Universal Gravitation

Chapter 8

Isaac Newton and Gravity

• Newton realized an apple falls because of force

• Moon follows circular path, force needed• Newton first to realize gravity causes

both.• Moon is actually “falling” towards earth• Tangential velocity (sideways) keeps it

going around earth instead of into it

Law of Universal Gravitation

• Newton extended gravitation theory to all objects,

• Realized gravitational attraction is property of all mass

• Attractive force is directly proportional to product of two masses

• Inversely proportional to square of distance between them

Law of Universal Gravitation

• F ~ m1 m2/d2

• Proportionality constant G must be used to make quantities equal

• F = Gm1 m2/d2

• G = force between 2 1-kg masses, 1 m apart

• Its value is G = 6.67 x 10-11 Nm2/kg2

• Henry Cavendish first to measure G

Law of Universal Gravitation

• Gravity very weak force• Force of attraction with earth is weight• Weight depends on distance from

center of earth• On top of high mountain, you weigh

less• Using G and radius of earth we can

calculate mass of earth = 6 x 1024 kg

An Inverse-Square Law

• Force of gravity varies inversely with square of distance between objects

• For a spherical body, distance is measured from its center

• If distance is doubled, force is one-fourth as much; if tripled, force is one-ninth (1/3)2

• If distance is halved, force is 4 times as much; if distance is 1/3 as much, force is 9 times as great.

An Inverse-Square Law

• Other inverse-square relationships: light and sound intensity, magnetic and electric force.

• Force of gravity weakens rapidly with distance due to spreading of force into surrounding space, but never really zero, even in deep space.

Gravitational Fields

• Gravitational force acts over a distance without contact

• Caused by interaction with gravitational field

• Field can be thought of as lines of force radiating from objects

• Strength of field is force per unit mass exerted on object: g = F/m

Gravitational Fields

• Earth’s grav. field strength is actually acceleration due to gravity at any location

• Combining g = F/m and F = G mM/R2, we find g = GMe/ R2, where Me = mass of the earth and R = distance from center of earth.

• g for any planet can be found using this formula.

Gravitation and the Universe

• Planets, moons, stars are spherical due to gravity

• All planets affect each other, can cause perturbations (wobbles) in orbits

• Neptune and Pluto were discovered because of perturbations in Uranus’ orbit.

• Perturbations of distant stars have led to discoveries of planets, double stars

Gravitation and the Universe

• “Big Bang” theory says universe hurled outward from big explosion

• Universe is expanding, only pull of gravitation can stop it.

• Unknown whether enough matter exists to stop expansion and begin contraction leading to “Big Crunch” and another big bang-expansion-contraction cycle.

Gravitational Field Inside Earth

• If hole were drilled through earth and you fell in, what would happen?

• As you fall, mass below you pulls down, mass above you pulls up.

• No force at center of earth, pull equal in all directions

• You would accelerate with decreasing acceleration until you reached the center, then decelerate as you approach opposite surface.

Satellites--Circular Orbits

• In circular orbit, gravity always pulls perpendicular to velocity

• Speed doesn’t change, only direction.• Time (period) for close orbit about 90

min., greater for higher orbits• If period of revolution is 24 hours,

satellite remains over same point on earth--communication satellites do this (geosynchronous orbit)

Satellites--Elliptical Orbits

• If speed is faster than needed for circular orbit, elliptical orbit can result

• Ellipse is oval with two focal points (foci)• Speed varies in elliptical orbit• Faster speed close to earth, slower

when farther away• All planets follow elliptical orbits around

sun

Tides

• Change in sea level due to difference in gravitational pull between earth and moon on opposite sides of earth.

• Causes elongation of earth and moon along line of force, bulging on near and far side.

• Earth turns underneath high water levels causing 2 high tides and two low tides per day.

Tides

• Sun also creates tides but effect is less than moon because distance difference is not as large a part of total distance.

• Extra large spring tides caused by sun-moon alignment on same side of earth

• Smaller neap tides occur when sun and moon are on opposite sides of earth

Eclipses

• When sun-moon-earth alignment is perfect, eclipses occur

• When earth is between sun and moon, moon is in shadow and lunar eclipse occurs.

• When moon is between sun and earth, parts of earth in moon’s shadow see solar eclipse.

Black Holes

• In stars, expansion forces due to nuclear fusion are balanced by contraction force of gravity.

• When solar fuel runs out, stars collapse

• Small stars will become dense, cool, black dwarfs.

Black Holes

• Large stars will continue collapse to enormous density with gravitational forces too strong for anything to escape

• Even light, radio communications are trapped--nothing can escape black hole (except small amounts of radiation).

• Edge of black hole is called event horizon

Black Holes

• Fabric of space-time is warped by tremendous gravity inside black hole

• Mass of black hole is same as star that created it so gravity outside black hole is same as if original star were still there.