Planetary Motion and Gravitation Notes

Planetary Motion and Gravitation

Focus Question

• What role does gravity play in planetary motion?

New Vocabulary

• Kepler's first law
• Kepler's second law
• Kepler's third law
• Gravitational force
• Law of universal gravitation

Review Vocabulary

• Newton’s third law: states that all forces come in pairs and that the two forces in a pair act on different objects, are equal in strength, and are opposite in direction

Early Observations

• In ancient times, it was assumed that the Sun, the Moon, the planets, and the stars revolved around Earth.
• Nicholas Copernicus proposed that Earth and other planetary objects revolved around the Sun.
• Tycho Brahe is credited with the most accurate measurements of his time, using instruments that he designed and built.

Kepler’s Laws

• Kepler’s first law states that the paths of the planets are ellipses, with the Sun at one focus.
• Kepler’s second law states that an imaginary line from the Sun to a planet sweeps out equal areas in equal time intervals.
• Kepler’s third law states that the square of the ratio of the periods of any two planets revolving around the Sun is equal to the cube of the ratio of their average distances from the Sun.
• If the periods of the planets are $T<em>A$ and $T</em>B$, and their average distances from the Sun are $r<em>A$ and $r</em>B$, Kepler’s third law can be expressed as follows:

Kepler's Third Law Equation

$\frac{T<em>A^2}{T</em>B^2} = \frac{r<em>A^3}{r</em>B^3}$

Kepler’s Laws - Example Problem

• Europa has a greater period than Io, so we would expect Europa to be farther from Jupiter than Io. This agrees with our answer.
• Problem: Europa, a satellite of Jupiter, has a period of 3.55 days. How many units is its radial distance?
  • Sketch the situation.
  • List the knowns and unknowns. The information for Io was taken from Example Problem 1.
    • KNOWN
      • $r_I = 4.2$ units
      • $T_I = 1.8$ days
    • UNKNOWN
      • $r_E = ?$
      • $T_E = 3.55$ days
  • Solve for the unknown using Kepler’s third law.

Newton’s Law of Universal Gravitation

• Beginning in 1666, Isaac Newton studied planetary motion and concluded that an attractive force must act between any two bodies with mass.
• The force of attraction between two objects, called the gravitational force, must be proportional to the objects’ masses.
• The law of universal gravitation states that objects attract other objects with a force that is proportional to the product of their masses and inversely proportional to the square of the distance between them.

Law of Universal Gravitation

• $F = G \frac{m<em>1m</em>2}{r^2}$

Universal Gravitation and Kepler’s Third Law

• Newton’s law of universal gravitation can be combined with the definition of a period to find the period (T) of an object orbiting the Sun.

Period of a Planet Orbiting the Sun

• $T = \sqrt{(\frac{4\pi^2}{GM})r^3}$
• Squaring both sides makes it apparent that this equation is Kepler’s third law of planetary motion: The square of the period is proportional to the cube of the distance that separates the masses.

Measuring the Universal Gravitational Constant

• In 1798, English scientist Henry Cavendish used equipment similar to the equipment shown to measure the gravitational force between two objects.
• The experimental value of G is: $G = 6.67 × 10^{−11} N⋅m^2/kg^2$

Quiz

• According to Kepler’s first law, what do the orbits of the planets look like?
  • ellipses
• According to Kepler’s first law, what is at the focus of a planet’s orbit?
  • Sun
• In the equation for Kepler’s third law, what should be in place of the x?
  • (There was no provided equation to base this off of in the transcript)
• Which two factors is the gravitational force between two objects dependent upon?
  • mass, distance
• This law graph applies to which law?
  • Kepler’s first law