Unit #1: Dynamics. Introduction. Introduction

Transcription

1 Unit #1: Dynamics Lesson #14: Universal Gravita7on 1/26 Introduction Before Newton developed the theory of universal gravita7on, there were two separate no7ons of gravity. Terrestrial gravity was thought to be the force holding things to the surface of the Earth. Astronomical gravity which was responsible for the mo7on of the planets, moons and comets. Newton showed that these two forces were actually the same; he unified terrestrial and astronomical gravity. 2/26 Introduction Newton, (perhaps upon observing an apple fall from an apple tree) thought: if the apple accelerates toward the ground, there must be a force that acts on it. Let's call this force "the force due to gravity", and its accelera7on we ll call the accelera7on due to gravity". Also, if the apple tree was twice as high, the apple would s7ll accelerate toward the ground, so the force due to gravity must reach to the top of the tallest apple tree. Perhaps, the force of gravity reaches even further, perhaps all the way to the Moon! This would mean that the orbit of the Moon around the Earth is caused by the force of gravity. 3/26 1

2 Newton s Thought Experiment If we fire a cannonball horizontally from a tall building, it will eventually fall to earth along the trajectory marked 1. As we increase the muzzle velocity of the cannon, the projec7le will travel further before returning to earth. Newton reasoned that if the cannon projected the cannonball with exactly the right velocity, the cannonball would travel completely around the Earth, always falling in the gravita7onal field but never reaching the Earth. That is, the cannonball is put into orbit around the Earth. 4/26 The Universal Force of Gravitation Newton concluded that the Moon con7nuously "falls" in its orbit around the Earth because of the force due to gravity. By such reasoning, Newton concluded that terrestrial and astronomical gravity were the same and that any two objects in the Universe exert gravita7onal a]rac7on on each other, with the force having a universal form: F G =G m 1 m 2 / r 12 2 Since G is the same everywhere, the force is universal 5/26 Weight and The Gravitational Force In everyday conversa7on, the terms mass and weight are oaen used to mean the same thing. In reality they are quite different. Weight is defined as: the gravita7onal force exerted on an object of a certain mass by another mass. Mathema7cally, the weight is: F G =G m 1 m 2 / r 12 2 where one of the masses is the mass of the Earth. 6/26 2

3 Weight and The Gravitational Force We can also define the accelera7on due to gravity as: g=g M/ r 2 So the weight of a mass m at the surface of the Earth is obtained by mul7plying the mass m by the accelera7on due to gravity, g, at the surface of the Earth. F g =mg So the mass of an object is a constant but the weight depends on its loca7on. If we transport it to the Moon, the gravita7onal accelera7on would change because the radius and mass of the Moon both differ from those of the Earth, resul7ng in a change of the object s weight. 7/26 Weighing the Earth In 1783, Henry Cavendish became interested in devising an experiment to "weigh the earth." Borrowing an idea from the French scien7st Coulomb who had inves7gated the electrical force between small charged metal spheres, his friend Rev. John Michell suggested using a torsion balance to detect the 7ny gravita7onal a]rac7on between metal spheres Michell set about construc7ng an appropriate apparatus however, he died in 1793 before conduc7ng experiments with it. 8/26 Weighing the Earth Cavendish later rebuilt most of the apparatus. His balance was constructed from a 2 metre wooden rod suspended by a metal fiber, with 5 cm. diameter lead spheres mounted on each end of the rod. These were a]racted to 160kg. lead spheres brought close to the enclosure housing the rod, roughly as shown below. 9/26 3

4 Weighing the Earth He published his result in 1798 that the average density of the earth is mes that of water. His work was done with such care that this value was not improved upon for over a century. The modern value for the mean density of the earth is mes the density of water. Cavendish's experiment is oaen described as the first experiment in modern physics. 10/26 Apparent Weight In yesterday s lesson, we discussed a rota7ng reference frame and introduced the idea of fic77ous forces that are used to account for mo7on in these non- iner7al reference frames. Today, we will consider the case of a reference frame that accelerates in a straight line, specifically, we will consider the case of an elevator that accelerates. 11/26 Apparent Weight In our daily experience, our apparent weight is equal to the Normal force applied by the floor upwards on our feet (This is what a bathroom scale measures). When you ride in an elevator, the normal force you feel varies which you interpret as your apparent weight. Three situa7ons are worth examining here: The elevator is at rest or moving at a constant speed The elevator accelerates upward The elevator accelerates downward 12/26 4

5 Elevator Case #1 When the elevator is at rest or moving at a constant speed, the normal force you feel is exactly equal to your weight and you feel normal This is a non- accelera7ng reference frame; an iner7al reference frame 13/26 Elevator Case #2 When the elevator accelerates upward, the normal force is greater than it usually is so you feel heavier. This is the first case of a non- iner7al reference frame. 14/26 Elevator Case #3 When the elevator accelerates downward, the normal force is less than it usually is so you feel lighter In the extreme case, the elevator cable snaps and you feel no normal force. This is oaen described as weightlessness 15/26 5

6 Example #1 Eg.#1 Determine the apparent weight of a 50. kg. person standing in an elevator, a) that is mo7onless In this case, the normal force is equal and opposite to the weight of the person. F N =mg =50.kg. 9.8 m. s. 2 =490N. 16/26 Example #1 b) That accelerates upward at 2.0m./s. 2. F NET =m a F N + F G =m a F N =m a m g =50.kg. 2.0 m. s. 2 [up] 50.kg. 9.8 m. s. 2 [down] =50.kg. 2.0 m. s. 2 [up]+50.kg. 9.8 m. s. 2 [up] =590N.[up] This is greater than the weight of the person so he will feel heavier as expected. 17/26 Example #1 c) That moves upward at 1.5m./s. In this case, the normal force is equal and opposite to the weight of the person. F N =mg =50.kg. 9.8 m. s. 2 =490N. Again, in this case, the person feels their normal weight 18/26 6

7 Example #1 d) That is in free- fall F NET =m a F N + F G =m a F N =m a m g =50.kg. 9.8 m. s. 2 [down] 50.kg. 9.8 m. s. 2 [down] =0N. This is the situa7on described earlier that is oaen described as weightlessness 19/26 Example #2 Eg.#2 What distance above the Earth s surface must a satellite be in order to be geosynchronous (the radius of the Earth is 6378 km.)? F NET =m a G mm/ r 2 =m v 2 /r G M/r = v 2 G M/r = ( 2πr/T ) 2 G M/r = 4 π 2 r/ T 2 2 GM T 2 /4 π 2 = r 3 3& GM T 2 /4 π 2 =r 20/26 Example #2 r= 3& N m. 2 kg kg. ( s.) 2 /4 π 2 = m. h=r r E = m m. = m. This is approximately 6 earth radii, a lot higher than the Shu]le ever goes (usually about 300 kilometers). 21/26 7

8 Global Positioning System GPS, which stands for Global Posi7oning System, is the only system today able to show you your exact posi7on on the Earth any7me, in any weather, anywhere to within about 10 metres. Even greater accuracy, usually within less than one metre, can be obtained with correc7ons calculated by a GPS receiver at a known fixed loca7on. GPS has 3 parts: the space segment (the satellites), the control segment (5 ground sta7ons to ensure the satellites are working properly) and the user segment (the handheld or car mounted receiver) 22/26 GPS Satellites The complete GPS space system includes 24 satellites (+ 5 spares), 20,000 km. above the Earth, which each take 12 hours to orbit the Earth. They are posi7oned so that signals from at least 4 of them can be received nearly 100 percent of the 7me at any point on Earth. Six signals are required in order to get the best posi7on informa7on. 23/26 GPS Satellites G.P.S. satellites are equipped with very precise clocks that keep accurate 7me to within three nano- seconds. This precise 7ming is important because the receiver must determine exactly how long it takes for signals to travel from each GPS satellite. The receiver uses this informa7on to calculate its posi7on. 24/26 8

9 Control Segment The control segment is composed of: a master control sta7on, located near Colorado Springs in El Paso County, Colorado an alternate master control sta7on, four dedicated ground antennas, and six dedicated U.S. monitoring sta7ons in Hawaii, Kwajalein Atoll, Ascension Island, Diego Garcia, Colorado Springs, Colorado and Cape Canaveral. Its purpose is to monitor each satellite s atomic clock and posi7on informa7on and adjust the posi7on of each satellite. 25/26 User Segment GPS receivers detect, decode, and process GPS satellite signals. Many different receiver models are in use. The typical hand- held receiver is about the size of a cellular telephone, and newer models are even smaller. 26/26 9