# Chapter 6: Uniform Circular Motion and ... 1 Chapter 6: Uniform Circular Motion and Gravity Brent...

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1

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

Δt

2

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

r

€

FC = 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 18o (no friction)?

• What does “centrifugal” mean? – A “fictitious force.”

3

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

Huh?

4

Gravity • gshuttle = 9.08 m/s2, 7.44% less than Earth’s

surface.

Gravity and Orbit Why are astronauts weightless in orbit?

Kepler’s Laws • The puzzle of retrograde motion

5

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

6

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

7

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

Kepler’s Esoterica

Kepler’s Esoterica

8

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

9

Black Holes

Tides

Retrograde Motion Explained

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