7.4 Universal Gravitation

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1 Circular Motion Velocity is a vector quantity, which means that it involves both speed (magnitude) and direction. Therefore an object traveling at a constant speed can still accelerate if the direction is changed. The change in direction is due to a net force on the object. This net force is called a centripetal force which results in centripetal acceleration. Centripetal means toward the center. Therefore both centripetal force, which results in circular motion, and centripetal accelerate are both directed toward the center of the circular motion. The reason that one feels pulled outwards while making a turn in a car is not due to a force but due to one s inertia. The inertia resists the change of motion and therefore the velocity of an object in circular motion is tangent to the curve, or a vector coming straight off of the curvature as shown in the picture below of a mass moving clockwise. The equation for centripetal force is shown below: Centripetal Force = F C = mv2 r Example: A student has a 0.5 kg mass on the end of a 2 m string. The student twirls the mass in a circular motion around their head at a constant speed of 4 m/s. What is the force of tension on the string? F C = mv2 r = (0.5 kg)(4 m/s)2 2 m = 4 N

2 Example: A car that has a mass of 1000 kg makes a turn at a radius of 30 m. If the force of friction needed to keep that car on the road is 2700 N. What speed should the car travel in order to not slip off the road? F C = mv2 r = (1000 kg)(v2 ) 30 m v 2 = (2700)(30) 1000 v = 9 m/s = 81 = 2700 N Period (T) The amount of time (in seconds) it takes for an object to make one full orbital rotation. Frequency (f) The amount of cycles an object makes in one second or Hertz (Hz). frequency = f = 1 Period (T) Example: If a ball on a string takes 5 seconds to make one orbital period, what is its frequency in cycles per second? f = 1 T = 1 5 cycles = 0.20 or 0.20 Hz second

3 Universal Gravitation According to a popular legend, Isaac Newton discovered that gravity was a universal force, or a force that is not just unique to Earth, while sitting under an apple tree on his mother s farm. Newton understood Galileo s concepts of inertia that objects would continue to move in a constant speed in a straight line unless acted on by a nonzero net force. What Newton observed was that in the same way that the apple accelerated downwards due to a gravitational force, the moon in a similar way does not follow a straight-line path, but instead circles the Earth. Therefore, the moon which exhibited circular motion must in fact experience a net force. And so Newton reasoned that the moon and the apple are both falling toward the Earth due to the Earth s gravity. And so Newton proposed that the moon was simply a projectile that was falling at the same curvature as the Earth under the gravitational attraction of the Earth. And the moon is traveling at the exact speed (tangential velocity) and is the correct distance (radius) away from the Earth so that it does not fall into the Earth or get cast out into space. Newton developed his Law of Universal Gravitation: F G = G m 1m 2 r 2 This law states that every object attracts every other object with a force called gravity. The force of gravity is directly proportional to each of the masses (m 1 and m 2 ), and inversely proportional to the square of the distance between them (r). According to Newton s 3 rd Law of Motion, the force of gravitational attraction is the same for each of the two masses (as shown below). The G value in the equation is the universal gravitational constant and is a constant of 6.67 x N m 2 /kg 2. Example: What is the force of gravity of a 50 kg object that is on the surface of the Earth according to Newton s Law of Universal Gravitation? (The Earth has a mass of 6 x kg and a radius of 6.4 x 10 6 m) F G = G m 1m 2 r 2 = (6.67 x )(50)(6 x ) (6.3 x 10 6 ) 2 = 500 N (Do not forget to use your E button on your calculator when putting in x10 # )

4 Example: If a planet has double the mass and double the radius of Earth, what is the force of gravity on that planet compared to Earth? F G = G m 1m 2 r 2 = (2) (2 2 ) = 2 4 = 1 2 The force of gravity on the planet would be ½ of that on Earth (g = 5 m/s 2 ). The mass was doubled but the radius was doubled to the second power or four. Example: Which diagram below best represents the gravitational forces between a satellite, S, and Earth? The third diagram (3) best represents the gravitational forces due to Newton s 3 rd Law of Motion and the Law of Universal Gravitation.

5 Kepler s Laws of Planetary Motion 7.4 Universal Gravitation Most of the scientific world in the 1600 s believed that the planets had circular orbits and the Earth was the center of the solar system (geocentric model or Ptolemaic model of the solar system). An astronomer named Johannes Kepler during that same time used data collected by his mentor Tycho Brahe, and developed three laws governing the motion of planetary bodies in a sun-centered solar system (heliocentric model or Copernican model of the solar system). These laws are still used today as an accurate description of the motion of any planet and any satellite. Kepler s Three Laws of Planetary Motion are: 1. The orbits of planetary bodies are ellipses with the sun at one of the two foci of the ellipse. (Law of Ellipses) 2. If you were to draw a line from the sun to the orbiting body, the body would sweep out equal areas in equal intervals of time. (Law of Equal Areas)

6 3. The ratios of the squares of the periods of any two planets are equal to the ratio of the cubes of their averages distances from the sun. (Law of Harmonies) The orbital period is given in the units of earth years and the average distance is given in astronomical units where 1 au is equal to the distance from the sun to the earth. Amazingly, every planet in our solar system has the same period 2 /distance 3 ratio. If one used exact numbers of orbital period and distance from the sun, the ratio would still be equal to each other. Example: What is the most accurate description of the shape of Mars orbit around the sun? An ellipse Example: Which planet takes the longest amount of time to make one complete revolution around the sun - Venus, Earth, Jupiter, or Uranus? Uranus is farthest from the sun, therefore according to Kepler s 3 rd Law of Planetary Motion it must have the longest orbital period. Example: Calculate the ratio of T 2 /R 3 for each of Jupiter s moons listed in the table below. Indicate whether this confirms or contradicts Kepler s 3 rd Law of Planetary Motion. a x 10-16, b x 10-16, c x 10-16, and d x All of these ratios are approximately the same and it confirms Kepler s 3 rd Law of Planetary Motion.

7 Phases, Eclipses, and Tides Motions of the Moon The moon revolves around Earth and also rotates on its own axis. As the relative positions of the moon, Earth, and sun change, it causes the phases of the moon, eclipses, and tides. The moon rotates once on its axis in the same amount of time as it revolves around the Earth, therefore a day and a year on the moon are approximately the same length. This is the reason why the same side of the moon always faces Earth. Phases of the Moon The moon does not produce light, but simply reflects the light from the sun. The different shapes of the moon visible from Earth are called phases. The moon goes through its whole set of phases each time it makes a complete revolution around Earth. These phases are caused by the relative positions of the moon, Earth, and the sun. The phase of the moon you see depends on how much of the sunlit side of the moon faces the Earth. The phases of the moon are as follows: New Moon The sun (and therefore the sunlit side of the moon) is behind the moon. Waxing Crescent The portion of the observable moon is waxing, or growing into a crescent shape. First Quarter Half of the sunlit side of the moon is observed. Waxing Gibbous As the moon continues to wax, or grow, the visible side is called gibbous. Full Moon The entire sunlit side faces the Earth. Waning Gibbous The visible side is waning or shrinking. Third Quarter Half of the sunlit side of the moon is observed. Waning Crescent Again a crescent is observed, but it is waning.

8 Eclipses The moon s orbit around the Earth is slightly tilted (about 5º) with respect to Earth s orbit around the sun. Therefore, most months neither the Earth s shadow nor the moon s shadow affects one another. However, an ellipse occurs when the moon s shadow is cast onto the Earth or the Earth s shadow is cast onto the moon. The two types of eclipses are a solar eclipse and lunar eclipse. During a new moon, the moon lies between the Earth and the sun and usually the moon travels a little above or below the sun it in the sky. A solar eclipse occurs when the moon passes directly between the sun and the Earth, blocking sunlight from the Earth and casting a shadow on the Earth. The moon s shadow during a solar eclipse is called the umbra, and it is a very dark, cone-shaped part which blocks the suns completely. During this time on Earth, people within the umbra experience a total solar eclipse and the sky is as black as night. Stars are visible in this total solar eclipse as well as the solar corona, the outer atmosphere of the sun. The moon also casts another part of its shadow during this time in a larger part called the penumbra. This is only a partial eclipse and since an extremely bright part of the sun remains visible, it is not safe to look directly at the sun during this partial solar eclipse. A lunar eclipse occurs at a full moon when the Earth is directly between the moon and the sun. The moon is then in the Earth s shadow and it looks dim from Earth. Similar to a solar eclipse, the Earth s shadow has an umbra and a penumbra. When the moon is in the Earth s umbra, you can see a total lunar eclipse. Unlike the precise nature of the solar eclipse, a lunar eclipse can be observed from anywhere on Erath that the moon is visible; therefore it is more likely to observe a lunar eclipse than a solar eclipse.

9 Tides The rise and fall of ocean water which is called tides occur every 12.5 hours (about six hours of rising and then about six hours of falling). This is caused by the combined effects of the gravitational forces exerted by the moon, sun, and rotation of the Earth. High tide occurs primarily when the moon s gravity is strongest due to the closer proximity to the moon. High tide also occurs on the far side of Earth from the moon. Low tides occur between the two high tides.

10 When the gravitational effects of the sun and the moon pull in the same direction (at a new moon and a full moon), their combined forces produce what is called a spring tide. During the moon s first and third-quarter phases, the sun s pull and moon s pull are at right angles and produce a neap tide, which produce very little difference between low and high tides. Both spring tides and neap tides occur twice a month.

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