Gravitation and Newton s Synthesis

Transcription

1 Gravitation and Newton s Synthesis Vocabulary law of unviversal Kepler s laws of planetary perturbations casual laws gravitation motion casuality field graviational field inertial mass gravitational mass principle of equivalence black hole 6-1 Newton s Law of Universal Gravitation Whenever a body has a force exerted on it, that force is exerted by some other body what exerts the force of gravity Newton concluded that it must be the earth itself that exerts the gravitational force on objects at its surface At the surface of the Earth, the force of gravity accelerates objects at 9.80 m/s^2 What about the moon? You could determine the a based on the concept of centripetal acceleration You would discover that the acceleration of the Moon toward the Earth is 1/3600 as great as the acceleration of objects at the Earth s surface Newton concluded that the gravitational force exerted by the Earth on any object decreases with the square of its distance r from the Earth s center Equation Box 6-1 Newton realized that the force of gravity on an object depends not only on distance but also on the object s mass In fact it is directly proportional to its mass Newtons third law- Earth exerts a force and the moon exerts a force on the Earth Because of the symmetry, Newton argued that the force of gravity must be proportional to both the masses Equation Box 6-2 Newton went a step further in his analysis of gravity 1 RoessBoss

2 He concluded based on the orbits of the planets that the force required to hold the different planets in orbit around the sun seems to diminish as the inverse square of their distance from the sun This led him to believe that it is also the gravitational force that acts between the Sun and each of the planets to keep them in their orbits And if gravity acts between these objects, why not in all Law of Universal gravitation- every particle in the universe attracts every other particle with a force that is proportional to the product of their masses and inversely proportional to the square of the distance between them. This force acts along the line joining the two particles The magnitude of the gravitational force can be written as Equation Box 6-3 G is the universal constant which must be measured experimentally and has the same numerical value for all objects The value os G must be very small, since we are not aware of any force of attraction between ordinary sized objects such as two baseballs The force was measured 100 years after Newton by Henry Cavendish in 1798 to detect and measure the incredibly small force, he used an apparatus with two masses on a string hanging from a rod which was hanging from another string He determined G to be 6.67 E -11 N m^2/ kg^2 usually r will be measured from the center, but some objects make it very hard to see that With two spheres that are uniform in shape the correct place to measure is from the center When dealing with particle- one has to use calculus 6-2 Vector Form of Newton s Law of Universal Gravitation Since you are dealing with a force you can still use the vector rules in determining resultants when you have various directions involved. It is also helpful when you have many particles involved 6-3 Gravity Near the Earth's Surface; Geophysical Applications Now lets look at the relationship between weight and Universal Gravity Law 2 RoessBoss

3 Equation Box 6-4 Thus the acceleration of gravity at the surface of the Earth is determined by the mass of the earth and the radius of the earth Once G was determined then we could determine g When you do the math you discover that g is 9.80 m/s^2 You can also determine the mass of the Earth which is 5.98 E24 Kg Keep in mind that g changes based on location because Earth is not a perfect sphere The higher you are the less g is Generally we use 9.80 since it is a good round about estimate overall Geophysicists use the value of g and how it changes based on the Earth s crust in order to determine where mineral deposits are, salt domes, oil, and other resources 6-4 Satellites and Weightlessness Artificial satellites circling the Earth are commonplace A satellite is placed in orbit by accelerating it to a sufficiently high tangential speed with the use of rockets If the speed it too high, the spacecraft will not be confined to Earth s gravity and will escape, never to return If the speed is too low, it will return to Earth Satellites are usually put into circular orbits because they require the least takeoff speed If a satellite stopped moving it would fall back to earth At high speed it would fly into space Earth s gravitational force pulls it into orbit A satellite is actually falling but it has high tangential speed which keeps it from hitting earth For satellites that move in circles, the needed acceleration is v^2/r The force that gives a satellite this acceleration is the force of gravity and since a satellite may be at a considerable distance from the earth- we must use the equation for the force acting on it Equation Box RoessBoss

4 Note that only one force- gravity- is acting on the satellite and that r is the sum of the Earth s radius plus the satellites height h above the earth People and other objects in a satellite circling the Earth are said to experience weightlessness You can feel this in an elevator If you dropped a pencil in an elevator that was falling- the pencil would be falling with acceleration g but so would the floor of the elevator and the person The pencil would hover in-front of the person- apparent weightlessness- because in fact gravity is still acting on the object and the weight is still mg It is only weightless in that it is in free fall Prolong effects of weightlessness can have adverse effects on the human body like diminished blood cells, bone structure etc 6-5 Kepler s Law and Newton s Synthesis Johannes Kepler describes the motion of the planets about the sun Kepler used Tycho Brahe s data of mars to derive these laws Keplers laws of planetary motion o First Law- The path of each planet about the sun is an ellipse with the sun at one focus o Second Law- Each planet moves so that an imaginary line drawn from the sun to the planet sweeps out equal areas in equal periods of time o Third Law- the ratio of the squares of the periods of any two planets revolving about the sun is equal to the ratio of the cubes of their semimajor axes (the semimajor axes is half the long axis of the orbit and represents the planets average distance from the sun) That is if T1 and T2 represent the periods for any two planets and s1 and s2 represents the semi major axes then Equation Box 6-6 The s^3/t^2 are the same for each planet Planetary data applied to Keplerʼs Third Law Planet Average Distance from sun, s (10^6) Period T (Earth Years) s^3/t^2 (10^24 km^3/y^2) Mercury Venus 4 RoessBoss

5 Planetary data applied to Keplerʼs Third Law Earth Mars Jupiter Saturn Uranus Neptune Pluto Kepler arrived at his laws through careful analysis of experimental data Newton showed that keplers laws could be derived mathematically from the law of universal gravitation and the laws of motion He also showed that for any reasonable form for the gravitational force law, only one that depends on the inverse square of the distance is fully consistent with Kepler s law He thus used Keplers laws as evidence in favor of his law of universal gravitation If you go through the work you will arrive at a relationship between Keplers and Newtons which is based on the period and the radius Equation Box 6-7 This is also another interpretation of Kepler s third law You can not compare Earth-Moon to Sun-Mars... The gravity attraction is based on separate things Measurements on the orbits of the planets indicated that they did not precisely follow Kepler s laws there are slight deviations from perfectly elliptical orbits This is due to bodies interacting with one another as they go through their orbit But because of these small forces each planetary orbit should depart from a perfectly ellipse orbit- these deviations are perturbations These perturbations also have lead to the discovery of other planets like Neptune and pluto Planets around other stars also wobble The laws formulated by Newton are referred to as casual laws Causality means the idea that one occurrence can cause another 5 RoessBoss

6 6-6 Gravitational Field most of the forces we meet in everyday life are contact forces- push or pull on a lawn mower etc gravitational force acts over a distance- there is a force even when the two objects are not in contact The idea of a force acting at a distance was a difficult one for early thinkers- Newton himself felt uneasy with this concept when he published his law Another point of view that avoids some of these conceptual difficulties is the concept of the field- developed by Michael Faraday to aid in the understanding of the electromagnetism Later on it was applied to gravity According to the field concept a gravitational field surrounds every body that has mass and this field permeates all of space A second body at a particular location near the first body experiences a force because of the gravitational mass is considered to act directly on this mass (we are closer to the idea of a contact force) We can quantitatively define the gravitational field as the gravitational force per unit mass at any point in space If we want to measure the gravitational field at any point- we place a small test mass m at that point and measure the force exerted on it Equation Box 6-8 The units of g are N/kg It is clear that the gravitational field which an object experiences has magnitude equal to the acceleration due to gravity at that point 1 N = 1 kg m/s^2- which is why we use the units m/s^2 6-7 Types of Forces in Nature Newton s law of universal gravitation depends on the distance between the masses of the objects involved Newtons second law on the other hand tells how a body will accelerate due to any type of force in the twentieth century physicist came to recognize four different fundamental forces in nature o the gravitational force o the electromagnetic force o the strong nuclear force o weak nuclear force 6 RoessBoss

7 Physicists have been working on theories that would unify these four forces- that is to consider some or all of these forces as different manifestations Electromagnetic and weak nuclear forces have been theoretically united to form successful electroweak theory in which the electromagnetic and weak forces are seen as two different manifestations of a single electroweak force Attempts to further unify the forces such as grand united theories are hot topics currently ordinary forces fit into the electromagnetic forces 6-8 Gravitational Versus Inertial Mass; the principle of equivalence Newton s second law relates the force acting on a body to its acceleration and its inertial mass as we call it Inertial mass- represents a resistance to any force we have dealt with mass as a property related to the gravitational force- that is mass as a quantity that determines the strength of a gravitational force between two bodies- gravitational mass Newtons and Cavendish s experiments indicated that these two types of mass are equal for a body and modern experiments have confirmed it The equivalence between gravitational and inertial masses was raised by Albert Einstein to a principle of nature Einstein called it simply the principle of equivalence He used it for his foundation for his general theory of relativity There is no experiment observers can perform to distinguish if an acceleration arises because of gravitational force or because their reference frame is accelerating How the body reacts to inertia or gravity is indistinguishable 6-9 Gravitation as Curvature of Space; Black Holes The principle of equivalence can be used to show that light ought to be deflected due to gravitational force of a massive body consider an elevator in free space where no gravity acts on it If there is a hole in the side of the elevator and a beam of light enters from outside, the beam travels straight across the elevator and makes a spot on the opposite side if the elevator is at rest If the elevator is accelerating upward then the beam still travels straight However our frame of reference since the elevator is moving it appears that the light has bent According to the equivalence principle an upward accelerating reference frame is equivalent to a downward gravitational field So we expect gravity to exert a force on a beam of light and to bend it out of a straight line Einstein in his general theory of relativity predicted that- light would be affected by gravity 7 RoessBoss

8 It was calculated that light from a distant star would be deflected by 1.75 of arc as it passes near the sun Such a deflection was measured in 1919 during an eclipse of the sun The eclipse reduced the brightness of the sun so that the stars in line with its edge at that moment would be visible A light beam travels the most direct path between two points If a light beam can follow a curved path then the curved path must be the shortest distance between two points Thus space must be curved and that it is the gravitational field that causes the curvature The curvature is greatest near very massive bodies The extreme curvature of space time could be produced by a black hole black hole is a hugely massive star that is so dense and so massive that gravity would be so strong that even light could not escape it Light would be pulled back in by the force of gravity since no light escapes it- it is black A body might pass by it and be deflected by its gravitational field, but if the body came too close, it would be swallowed up never to escape one theory is that there is a giant black hole in the center of our universe 8 RoessBoss