Ch 8 Universal Gravitation

Black hole-

•an extremely massive object that can bend light back to the object

Centripetal acceleration

•-acceleration toward the center of a circular path

Freefall—

•accelerating downward b/c of an unbalanced gravitational force

General theory of relativity—

•Einstein’s theory that mass causes space to be curved, accelerating bodies

Gravitational field—

•the area around a mass that acts on other masses, causing them to be attracted to each other

Gravitational force—

•an attractive force that exists between all objects

Inverse square law—•a relationship in which one variable is inversely proportional to the square of another variable

Kepler’s laws of planetary motion—•Laws that describe the motion of planets and satellites

•

Law of universal gravitation—

•The attraction between two bodies depends on the masses of the two bodies and the distance between their centers

Newton’s second law—•the acceleration of a body is directly proportional to the net force on it and inversely proportional to its mass

Satellite—•a body that is in orbit around another body

Universal gravitational constant

—•G- needed in calculating the gravitational force between 2 objects

Weightlessness-

•the apparent loss of gravitational force on an object in orbit or in freefall

•The force of gravity on a satellite, its weight, provides the centripetal force to maintain its circular motion.

•A satellite farther from the earth has a larger velocity

•The Velocity of a satellite is independent of its mass

Equation; the velocity a satellite must have

to orbit the earth• __ •V = V² or V = g r

• r

•g = acceleration of gravity at distance r from the center of the earth

•r is the average radius of its orbit from the center of the earth.

•The mass of the satellite does not affect its orbital velocity.

• The orbital velocity is independent of the mass of the satellite

•A more massive satellite requires a greater centripetal force to keep it in orbit.

• However, a more massive satellite also has a greater weight.

•The greater weight provides the greater centripetal force.

Astronauts are weightless

because?_____•They and the satellite are in free-fall, accelerating toward the earth

• 1. Calculate the velocity at which a satellite must be launched in order to achieve an orbit about the earth. Use 9.8 m/s²

• As the acceleration of gravity and 6.5 X 10³ km as the earth’s radius.

• V = gr __________________

• V = (9.8 m/s²) (6.5 X 106 m)

• = 8.0 X 10³ m/s

• 2. During the lunar landings, the command module orbited close to the moon’s surface while waiting for the lunar module to return from the moon’s surface. The diameter of the moon is 3570 km and the acceleration of gravity on the moon is 1.60 m/s².

• a. at what velocity did the command module orbit the moon?

•2. During the lunar landings, •a. at what velocity did the

command module orbit the moon?

•V = gr• • = (2.9 X 106 m²/s²• = = 1.7 X 10³ m/s

• = 1.7 km/s

• 2. During the lunar landings, the command module orbited close to the moon’s surface while waiting for the lunar module to return from the moon’s surface. The diameter of the moon is 3570 km and the acceleration of gravity on the moon is 1.60 m/s².

• b.In how many minutes did the module complete one orbit?

•2. During the lunar landings, the

• b. In how many minutes did the module complete one orbit?

•b. V = 2 r t = 2 r = t V

•= 6.6 X 10³ s = 6.6 X 10³ s 60 S/min

•= 1.1 X 10² min

• 3. Calculate the velocity at which a satellite orbits Jupiter. The acceleration of gravity on Jupiter is 5.8 X 10³ m/s² . The diameter of the planet is 1.422 X 10 5 km.

• V = gr• V = (5.8 X 10³ m/s²)(7.11 X 107

m)• = 6.42 X 105 m/s = 642 km/s

•Greeks used terms levity (rise) & gravity (fall)

•Galileo & Newton- stated gravity to the force that exists between Earth & objects

•Newton- stated same force exists between all bodies

•Einstein- gave different & deeper description of the gravitational attraction

•We Still do not know WHY things fall. (only how)

•Early scientist watched the skies

•Notices stars moved in regular paths

•Planets (wanderers) had complicated paths

•Astrologist claimed the motion of the bodies contolled events in life

•Comets- erratic movement, appeared w/o warning, w/bright lights.- considered bearers of evil omens

• It Took Galileo, Kepler, Newton & others to understand that the all of them follow the same laws that govern the motion of objects on Earth.

Tycho Brahe (1546-1601)

•Was interested in astronomy After observing eclipse at 14.

•Decided to learn how to make accurate prediction of astronomical events when a predicted planets in conjuncture occurred 2 days late.

Tycho Brahe•Studied throughout Europe for

5 years then set up observatory on island. (Believed Earth was center of Universe)

•Next 20 years spend recording positions of planets & stars.

•Moved to Prague where Kepler became one of his assistants.

Tycho Brahe

•Made very accurate measurements of the positions of planets & stars which were used by Kepler to formulate his laws.

Kepler•Believed measurements on number, distance and motion of the planets could be explained with a sun-centered system using geometry & mathematics.

Kepler’s theories are no longer considered

correct( still describe the behavior of every planet &

satellite)

Kepler•Was driven to find the true

paths of planets after finding that Tycho’s predictions were wrong by eight minutes of arc (1/4 the width of the moon).

•Kepler’s Laws are now known to be the result of the conservation of energy and angular momentum.

Kepler’s laws•still apply to motion in any conic section (ellipse, parabola, hyperbola and circle. (the behavior of every planet and satellite) even though they are no longer considered correct.

Kepler’s laws •1.The paths of the

planets are ellipses with the center of the sun at one focus.

Kepler’s laws •2.An imaginary line from

the sun to a planet sweeps out equal areas in equal time intervals. Thus, planets move fastest when closest to the sun, slowest when farthest away.

Kepler’s laws

•3.The ration of the squares of the periods of any 2 planets revolving about the sun is equal to the ratio of the cubes of their avg distances from the sun.

Kepler’s laws•3. Thus, if Ta & Tb are

their periods and ra & rb their avg distances,

• [T a ] ² = [ r a ] ³• [T b ] [ r b ]

Kepler’s Laws• Or The ration of the avg radius of

a planet’s orbit about the sun, r cubed & the planet’s period,

• T (the time for it to travel about the sun once) squared,

• is a constant for all the planets. This law can be expressed as

•or r³ = k• T²

•Laws 1 & 2 apply to planet, moon or satellite individually

•3rd – movement of satellites about a single body (planets around sun) & compare distances and periods of the moon & artificial satellites around early.

Problem Solving•For preciseness keep at least 1

extra digit in your calculations until you reach the end.

•You do not need to convert all units to meters & seconds- just use the same units throughout the problem

Problem Solving•For 3rd law- to find radium of the orbit- solve for the cube of the radius then take the cube root.

•Use cube-root key or Yx or Xy. •Enter the cube of the radius- press the Yx key then enter 0.33333333 and press =

Jules Verne- (1828 – 1905)

•Wrote about airplanes, submarines, guided missiles and space satellites, accurately predicting their uses.

In 1666, Newton•Used math to show that if the path of a planet were an ellipse then the force on the planet must vary inversely w/the square of the distance between the planet and the sun.

Newton•the force on the planet must vary inversely w/the square of the distance between the planet and the sun.

•F= force: • α = is proportional to: • d = avg distance between the centers of the 2 bodies

Newton

• F= force: • α = is proportional to: • d = avg distance between the centers of the 2 bodies

•F α 1• d²

• the force acted in the direction of a line connecting the centers.

•Could go no farther - b/c he Could not determine the force

•Recognized that the force which pulled the apple down must be proportional to its mass

According to 3rd law-

•The apple attracts earth also- • the laws would work anywhere•He assumed that the laws that

governed the motion on earth would work anywhere. That G is a universal constant- the same everywhere.

LAW OFUNIVERSAL GRAVITATION

Every object attracts every other object with a force that is directly proportional to the mass of each object

& is inversely proportional to the square of the distance between their centers.

How does the earth's pull on Dr. J's craft compare to that on Tripod's?

•Inverse square law- If the planet’s mass is doubled, the force is doubled. If the distance is doubled, the force would be only ¼ as strong

The force on Timex's craft is only 1/4 of that on Tripod's because Timex is 2x as far from the earth's center.

Law of Universal Gravitation

•Newton couldn’t determine the proportionality constant G because the mass of the earth was unknown.

The force of gravitational attraction between the earth & each spaceship can be found using: FG = MEMs d2

where G is the universal constant of gravitation

(6.67 x 10-11 N-m2/kg2).

Cavendish•determined the value of G by experimentation

Einstein•According to the general theory of relativity

• mass causes space to be curved

•For an object to escape earth’s gravitational pull, the sum of the potential and kinetic energies must be zero.

•You can show that a projectile needs to attain a horizontal speed of 11.2 km/s to escape from earth’s gravitational pull.

• Orbital velocity increases as a planet moves closer to the sun.

•As the distance between 2 bodies increases the force of attraction between them decreases.

•Anything that has mass is surrounded by a magnetic field

Inverse square law- ex. The force of gravitation

depends on 1/d²• Gravitational force between any 2

bodies varies directly as the product of their masses and inversely as the square of the distance between them.

•Rp- radius of the planet•Tp- time required for the planet to make one complete revolution

•Keplers 3rd law- the squire of the period is proportional to the cube of the distance.

Henry Cavendish-•verified the existence of

gravitational forces between masses.

•Calculated the attractive force between masses & found that the force agreed with Newton’s law of gravitation.

Henry Cavendish-

•He found the value of G= 6.6.7 X 10 -11 N-m ²/kg² by experimentation.

8.2•When the planet Uranus was

discovered 1741 , some believed that Newton’s law of gravitation didn’t correctly predict its orbit.

•Others searched for an unknown planet which may be affecting its orbit and found Neptune

Newton-Hypothetically• Use a mountain with a cannon on the

top w/a cannonball as a projectile. During the 1st second the ball drops 4.9 meters. And even if it travels farther- it still falls another 4.9 meters.

• If the ball goes just fast enough after one second it may reach a point where the earth has curved 4.9 m away from the horizontal.

Newton-Hypothetically• If the curvature of Earth will just

match the curvature of the trajectory, the ball will orbit earth.

• Earth curves away from a line tangent to its surface at a rate of 4.9 m for every 8 km.

•

Newton-Hypothetically• .The cannonball will fall toward

Earth’s surface at the same rate that the Earth’s surface curves s away.

• An object with a horizontal speed of 8 km/s will keep the same altitude and circle Earth as an artificial satellite.

The orbital velocity and period are independent of the mass of the satellites.

• A more massive satellite requires more force to put it into orbit.

• Weight and weightlessness• As we move farther from Earth’s

center, the acceleration due to gravity is reduced according to this inverse square relationship

Zero-g = weightlessness• In orbit the force of gravity still

applies, but the shuttle and astronauts inside are all falling freely toward Earth as they orbit around it.

• If in an elevator going downward at 9,8 m/g2 then a scale would exert no force on you and you would feel weightless.

Gravitational Field-• Anything that has mass is surrounded

by a gravitational field which acts over a distance and does not require any touch (friction) to start it moving.

• To find the strength- force divided by a unit of mass; measured in newtons/kilogram; same as gravity = 9.8N/kg

Gravitational Field-•It is independent on the size

of the mass. •The strength of the field

varies inversely w/the square of the distance from the center of earth

Albert Einstein-• Proposed that gravity is an

effect of space• Mass changes the space about it• Mass causes space to be curved

and other bodies are accelerated b/c they move in this curved space/

Albert Einstein-• General Theory of relativity- in every

test- gives correct results. Slightly different from Newton’s laws

• Predicted the effects of a black hole- capturing light due to gravity- no light escapes.

• Theory is not yet complete- doesn’t explain how masses curve space.