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# PH125 Ch. 13&15 Notes Physics 125

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This 9 page Class Notes was uploaded by Nora Salmon on Sunday March 29, 2015. The Class Notes belongs to Physics 125 at University of Alabama - Tuscaloosa taught by Prof. Andreas Piepke in Fall2015. Since its upload, it has received 223 views. For similar materials see Physics 1 w/Calculus in Physics 2 at University of Alabama - Tuscaloosa.

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Date Created: 03/29/15

131 132 0 0 Chapter 13 What is Physics The gravitational force doesn t just hold us to earth It holds the moon in orbit around the earth our solar system in orbit around the sun and even holds the Local Group of galaxies including the Milky Way and Andromeda together Gravitation is also responsible for black holes when a huge star burns out the gravitational forces between its particles can make the star collapse in on itself and form a black hole Newton s Law of Gravitation Newton theorized that every body in the universe attracted every other body This is called universal gravitation and causes a tendency of objects to move toward one another Newton39s law of gravitation states that the attraction two particles feel toward one another is a quanti able force and has magnitude given by the equann mlm2 2 r 0 Here m1 and m2 are the masses of the two particles r is the distance between them and G is a gravitational constant G6671011Nm2kg2 The force of gravitation is a vectorial quantity directed along an axis r Imagine an apple sitting on the earth s surface The apple experiences an attraction force to the earth along the raxis and the earth experiences an attraction force to the apple of equal magnitude but opposite direction 0 Note this follows Newton s 3rCI law of equal but opposite force pairs Particles vs Objects We can apply the law of gravitation not only to particles but also to objects as long as the objects are small in comparison to the distance between them 0 For example the moon and the earth very far apart so we can treat them like particles 0 But what about the appleearth relationship o FG Chapter 13 0 Newton also had a solution for this the shell theorem 0 The shell theorem states that a uniform spherical shell of matter attracts a particle or object outside that shell as though the shell s mass were concentrated at its center 0 Therefore the earth acts as a particle with all its mass located at its center 0 Gravitational force vs acceleration Recall Fma The apple and the earth produce equal and opposite forces on one another however their masses are quite different If the apple and earth are separated the acceleration of the earth toward the apple is therefore much smaller in magnitude than the acceleration of the apple toward the earth 0 This is because the system must obey Newton s 2ncl and 3rd laws 0 59 e 5 mapple aapple mearth aearth o The system has a proportionality o 133 Gravitation and Superposition o In a system of a group of particles we can nd the net gravitational force on any one of the particles from the other particles with something called the principle of superposition This basically says that the quotnet effect on an object is the sum of individual effects As with any other net force we can compute individual forces and then add them vectorially to nd the net gravitational force The equation given is 17 1t277127713l71n Here F12 is the force on particle 1 by particle 2 F13 is the force on particle 1 by particle 3 and so on 0 We can use this principle to quantify the gravitational force on a particle by a large object Divide the object into small parts treating each part like particles we can nd the differential sum of the vector forces of each part of the object acting on the particle We then have an integral 51de the book drops the subscript quotnetquot 0 134 Gravitation Near Earth s Surface Chapter 13 0 When calculating gravitation near earth s surface assume that Earth is a uniform sphere Particles near earth when released fall toward the center of the earth with gravitational acceleration 98 ms2 o If we substitute gravitational acceleration into Newton s 2nCI law we get Fmag Now solve for acceleration agFm If we take F to be Fgrav and solve for ag in the GM graVItational force equation we get ag r2 where M is the mass of the earth 0 The gravitational acceleration of a particle is not the same as its freefall acceleration g This is for three reasons 0 1 The earth s mass is not uniformly distributed the density of the earth varies from its care to its mantle and over different regions of the surface 0 2 The earth is actually not a sphere it is an quotellipsoidquot the radius at the equator is larger than the radius from the center of the earth to one of the poles 3 The earth is rotating a particle or object located on the earth s surface anywhere but one of the poles rotates around a rotation axis and thus has a centripetal acceleration also directed toward the center of the earth 0 135 Gravitation lnside Earth 0 Newton s shell theorem can be expanded for particles that exist inside rather than outside a uniform shell A uniform shell of matter exerts no gravitational force on a particle inside the shell This means that the sum of the force vectors on the particle from the various elements of the shell is zero 0 As an object descends through the surface of the earth the gravitational force acting on the object increases to a maximum point then decreases after that as the object continues descending 136 Gravitational Potential Energy 0 Suppose that the distance between two objects was so large we could consider it in nite Chapter 13 Then taking gravitational potential energy to be U GMm r be negative for any nite distance between the two particles becoming more and more negative as the distance between the particles decreases o If a system contains more than two particles we can calculate the gravitational potential energy of each pair of particles and then algebraically sum the results Gm1m2Gm2m3Gm1m3 UO for an in nite distance then U will We have U r12 r23 r13 0 Path Independence The gravitational force is a conservative force therefore the work done on a particle by F9 doesn t depend on the path between the points 0 We can then use AUz W 0 Potential Energy and Force dU d th n F dr e dr Since F 2 r r GMm GMm 0 Escape Speed The minimum initial speed an object must have to cause it to escape a planet s gravitational force and thus travel for in nity is called the escape speed 0 At in nity the projectile s kinetic energy is zero because it has no velocity at in nity it has no potential energy either so EO 0 Therefore rearranging the conservation of energy equation we get an equation for escape speed 2GM 0 v R o 137 Kepler s Laws 0 Kepler s three laws hold for planets orbiting the sun as well as satellites orbiting any massive central body like the sun or the earth etc Law 1 The Law of Orbits o All planets move in elliptical orbits with the sun at one focus Explained Assume that the mass of the sun is larger than the orbiting body m then the center of mass of the system is approximately at the center of the sun Chapter 13 o The body m moves around the sun in an ellipse with a certain eccentricity that describes the ellipses deviance from the path of a perfect circle An eccentricity of 0 corresponds to a circle Law 2 The Law of Areas A line that connects a planet to the sun sweeps out equal areas in the plane of orbit in equal time intervals 0 AKA dAdt is a constant This law tells us that a planet moves most slowly when it is farthest away from the sun and most quickly closest to the sun o It is another demonstration of the law of conservation of angular momentum see p 343 for an algebraic proof For a planet of mass m Lmr2w where L is angular momentum 0 Given Kepler s 2nCI law we have d Azi and L is then constant if dAdt dt n is constant which con rms conservation of angular momentum Law 3 The Law of Periods The square of the period of a planet is proportional to the cube of the semimajor axis of its orbit 4H2 3 r 0 T2 GM 138 Satellites Orbits and Energy 0 As a satellite orbits Earth in an elliptical path both its speed and its distance from the center of earth uctuate the mechanical energy of the satellite however remains constant To nd KE use Newton s 2nCI Law 1 2 KEz mv 2 GMm v2 GMm 2 Fmw 2 m 9 mv r r r 1 2 Emv 2r Chapter 13 This shows us that K05U for a satellite in a circular orbit Then since E K U then GMm GMm GMm 2r r 2r So total energy EK for a satellite in a circular orbit o For a satellite in an elliptical orbit we can substitute a for GMm 2a This shows us that the energy of an orbiting satellite does not depend on the eccentricity of its orbit only on the semimajor axis 0 139 Einstein and Gravitation o Einstein quotIf a person falls freely he will not feel his own weightquot He used this thought to begin forming the general theory of relativity It is a theory about gravitation with one fundamental point the principle of equivalence which states that gravitation and acceleration are equivalent 0 Curvature of space Einstein postulated that gravitation is due to a curvature of space caused by the two masses gravitating toward one another E the semimajor axis for r E IDS Chapter Fifteen What is Physics Two primary goals of physics and engineering are studying and controlling oscillations In this chapter we ll discuss a kind of oscillation called simple harmonic motion Simple Harmonic motion All oscillatory motion have a frequency or number of oscillations per second Measured in Hz where 1 Hz 1 oscillations The inverse of the frequency is the period T of the motion which is the time it takes to complete one oscillation Periodic motion Motion that repeats itself at regular intervals is called periodic or harmonic motion For harmonic motion of a particle we can model the displacement of the particle at a function of time 0 Such functions have an amplitude which is a constant that depends on how the motion was started it is the maximum displacement of the particle in either direction 0 For harmonic motion we have the equation xtxm cos wt Here xt is the displacement at time t xm is the amplitude w is the angular frequency t is the time and phi is the phase constant or phase angle The phase constant depends on the displacement and velocity of the particle at time t0 The angular frequency of the motion is also a constant based on the frequency of the motion oo2nf 0 Velocity We can find the velocity of a particle moving in harmonic motion by differentiating our equation for displacernent dx t vt dt 0 This function has an amplitude called the velocity amplitude 2 wxm sin hunch o Acceleration We can also nd acceleration of harmonic motion we just differentiate the velocity function Chapter Fifteen 39 tdltt2 002meOSUOtI 002xt The acceleration is proportional to displacement but opposite in sign 0 153 The Force Law for Harmonic Motion 0 Using Newton s 2ncl law Fma m002x Looks familiar right That s because we ve already learned about a similar function Hooke s law which is an alternate de nition of simple harmonic motion 0 The blockspring system we learned about with Hooke s law is called a linear simple harmonic oscillator The angular frequency of the motion of the block in the system is related to the spring constant lc JoE m 0 We can combine the equation for angular frequency and period and write T2211 o 154 Energy in Simple Harmonic Motion 0 The potential energy of a linear oscillator depends entirely on how much the spring is stretched or compressed The potential energy of a spring Ut is Utkx2kxfncoszoot 0 Conversely the kinetic energy of such a system depends entirely on the velocity of the block I Ktmv2 mw2xisin2oot 0 Using these two equations we can nd an equation for mechanical energy EUK kxi The energy of a linear oscillating system is independent of time o 155 An Angular Simple Harmonic Oscillator o The torsion pendulum A torsion pendulum is a pendulum system where the elasticity of the system is associated with the twisting of a suspension wire rather than the extension and compression of a spring Chapter Fifteen In a torsion system the oscillation is called angular simple harmonic motion 0 The disc is given a torque by rotating the disc through an angle in either direction We have tz KO This is the angular form of Hooke s law 0 The period of torsion pendulums 39 T2H l k 0 156 Pendulums o The simple pendulum A simple pendulum is a mass m suspended from one end of an unstretchable massless string of length L that is xed at the other end The forces acting on this mass called the bob are the tension force from the string and the gravitational force which can be broken up into trigonometric vertical and tangential components 0 The component of the gravitational force tangent to the path the bob takes causes what is called a quotrestoring torquequot because it tends to restore the bob to an equilibrium position 0 Restoring torque T Lmgsin9 W

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