Chapter 6: Uniform Circular Motion and ... 1 Chapter 6: Uniform Circular Motion and Gravity Brent...
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Chapter 6: Uniform Circular
Motion and Gravity Brent Royuk
Phys-111 Concordia University
Angular Measure • Angular distance: Δθ = θ - θo
– Analogous to linear distance – Rotation instead of translation – How far did you run? 33 m. How far did you turn?
33o • Radians
– All thetas are measured in radians – There are 2π radians in a circle – A radian is dimensionless
• 180° = πR • mR = (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 – 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 – vt = rω
ω = Δθ
Angular & Tangential Speeds • How fast would a bug have to run to
stay motionless on a record? • vt = 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?
ac = v2
FC = mv2
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 18o (no friction)?
• What does “centrifugal” mean? – A “fictitious force.”
The Law • Newton’s Law of
Universal Gravitation • What’s universal about
Newton’s Universal Law of Gravitation?
• The Law:
F = G m1m2 r 2
Gravity • G = 6.67 x 10-11 N m2/kg2 • 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
– gEverest = 9.78 m/s2
• 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
Gravity • gshuttle = 9.08 m/s2, 7.44% less than Earth’s
Gravity and Orbit Why are astronauts weightless in orbit?
Kepler’s Laws • The puzzle of retrograde motion
Kepler’s Laws • The puzzle of retrograde motion • The geocentric solution: Epicycles
Kepler’s Laws • Tyco Brahe (1546-1601)
– 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 Laws • Johannes Kepler (1571-1630)
– “By the study of the orbit of Mars, we must either arrive at the secrets of astronomy or forever remain in ignorance of them.”
• First two laws: 1609 • Third law: 1619
Kepler’s House in Regensburg, Germany
Kepler’s First Law • Planets follow elliptical orbits, with the
Sun at one focus of the ellipse
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.
Kepler’s Second Law
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
Actually, 4π 2mr
T 2 = G mMS
r 2 , so
T = 2π
GM r 3 /2
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
Orbital Maneuvers • The Hohmann transfer
Retrograde Motion Explained