Physics 103: Lecture 14 Chapter 7 Rotational Motion, Law of Gravity

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Physics 103: Lecture 14 Chapter 7 Rotational Motion, Law of Gravity. Today’s lecture will cover Center of Gravity Newton’s Law of Gravitation Kepler’s Laws. mg. N. N. Preflight 1, 2 & 3.
Physics 103: Lecture 14Chapter 7Rotational Motion,Law of Gravity
  • Today’s lecture will cover
  • Center of Gravity
  • Newton’s Law of Gravitation
  • Kepler’s Laws
  • Physics 103, Fall 2009, U. WisconsinmgNNPreflight 1, 2 & 3
  • You are driving a car with constant speed around a horizontal circular track. On a piece of paper draw a Free Body diagram for the car.
  • How many forces are acting on the car? 1 23 45 gravitynormal force of roadcentripetal forceFcPhysics 103, Fall 2009, U. WisconsinPreflight 4 & 5
  • The net force on the car is
  • ZeroPointing radially inward, toward the center of the circlePointing radially outward, away from the center of the circleTo travel in a circle something must supply the centripetal acceleration.Physics 103, Fall 2009, U. WisconsinVdownFn< MgFn= MgFn> MgPreflight 6 & 7
  • Suppose you are driving through a valley whose bottom has a circular shape. If your car has mass M, what is the normal force exerted on you by the car seat as you drive past the bottom of the hill?
  • Normal force must also supply centripetal accelerationPhysics 103, Fall 2009, U. WisconsinA SPECIAL POINTCenter of mass (or center of gravity)For symmetric object with uniform density the center of mass is easy :Center of mass is the same as the center of the volumeSymmetryPhysics 103, Fall 2009, U. WisconsinCenter of Gravity
  • Often all of the weight of the object can be considered to be concentrated at one point when determining motion or equilibrium
  • mxmySSxiiandyii==cmcmmmSSiiParticularly useful when treating it with external problem. Physics 103, Fall 2009, U. WisconsinExperimentally Determining the Center of Gravity
  • The wrench is hung freely from two different pivots
  • The intersection of the lines indicates the center of gravity
  • A rigid object can be balanced by a single force equal in magnitude to its weight as long as the force is acting upward through the object’s center of gravity
  • Physics 103, Fall 2009, U. WisconsinQuestion 1Can a body’s center of gravity be outside its volume?a) yesb) noCenter of gravity (CG) is defined as:where m1 is the mass of element at coordinate (x1,y1,z1) …CG can be outside the volume.Physics 103, Fall 2009, U. WisconsinQuestion 2Where is the center of gravity of a “yummy” donut?It is at the origin of the circular ring, half way from the bottom of the donut - where there is no dough.Physics 103, Fall 2009, U. WisconsinA moment later……C G has shifted along the line of symmetry away from the bite.Physics 103, Fall 2009, U. WisconsinNewton’s Law of Gravitation
  • Every particle in the universe attracts every other particle with a force along the line joining them. The force is directly proportional to the product of their masses and inversely proportional to the square of the distance between them.
  • Note: “particle”! --- point-like.If an extended object you must treat the vector sum of all the forces. This is done automatically by considering the object as if it were of the same mass concentrated at the “center of mass” (or the center of gravity!?)If a system of extended objects you must still find/consider the center of mass.Physics 103, Fall 2009, U. WisconsinmrMmMFg = Gr2Newton’s Law of GravityMagnitude:G = 6.67 x 10-11 N m2/kg2Direction: attractive (pulls them together) force on M due to m is away from M center and force on m due to M is away from m centerWork done to bring mass m from infinity to the proximity of mass MOnly differences in potential energy matterZero point is arbitrary.Physics 103, Fall 2009, U. Wisconsinmga=g = GmM MFg = ma = Gr2RE2Close to the Surface of the Earth
  • Consider an object of mass m near the surface of the earth.
  • =9.8 m/s2r~REPhysics 103, Fall 2009, U. WisconsinPreflight 8 & 9
  • Two satellites A and B of the same mass are going around Earth in concentric orbits. the distance of satellite B from Earth’s center is twice that of satellite A. What is the ratio of the centripetal acceleration of B to that of A?
  • Since the only force is the gravitational force, it must scale as the inverse square of their distances from the center of the Earth.1/81/41/20.7071.0Physics 103, Fall 2009, U. WisconsinPreflight 10 & 11
  • Suppose Earth had no atmosphere and a ball were fired from the top of Mt. Everest in a direction tangent to the ground. If the initial speed were high enough to cause the ball to travel in a circular trajectory around the earth, the ball’s acceleration would
  • be much less than g (because the ball doesn’t fall to the ground)
  • be approximately g
  • depends on the velocity of the ball
  • Physics 103, Fall 2009, U. WisconsinOrbits
  • Acceleration is provided by gravity
  • Orbits not necessarily circular
  • Physics 103, Fall 2009, U. WisconsinEscaping Gravity
  • Kinetic energy of the rocket must be greater than the gravitational potential energy
  • Defines minimum velocity to escape from gravitational attraction
  • Physics 103, Fall 2009, U. WisconsinT2= constantR3Kepler’s Emperical Laws Based on Tycho Brahe’s astronomical measurements
  • 1st Law: Orbit of a planet is an ellipse with the Sun at one focus
  • 2nd Law: Equal areas swept out in equal times.
  • 3rd Law:
  • Newton:Physics 103, Fall 2009, U. Wisconsin
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