝐸1

𝑟𝑟1 = 𝑅𝑅𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 = 6.4 × 103 km 𝑟𝑟2 = 𝑟𝑟𝑔𝑔𝑔𝑔𝑔𝑔𝑀𝑀𝐸𝐸𝐸𝐸𝐸𝐸 = 42.2 × 103 𝑘𝑘𝑚𝑚

𝑊𝑊𝑔𝑔𝑒𝑒𝐸𝐸 = −𝑊𝑊𝑔𝑔𝐸𝐸𝐸𝐸𝑔𝑔 = 𝑈𝑈𝐸𝐸3 − 𝑈𝑈𝐸𝐸2 = − 𝐺𝐺 𝑀𝑀𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 𝑚𝑚1𝑟𝑟3−

1𝑟𝑟2

𝑟𝑟2 = 𝑟𝑟𝑔𝑔𝑔𝑔𝑔𝑔𝑀𝑀𝐸𝐸𝐸𝐸𝐸𝐸 = 42.2 × 103 𝑘𝑘𝑚𝑚 𝑟𝑟3 = 𝑟𝑟𝑚𝑚𝑔𝑔𝑔𝑔𝑚𝑚 = 384.4 × 103 𝑘𝑘𝑚𝑚

𝑚𝑚 = 11 × 103 𝑘𝑘𝑔𝑔

= 5.842 × 1011 𝐽𝐽

= 0.925 × 1011 𝐽𝐽

𝑊𝑊𝑔𝑔𝑒𝑒𝐸𝐸 = = − 𝐺𝐺 𝑀𝑀𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 𝑚𝑚1𝑟𝑟2−

1𝑟𝑟1


𝑟𝑟1 = ∞

𝑈𝑈𝐸𝐸1 = 0

𝑈𝑈𝐸𝐸 = −𝐺𝐺𝑀𝑀 𝑚𝑚𝑟𝑟

𝑈𝑈𝐸𝐸2 − 𝑈𝑈𝐸𝐸1 = − 𝐺𝐺𝑀𝑀𝑚𝑚1𝑟𝑟2−

1𝑟𝑟1

𝑈𝑈𝐸𝐸1 = −𝐺𝐺𝑀𝑀 𝑚𝑚𝑟𝑟1

A Simplifying and Meaningful Reference Point

𝑟𝑟2 → 𝑟𝑟


Mechanical Energy (Kinetic + Potential Energies)

𝐸𝐸 = 𝐾𝐾 + 𝑈𝑈 =12𝑚𝑚 𝑣𝑣2 −

𝐺𝐺𝑀𝑀 𝑚𝑚𝑟𝑟


Example 8.5𝑣𝑣𝑔𝑔 = 3.1 𝑘𝑘𝑚𝑚/𝑠𝑠

ℎ = ? ? 𝑟𝑟 = ℎ + 𝑅𝑅𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸



12𝑚𝑚 𝑣𝑣𝑔𝑔2 −

𝐺𝐺𝑀𝑀 𝑚𝑚𝑟𝑟𝑔𝑔

=12𝑚𝑚 𝑣𝑣2 −


𝑟𝑟𝑔𝑔 = 𝑅𝑅𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸

𝑣𝑣 = 01𝑟𝑟

=1𝑟𝑟𝑔𝑔−

𝑣𝑣𝑔𝑔2

2 𝐺𝐺 𝑀𝑀


𝑚𝑚: unknown


Closed and Opened Orbits


Escape Velocity



≥ 0

𝑣𝑣 ≥2 𝐺𝐺 𝑀𝑀𝑟𝑟

𝐸𝐸 ≥ 0


Gravitational Field

close to the surface of a spherical object

a generalized and more precise description

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Gravitation - physics.louisville.edu 298 summer 18/lectures/0… · Summer 2018 Prof. Sergio B....

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