Chapter 6: Uniform Circular Motion and Gravity

Transcription

1 Chapter 6: Uniform Circular Motion and Gravity Brent Royuk Phys-111 Concordia University Angular Measure Angular distance: Δθ = θ - θ o Analogous to linear distance Rotation instead of translation How far did you run? 33 m. How far did you turn? 33 o Radians All thetas are measured in radians There are 2π radians in a circle A radian is dimensionless 180 = π R m R = (m π)/180 Importance The radian equation: s = rθ Example: If you run 32 m on a circular track with r = 200 m, what is your Δθ? Angular Speed & Velocity Definition: θ = ωt ω = Δθ Δt Average vs. instantaneous Example: Make half a turn in 2.0 s: ω =? MKS Units Other units What is the angular speed for a 45 rpm record? How far would above object turn in 85 s? How many revolutions is this? Angular Velocity Velocity implies? Two options only Right-hand rule also used: (+), ( ), cw, ccw Tangential Velocity v t = rω Angular & Tangential Speeds How fast would a bug have to run to stay motionless on a record? v t = r ω What is tangential speed of a point at the edge of an 8.0 cm disk spinning at ω = 3.4 s -1? Other measurable quantities: Period (T), frequency (f, ν) Circular Motion Uniform Circular Motion What is the nature of the speed, velocity and acceleration? What does centripetal mean? Derivation Small-angle approximation What is a centrifugal force? a c = v2 r F C = mv2 r Circular Motion Swing a mass on a string, calculate and measure the centripetal force. A 1200-kg car rounds a corner of radius 45 m. If the coefficient of friction between the tires and road is 0.82, what is the greatest speed the car can have in the corner without skidding? What if the roadway is banked inward 18 o (no friction)? What does centrifugal mean? A fictitious force. 1

2 The Law Newton s Law of Universal Gravitation What s universal about Newton s Universal Law of Gravitation? The Law: F = G m 1 m 2 r 2 Gravity G = 6.67 x N m 2 /kg 2 Superposition Gravitational Attraction of Spherical Bodies The Shell Theorem Why don t we ever notice attractions between two terrestrial bodies? Physical attraction on a love seat. Let each person be a uniform-density sphere Big G, Little g g Everest = 9.78 m/s 2 Gravity is an Inverse Square Law What if you halve the distance between objects? Double it? The silly-looking platform illustration. Cavendish s Torsion Balance Weighing the Earth Huh? Gravity g shuttle = 9.08 m/s 2, 7.44% less than Earth s surface. Gravity and Orbit Why are astronauts weightless in orbit? The puzzle of retrograde motion 2

3 The puzzle of retrograde motion The geocentric solution: Epicycles Johannes Kepler ( ) By the study of the orbit of Mars, we must either arrive at the secrets of astronomy or forever remain in ignorance of them. Tyco Brahe ( ) Brahe s observatory: 1582, the Island of Hveen, near Copenhagen Non-telescopic measurements with a precision of one arc-minute Brahe s unusual solar-system model The Tychonian Society Kepler s First Law Planets follow elliptical orbits, with the Sun at one focus of the ellipse First two laws: 1609 Third law: 1619 Kepler s House in Regensburg, Germany Kepler s Second Law Kepler s Second Law As a planet moves in its orbit, it sweeps out an equal amount of area in an equal amount of time. Why? Conservation of L. There s no torque about the Sun. 3

4 Kepler s Third Law The period, T, of a planet increases as its mean distance from the Sun, r, raised to the 3/2 power. That is, T α r 3/2 Kepler s Esoterica Actually, 4 π 2mr T 2 mm S =G T = r2 2π GM r, so 3 /2 Kepler s Esoterica Orbits What is the orbital speed of a satellite orbiting 100 km above the Earth? Find the radius of an orbit above the equator such that the satellite completes one orbit in one day. Geosynchronous Orbit Compare with the space shuttle, which orbits at an altitude of about 150 km. What is the space shuttle s orbital period? GPS satellites orbit at an average altitude of 12,400 miles = 20,000 km. Which travels faster, a GPS satellite or a geosynchronous satellite? Geosynchronous/GPS Orbital Maneuvers The Hohmann transfer 4

5 Black Holes Tides Retrograde Motion Explained 5