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Transcript
Circular Motion & Gravity Circular Motion • Objects travel in a circle • Rotate about an axis of rotation • Tangential speed (vt) describes the rate at which the object moves around the circle 2r circumfere nce vt T period • Direction is tangential to the circular path vt depends upon radius • Given the object is rigid, e.g. a CD • Object B must travel a greater distance to keep up with object A • SB > SA • But ΔtB = ΔtA • Therefore, vB > vA Comparison of Translational Motion & Uniform Circular Motion UCM = motion of an object traveling in a circle at a constant speed, vt Type of Motion Translational Displacement Linear Δx Time Δt Formula vavg = Δx/Δt Uniform Circular Circumference 2πr Period T vt = 2πr/T Uniform Circular Motion • Tangential speed vt is constant • Because direction is changing, there is acceleration • Centripetal acceleration Centripetal Acceleration • a = Δv/Δt • When subtracting vectors, reverse the direction of vi • Centripetal acceleration is, therefore, directed toward the center (axis of rotation) when θ is small Centripetal Acceleration • Centripetal means “center seeking” and is always directed toward the center • Due to a change in direction of vt • Phet simulation 2 vt ac r Tangential Acceleration • Tangential acceleration occurs when there is a change in tangential speed. • For example, if a car is speeding up as it goes around a curve, – It has tangential acceleration and – Centripetal acceleration Centripetal Force Because Fc acts at right angles to the object’s circular motion, it changes the direction of the objects velocity Centripetal Force • Is the cause of centripetal acceleration • It is directed toward the axis of rotation • It is the net force acting on an object in uniform circular motion, i.e. it is the cause of circular motion • Centrifugal force is a misunderstanding of inertia Centripetal Force & Newton’s 2nd Law F ma 2 vt Since ac r Fc mac 2 vt Fc m r Centripetal Force • Is just the name of any net force acting on an object in uniform circular motion • Fc could take any form…. • It could be frictional force, tension force, gravitational force, etc. Motion of a Car Around a Curve • On a horizontal turn, the centripetal force is friction Circular Motion About a Banked Curve Conical Pendulum Vertical Circular Motion Centifugal Force? • If Fc is insufficient to maintain circular motion, the object will leave it’s circular path due to its own inertia, not because some force is pulling it away from the axis of rotation • Thus, inertia is often mistaken for “centrifugal force” Gravity Gravitational Force • • • • • Force of attraction between two masses Attractive only One of four fundamental forces Very weak (the weakest) When one object orbits another, gravitational force is a centripetal force Newton’s Law of Universal Gravitation • Gravitational force is… – directly proportional to the product of the masses of the two bodies – inversely proportional to the square of the distance between the centers of the two masses – If the objects are large (e.g. planets, moons) then the radii would be included in r m1m2 Fg G 2 r G universal gravitatio n constant 2 N m G 6.673 1011 kg 2 Gravitational Force Exists Between Any Two Masses Newton’s Cannon http://spaceplace.nasa.gov/en/kids/orbits1.shtml http://galileoandeinstein.physics.virginia.edu/more_stuff/Applets/n ewt/newtmtn.html Importance of Gravitational Force • Keeps you from floating away into space • Gravitational force keeps the Moon and planets in orbit • Keeps earth in orbit around sun • Causes ocean tides Black Holes: Extreme Gravity Extreme density Escape velocity > speed of light Detect by effects on surrounding matter Gravitational Field Strength • Increases as distance from mass center decreases • Because gravitational field strength varies, weight varies with location Gravitational Field Strength • Describes the amount of gravitational force per unit mass at any given point • Equals free-fall acceleration g Fg m Weight Changes with Location • Because gravitational field strength varies, ag varies (acceleration of gravity). • Since w = mag, weight must vary as ag varies • Fg is an example of an inverse square law m1m2 Fg G 2 r 7.3 Motion in Space Astronomer Ptolomey Planets orbit… Earth Type of orbit Epicycles Copernicus Sun Circular Kepler Sun Elliptical Kepler’s Laws of Planetary Motion 1. The Law of Orbits: All planets move in elliptical orbits, with the sun at one focus. 2. The Law of Areas: A line that connects a planet to the sun sweeps out equal areas in equal times. 3. The Law of Periods: The square of the period of any planet is proportional to the cube of the average distance from the sun, T r 2 3 Kepler’s st 1 Kepler's Law Simulation and nd 2 Laws Kepler’s 3rd Law Describes Orbital Period Period and speed of an object in Circular O rbit Orbital Period Orbital speed r3 m T 2 vt G Gm r whe re m is the mass of the orbited mass Actual and Apparent Weight • A bathroom scale records the normal force of scale acting on your body • Step on the scale … the normal force equals your weight Actual and Apparent Weight • Now try this • Step on the scale and have someone press down on your shoulders – Predict and explain the result • Step on the scale and have someone lift you slightly • Predict and explain the result Actual and Apparent Weight • How does this relate to your experiences in an elevator? • What would the scale read if, in an elevator, it descended with an acceleration of g? Weight and Apparent Weightlessness Torque • a quantity that measures the ability of a force to rotate an object about an axis • is not a force • “rotating ability” • the product of force and “lever arm” • τ = F · d sinθ • Lever arm (d) is distance perpendicular to direction of force to axis of rotation Fd sin Torque • Sign (+) is counterclockwise (-) is clockwise • Net Torque and when 2 or more forces act to rotate the same object, τnet = Στ τnet = τ1 + τ2 = F1d1 + F2d2 Torque Equilibrium • Torque Equilibrium: Στ = 0 Torque Equilibrium The torque due to the boy is equal and opposite to that of the girl. Net Torque Center of Mass (COM) • Point mass vs. extended object • The point in a body at which all the mass can be considered to be concentrated when analyzing translational motion • Unless an object rotates about a fixed point, (e.g. a hinge)… – The point about which a mass or system of mass rotates during rotational motion Center of Mass • The extended object rotates about the CoM • CoM follows the expected parabolic path Center of Mass • May not lie within the mass or system of masses Simple Machines • All machines are combinations of simple machines • Purpose is to change magnitude or direction of an input force • Mechanical Advantage describes the ratio of output and input forces Fout MA Fin Ideal vs. Actual Mechanical Advantage • Ideal MA MA if there were no friction d in IMA d out • Actual MA MA that takes friction into account Fout AMA Fin Machines and Work • Machines do not change the amount of work • Machines make work easier Efficiency • A measure of how well a machine works • A less efficient machine produces less output per input • A less efficient machine requires more input to get the same output Wout eff Win