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**UCM & Gravity – Kepler’s Laws**

Unit #5 UCM & Gravity

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**Objectives and Learning Targets**

Explain the acceleration of an object moving in a circle at constant speed. Define centripetal force and recognize that it is not a special kind of force, but that it is provided by forces such as tension, gravity, and friction. Solve problems involving calculations of centripetal force. Determine the direction of a centripetal force and centripetal acceleration for an object moving in a circular path. Calculate the period, frequency, speed and distance traveled for objects moving in circles at constant speed. Analyze and solve problems involving objects moving in vertical circles. Determine the acceleration due to gravity near the surface of Earth. Utilize Newton’s Law of Universal Gravitation to determine the gravitational force of attraction between two objects. Explain the difference between mass and weight. Explain weightlessness for objects in orbit. Explain how Kepler’s Laws describe the orbits of planetary objects around the sun. Unit #5 UCM & Gravity

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**Kepler’s Laws of Planetary Motion**

In the early 1600s, most of the scientific world believed that the planets should have circular orbits, and many believed that the Earth was the center of the solar system. Using data collected by Tycho Brahe, German astronomer Johannes Kepler developed three laws governing the motion of planetary bodies, which described their orbits as ellipses with the sun at one of the focal points (even though the orbits of many planets are nearly circular). These laws are known as Kepler’s Laws of Planetary Motion. Unit #5 UCM & Gravity

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**Kepler’s Laws of Planetary Motion**

A. Tycho Brahe, Danish, ( ) 1. Owned an island 2. Accurate observer of the heavens 3. Lost his nose in a dual B. Johannes Kepler, German, ( ) 1. Assistant to Brahe 2. Sun at center of our system 3. Three Laws Unit #5 UCM & Gravity

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Kepler’s 1st Law Kepler’s First Law of Planetary Motion states that the orbits of planetary bodies are ellipses with the sun at one of the two foci of the ellipse. Unit #5 UCM & Gravity

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Kepler’s 2nd Law 2. Kepler’s Second Law of Planetary Motion states that if you were to draw a line from the sun to the orbiting body, the body would sweep out equal areas along the ellipse in equal amounts of time. This is easier to observe graphically. In the diagram, if the orbiting body moves from point 1 to point 2 in the same amount of time as it moves from point 3 to point 4, then areas A1 and A2 must also be equal. Unit #5 UCM & Gravity

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Kepler’s 2nd Law Earth does not move the same speed each month during its orbit. It moves the fastest in January and the slowest in July because the Kepler’s 2nd Law and the elliptical orbit of Earth. Also, the Earth is the closest to the Sun in January, which has no effect on the winter season, the amount of daylight and energy absorbed by the surface is a function of the tilt of the Earth on its axis. Unit #5 UCM & Gravity

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**Kepler’s 3rd Law (TA/TB)2 = (rA/rB)3 2 = a3**

3. Kepler’s 3rd Law of Planetary Motion, described several years after the first two laws were published, states that the ratio of the squares of the periods of two planets is equal to the ratio of the cubes of their semi-major axes. If this sounds confusing, don’t worry, once again it’s not as bad as it looks. The semi-major axis is the distance of the planet from the sun. What this is really saying, then, is that planets that are closer to the sun (with a smaller semi-major axis) have much shorter periods than planets that are farther from the sun. For example, the planet Mercury, closest to the sun, has an orbital period of 88 days. Neptune, which is 30 times farther from the sun than Earth, has an orbital period of 165 Earth years. (TA/TB)2 = (rA/rB)3 2 = a3 = orbital period (Earth years) a = semi-major axis in astronomical units (AU) Unit #5 UCM & Gravity

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**Kepler’s 3rd Law Example**

See Ex. Prob. P 180 TI = 1.8 days rI = 4.2 units TC = 16.7 days rC = ? Unit #5 UCM & Gravity

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**Kepler’s 3rd Law Complete derivation together to get… T2 = (4π2/GM)r3**

M is the mass at the center of the orbit. This equation combines Kepler’s 3rd Law w/Newton’s Law ofUniversalGravitation. Unit #5 UCM & Gravity

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**Kepler’s Orbital Drawing**

Unit #5 UCM & Gravity

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Sample Problem #1 Question: Given the elliptical planetary orbit shown above, identify the interval during which the planet travels with the highest speed. 1. Interval P1 to P2 2. Interval P3 to P4 3. They are the same. Answer: (1) Interval P1 to P2. Because Area 1 is equal to Area 2, we know that the time interval from P1 to P2 must be equal to the time interval from P3 to P4 by Kepler’s 2nd Law of Planetary Motion. Since the planet travels a greater distance from P1 to P2, it must have the higher speed during this portion of its journey. Unit #5 UCM & Gravity

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Sample Problem #2 Question: The shape of Mars’ orbit around the sun is most accurately described as a: 1. circle 2. ellipse 3. parabola 4. hyperbola Answer: (2) ellipse. The orbits of planets are ellipses with the sun as one of the foci of the ellipse. Note that even though the orbits are best described as ellipses, many of the planetary orbits are “nearly circular.” Unit #5 UCM & Gravity

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Sample Problem #3 Question: Which planet takes the longest amount of time to make one complete revolution around the sun? 1. Venus 2. Earth 3. Jupiter 4. Uranus Answer: (4) Uranus must have the longest orbital period since it is farthest from the sun according to Kepler’s 3rd Law of Planetary Motion. Unit #5 UCM & Gravity

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