Newton’s Law of Universal Gravitation

Warm-up: Answer the following questions. Gravitational force is always attractive / repulsive. (circle) Newton’s 3rd Law tells us that if a gravitational force exists between two objects, one very massive and one less massive, then the force on the less massive object will be greater than / equal to / less than the force on the more massive object. The distance between masses is measured from their edges between them / from their centers / from the edge of one to the center of the other.

As the distance between masses decreases, force increases / decreases. Doubling the mass of both masses would result in a change of force between the masses of 8x / 6x / 4x / 2x / no change / ½x / ¼x / 1/6x / 1/8x. Reducing the distance between two masses to half while doubling the mass of one of the masses would result in a change of force between the masses of 8x / 6x / 4x / 2x / no change / ½x / ¼x / 1/6x / 1/8x. What is the gravitational force between two students, Dylan and Sarah, if Dylan has a mass of 75 kg, Sarah has a mass of 54 kg, and their centers are separated by a distance of 0.45 m? ________________ N What is the gravitational force between two students, John and Mike, if John has a mass of 81 kg, Mike has a mass of 93 kg, and their centers are separated by a distance of 0.62 m? ________________ N

Objective Students will analyze the factors affecting the motion of the planet revolving around the sun or satellite revolving around a planet in order to explain planetray motion and satellites motion.

Newton’s Law of Universal Gravitation http://scienceblogs.com/startswithabang/upload/2009/07/meet_our_second_moon/400px-NewtonsLawOfUniversalGravitation.svg.png

Universal Gravitation G is the constant of universal gravitation G = 6.673 x 10-11 N m² /kg² This is an example of an inverse square law Determined experimentally Henry Cavendish in 1798 11/28/2018

Universal Gravitation The force that mass 1 exerts on mass 2 is equal and opposite to the force mass 2 exerts on mass 1 The forces form a Newton’s third law action-reaction The gravitational force exerted by a uniform sphere on a particle outside the sphere is the same as the force exerted if the entire mass of the sphere were concentrated on its center 11/28/2018

The value of g (acceleration due to gravity) Fgrav = m*g mass of the earth (approx. 5.98x1024 kg) 6.38x106 m (a typical earth radius value)

Location Distance from Earth's center (m) Value of g m/s2 Earth's surface 6.38 x 106 m 1000 km above surface 7.38 x 106 m 2000 km above surface 8.38 x 106 m 3000 km above surface 9.38 x 106 m 4000 km above surface 1.04 x 107 m 5000 km above surface 1.14 x 107 m 6000 km above surface 1.24 x 107 m 7000 km above surface 1.34 x 107 m 8000 km above surface 1.44 x 107 m 9000 km above surface 1.54 x 107 m 10000 km above surface 1.64 x 107 m 50000 km above surface 5.64 x 107 m

Planet Radius (m) Mass (kg) g (m/s2) Mercury 2.43 x 106 3.2 x 1023 Venus 6.073 x 106 4.88 x1024 Mars 3.38 x 106 6.42 x 1023 Jupiter 6.98 x 107 1.901 x 1027 Saturn 5.82 x 107 5.68 x 1026 Uranus 2.35 x 107 8.68 x 1025 Neptune 2.27 x 107 1.03 x 1026 Pluto 1.15 x 106 1.2 x 1022

Orbital Motion http://www.physicsclassroom.com/shwave/orbits.cfm

Questions What is eccentricity? What is the relationship between eccentricity and “ovalness” of the orbit? What Kepler’s first law of planetary motion? Which planet has the most eccentric orbit?

Click the velocity. Compare the motion (velocity) of the planet when the eccentricity is 0.0 and when the eccentricity is 0.60. Use the length of the green arrow in comparing the velocities.  What is Kepler’s second law of planetary motion?  For an eccentric orbit, how do you compare the amount of sunlight received by the planet when it is closest to the sun and when the planet is away from the sun? Explain your answer. Considering your answer for number 5, what is the season when the planet is closest to the sun and what is the season when it is away from the sun? Explain. What is the difference between aphelion and perihelion?

What is the difference between aphelion and perihelion? Click the velocity and force. When the eccentricity is 0.0, describe the force and the velocity vector? Describe their magnitudes (length of arrow) and the directions. Click the velocity and the force. When the eccentricity is 0.6, DRAW the force and velocity vector when The planet is at aphelion The planet is at perihelion The planet is approaching the aphelion The planet is leaving the aphelion The planet is approaching the perihelion The planet is leaving the perihelion   What is Kepler’s 3rd of planetary motion? Do you think planetary collision is possible? Explain.

Kepler’s Laws All planets move in elliptical orbits with the Sun at one of the focal points. A line drawn from the Sun to any planet sweeps out equal areas in equal time intervals. The square of the orbital period of any planet is proportional to cube of the average distance from the Sun to the planet. 11/28/2018

Kepler’s First Law All planets move in elliptical orbits with the Sun at one focus. Any object bound to another by an inverse square law will move in an elliptical path Second focus is empty 11/28/2018

Kepler’s Second Law A line drawn from the Sun to any planet will sweep out equal areas in equal times Area from A to B and C to D are the same 11/28/2018

Kepler’s Third Law The square of the orbital period of any planet is proportional to cube of the average distance from the Sun to the planet. T is the period of the planet a is the average distance from the Sun. Or a is the length of the semi-major axis For orbit around the Sun, K = KS = 2.97x10-19 s2/m3 K is independent of the mass of the planet 11/28/2018

The Mass of the Sun Calculate the mass of the Sun noting that the period of the Earth’s orbit around the Sun is 3.156107 s and its distance from the Sun is 1.4961011 m. 11/28/2018

Geosynchronous Orbit From a telecommunications point of view, it’s advantageous for satellites to remain at the same location relative to a location on the Earth. This can occur only if the satellite’s orbital period is the same as the Earth’s period of rotation, 24 h. (a) At what distance from the center of the Earth can this geosynchronous orbit be found? (b) What’s the orbital speed of the satellite? . 11/28/2018

Planetary and Satellite Motion Fnet = ( Msat • v2 ) / R Fgrav = ( G • Msat • MCentral ) / R2 (Msat • v2) / R = (G • Msat • MCentral ) / R2 v2 = (G • MCentral ) / R g = (G • Mcentral)/R2

Practice Problem #1 A satellite wishes to orbit the earth at a height of 100 km (approximately 60 miles) above the surface of the earth. Determine the speed, acceleration and orbital period of the satellite. (Given: Mearth = 5.98 x 1024 kg, Rearth = 6.37 x 106 m)

Problem 2 The period of the moon is approximately 27.2 days (2.35 x 106 s). Determine the radius of the moon's orbit and the orbital speed of the moon. (Given: Mearth = 5.98 x 1024 kg, Rearth = 6.37 x 106 m)

Problem 3 A geosynchronous satellite is a satellite that orbits the earth with an orbital period of 24 hours, thus matching the period of the earth's rotational motion. A special class of geosynchronous satellites is a geostationary satellite. A geostationary satellite orbits the earth in 24 hours along an orbital path that is parallel to an imaginary plane drawn through the Earth's equator. Such a satellite appears permanently fixed above the same location on the Earth. If a geostationary satellite wishes to orbit the earth in 24 hours (86400 s), then how high above the earth's surface must it be located? (Given: Mearth = 5.98x1024 kg, Rearth = 6.37 x 106 m)

Homework http://www.physicsclassroom.com/curriculum/circles/circle7.pdf http://www.physicsclassroom.com/curriculum/circles/circle8.pdf http://www.physicsclassroom.com/curriculum/circles/circle9.pdf