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Physics I Class 17 Newton’s Theory of Gravitation Rev 03-Nov-04 GB 17-1 Forces Known to Physics There are four fundamental forces known to physics: • Gravitational Force (today) • Electromagnetic Force (later in Physics and 2) • Weak Nuclear Force • Strong Nuclear Force (All forces we observe are comprised of these fundamental forces Most forces observable in everyday experience are electromagnetic on a microscopic level.) 17-2 Forces in Physics (so far) We have encountered the following forces in Physics 1: • Gravity • Ideal Springs (Hooke’s Law) • Pushes and Pulls • Friction What makes gravity different from the other three? (Hint: The ideal spring force is also conservative, so that isn’t the answer.) 17-3 Newton’s Theory of Gravitation Isaac Newton, 1642-1727 In 1666, our old friend, Isaac Newton, was musing on the motions of heavenly bodies while sitting in a garden in Lincolnshire England, where he had gone to escape the plague then ravaging London What if the force of gravity, the same force that causes an apple to fall to the ground in this garden, extends much further than usually thought? What if the force of gravity extends all the way to the moon? Newton began to calculate the consequences of his assumption… 17-4 Newton’s Law of Universal Gravitation  m1 m F = G rˆ r The meaning of each term:  F: G: m1 : m2: r 2: rˆ : Gravitational force on object from object –11 2 Universal gravitational constant = 6.673 x 10 N m /kg Mass of object Mass of object Center distance from object to object 2, squared Unit vector from object to object 17-5 Properties of Gravity Object Gravitational Force on from Object • • • • • Every object with mass is attracted by every other object with mass Gravity is a force at a distance (through occupied or empty space) Gravity is a “central” force (center-to-center for spherical bodies) Gravity varies as the inverse square of the center distance Gravity varies as the product of the masses 17-6 If Gravity Varies As 1/r2, Where Does g = 9.8 m/s2 Fit In? Consider the force on an object near the surface of the earth (Assume the earth is a sphere and ignore rotation effects.) R = radius of the earth M = mass of the earth m = mass of the object  mM GM  F = G rˆ = m rˆ = m g (What is the direction?) R R g = 9.8 m/s2 only seems constant because we don’t go very far from the surface of the earth 17-7 Gravity is a Conservative Force Both the mathematical form of Newton’s Law of Universal Gravitation and experimental evidence show that gravity is a conservative force Therefore, we can find a gravitational potential energy for an object with mass m being attracted by another object with mass M The gravitational potential energy is defined (for convenience) to be zero at infinity We can calculate it by finding the positive work from any point to infinity – you can find the details in the book in section 13-6 ∞ ∞   GmM GmM ′ U g (r ) = ∫ F ⋅ d r ′ = − ∫ d r = − ′ ( r ) r r r 17-8 We Have Two Formulas for Gravitational Potential Energy! Old: New: U g ( y) = m g ( y − y ) GmM U g (r ) = − r How could these be the same? Consider a location near the surface of the earth, y0 = R, y = R+h The only thing that matters is ∆U, not U itself Old: New: (h [...]... wrong with Newton’s theory of gravity, because it implied that the force of gravity is transmitted instantaneously to all points in the universe This contradicts the fundamental limitation in the Theory of Special Relativity that the fastest speed information or energy of any type can travel is the speed of light To overcome this problem Einstein postulated a third principle, the Principle of Equivalence,... The Law of Periods: The square of the period of any planet about the Sun is proportional to the cube of the semi-major axis of its orbit Newton showed through geometrical reasoning (without calculus) that his Law of Universal Gravitation explained Kepler’s Laws 17-15 Kepler’s Three Laws of Planetary Motion Try this link to see an animation: http://home.cvc.org/science/kepler.htm 17-16 The Law of Areas... Newton’s Law of Universal Gravitation Consider the implications of Newton’s formula Practice calculating gravitational force vectors 17-13 Class #17 Optional Material Part A - Kepler’s Laws of Orbits Material on Kepler’s Laws thanks to Professor Dan Sperber 17-14 Kepler’s Three Laws of Planetary Motion 1 The Law of Orbits: All planets move in elliptical orbits having the Sun at one focus 2 The Law of Areas:... that of earth and the elevator is accelerating upward at 4.9 m/s2 According to Einstein, there is no experiment that could detect a difference 17-22 The Principle of Equivalence 17-23 General Theory of Relativity By 1915, Einstein had worked through all the math (with some help) to show that his postulates led to a new theory of gravity based on the effect of mass and energy to curve the structure of. .. of Earth, the mass of the planet is: A 2 M B 4 M C 8 M D 16 M E 32 M 17-11 Class #17 Problems of the Day 2 Calculate the acceleration due to gravity at the surface of the planet Mars Assume Mars is a perfect sphere and neglect effects + 23 +6 due to rotation Use M = 6.4 ×10 kg and R = 3.4 ×10 m 17-12 Activity #17 Gravitation (Pencil and Paper Activity) Objective of the Activity: 1 2 3 Think about Newton’s. .. space and time His theory has some startling implications, one being the existence of “black holes” – regions of space where the gravity field is so high that even light cannot escape The predictions of General Relativity, including the existence of black holes, have been confirmed by all experiments to date 17-24 Black Holes Black holes are detected by the characteristic x-rays given off by matter falling...Class #17 Problems of the Day _1 To measure the mass of a planet, with the same radius as Earth, an astronaut drops an object from rest (relative to the surface of the planet) from a height h above the surface of the planet (h is small compared to the radius.) The object arrives at the surface with a speed that is... problem Einstein postulated a third principle, the Principle of Equivalence, to go with his two principles of Special Relativity (1907) 17-21 The Principle of Equivalence In broad terms, the Principle of Equivalence states that there is no experiment that one can perform to distinguish a frame of reference in a gravitational force field from one that is accelerating with a corresponding magnitude and... Motion Try this link to see an animation: http://home.cvc.org/science/kepler.htm 17-16 The Law of Areas ∆A = 21 (r ∆θ )r dA 1 2 dθ 1 2 = 2r = 2r ω dt dt L = constant L = rmv⊥ = rmω r L = mr ω 2 The Law of Periods F = ma GMm 2 = mω r 2 r 2 GM 2π   2 =ω = 3  T  r 2 ( 2 π ) T2 = r3 GM ENERGY IN CIRCULAR ORBITS GM K = mv = m r GMm K= 2r GMm U =− r GMm E =U + K = − 2r 1 2 2 1 2 Class #17 Optional Material... including the existence of black holes, have been confirmed by all experiments to date 17-24 Black Holes Black holes are detected by the characteristic x-rays given off by matter falling into them 17-25 If Newton’s Gravity isn’t true, why do we still use it? It’s a good approximation for most engineering purposes Massive Black Holes In Galaxies NGC 3377, NGC 3379 And NGC 4486B 17-26
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