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Gravity Don’t Let It Get You Down!

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The Truth About Gravity u Gravity is a phenomenon u The phenomenon results in a force which can accelerate objects with mass u “g” represents the acceleration due to the force caused by the phenomenon of gravity

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Back to the Beginning u Astronomy drove our understanding and discovery u It did so without knowing it, however

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The Importance of Being Gravity u Gravity has basic properties that set it apart from the other forces: (1) it is long-ranged and thus can act over cosmological distances; (2) it always supplies an attractive force between any two pieces of matter in the Universe. u Thus, although extremely weak, it always wins over cosmological distances and is the most important force for the understanding of the large scale structure and evolution of the Universe.

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So, let us deal with GRAVITY We’ll need a bit of a history lesson: Brahe Kepler Newton Einstein Pay close attention, gravity has many implications!

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Tycho Brahe A wild Dane, but he made and recorded large quantities of accurate measurements of the motions of the planets around the Sun. 1546 - 1601 Began working with Johannes Kepler in 1600.

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1) The planets move abort the sun in elliptical orbits with the Sun at one focus. 2) The radius vector joining a planet to the sun sweeps over equal areas in equal intervals of time. The empirical discovery of these laws from Tycho Brahe's mass of data constitutes one of the most remarkable inductions ever made in science. T 1 2 / T 2 2 =R 1 3 / R 2 3 or T 2 =k. R 3 3) The square of the time of one compete revolution of a planet about its orbit is proportional to the cube of the orbit's semi-major axis Kepler’s “Laws” of Planetary Motion

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Isaac Newton u Born 1642, the year Galileo died u Loner, tinkerer, paranoid u 1665-1666 Plague was very good for him u Suffered mental breakdown 1675 u Math, Chemistry, Theology, Parliament u Died 1727 u Has his picture on the British pound note He put the physics and mathematics to Kepler’s Laws!

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Newton’s Laws of Motion u First Law - A body remains in its state of motion unless acted upon by an outside force u Second Law - A body acted upon by an external force will change its momentum in the direction of the force such that the greater the force the greater the change in momentum (F= ma). u Third Law - Forces always occur in pairs, i.e. for every action there is an equal and opposite reaction

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Universal Law of Gravitation u All objects in the Universe attract each other with a force that varies directly as the product of their masses and inversely as the square of their separation from each other. F = G m m F = G m m r gravity 1 2 2

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General Relativity u Einstein’s Theory of Gravity, published 1915 u Principle of Equivalence: Accelerations are indistinguishable from gravitational fields. They are equivalent. u So, for example, when you are in freefall (like in an orbiting Shuttle), your downward acceleration is just enough to cancel the gravitational force.

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Einstein’s View of Gravity u Gravity is due to the curvature of spacetime. u Spacetime is curved by mass.

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GR Made Predictions Light would be bent by gravity S Tested by Arthur Eddington during solar eclipse. S Confirmed! u Emission of gravitational radiation by accelerating objects S Tested by observations of binary pulsars. S Confirmed to 14 decimal places!

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Applications of Newton’s 2nd Law and Einstein’s GR u Projectile Motion u Pendulums u Black Holes

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Projectile Motion Projectile - any object given an initial velocity which subsequently follows a path determined by the gravitational force acting on it, and by the frictional resistance of the atmosphere bullet shot from a gun, rocket after the fuel is exhausted, thrown or batted baseball

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Trajectory Trajectory - the path followed by a projectile

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Our Assumptions We will consider only short length trajectories so that: u Gravitational force is considered constant in magnitude and direction u Earth is an inertial system u Air resistance is negligible In other words, we will examine motion in a vacuum on a flat, non-rotating Earth. In physics, we call this creating an

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First, the Forces u Only force acting on the projectile is its weight…remember we are in the Ideal Case. u X-axis is horizontal; y-axis is vertical; origin is point where projectile starts its free flight u So x-component of the force on the projectile is zero and y-component of the force is the weight, mg.

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Acceleration Netwon’s 2nd law then tells us that the x- component of the acceleration is zero and the y- component is -g. In other words, trajectory is a combination of a horizontal motion with constant velocity and a vertical component with constant acceleration.

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Result u Under these conditions, projectiles travel in trajectories which are parabolas. u You can derive the equation for the parabolic motion from Newton’s Laws!

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Let’s Play

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Pendulums mg T mg sin For small angles, sin Simple Harmonic Motion L x Period = 2 L/g Measure period of oscillation and length of pendulum, determine g!

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Black Holes A huge great enormous thing, like — like nothing. A huge big — well, like a — I don’t know — like an enormous big nothing … Piglet describes the Heffalump, Piglet describes the Heffalump, in Winnie the Pooh by A.A. Milne in Winnie the Pooh by A.A. Milne

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History u Based on Newton’s theory of gravity u Proposed independently by: –1783 Rev. John Mitchell –1796 Pierre Simon Laplace

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Structure of a Blackhole

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Artist’s View

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HST View

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Calculation of Critical Radius

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Calculation Continued

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Earth: normal size Earth: Normal Size

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Earth: half size Earth: Half Size

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Earth: Quarter Size

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Earth: Black HoleR bh = 2GM/c 2 R = 9 mm

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Earth as Depression in Spacetime

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Blackhole is Bottomless Abyss in Spacetime

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Bending of Light D D R photon d d is the approximate distance the photon falls over the diameter D of an object with mass M

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From Newton, We Know... F = ma = GMm/R 2 Solving for a, a = GM/R 2 If you accelerate at rate a for time t, you move a distance d of d = 1/2 at 2 Recall that t = D/c. Putting everything together, we see: d = 1/2 GMD 2 /(c 2 R 2 )

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Solving the Equations D D R d d = 1/2 GMD 2 /(c 2 R 2 ) but D=2R and = d/D RESULT: = GM/c 2 R Using GR, we would get 4 GM/c 2 R!

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Making the Prediction When you plug in the values for G, c, and the mass and radius of the Sun, you predict that light should be bent by 8.5 x 10 -6 radians. u This corresponds to 1.75 seconds of arc. u Eddington measured 1.75…probably.

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Light Benders, Mind Benders

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Mass and Weight u Weight is a force. It is the resultant gravitational force exerted on a body with mass m by all the other bodies on the Universe. u Near Earth’s surface, the gravitational force from our planet dominates all others, so according to Newton’s Laws, we can write: Weight = F g = G m M e / R 2 = mg where M e is the mass of the Earth and R is the radius of the Earth.

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But also... u Since weight is a force, it is measured in units of force, namely Newtons. Remember what mass is measured in units of?

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Remember This! u Mass is a fundamental, universal property. You have the same amount of mass no matter where you are in the Universe. The only thing that can change your mass is velocity…according to General Relativity. As you approach the speed of light, you become infinitely heavy. u Weight is not fundamental; its value depends on what your circumstances are in the Universe.

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© 2004 Pearson Education Inc., publishing as Addison-Wesley 5. Universal Laws of Motion “ If I have seen farther than others, it is because I have stood.

© 2004 Pearson Education Inc., publishing as Addison-Wesley 5. Universal Laws of Motion “ If I have seen farther than others, it is because I have stood.

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