Gravity

9.1

The Universal Law of Gravity:

• The circular motion of heavenly bodies used to be considered natural; not-requiring any explanation.
• When an apple fell from a tree and struck Isaac Newton, he considered that a force may have pulled it down.
  • He considered that this force might’ve been what pulls everything in the universe (moons, planets, etc).
• This led to the notion that there are two sets of natural laws:
  • One for earthly events
  • One for motion in the heavens
• This union of territorial and cosmic laws is called Newtonian Synthesis.

How Newton tested his hypothesis:

• He compared the fall of an apple with the fall of the Moon.
• He realized that the Moon falls away from the straight line it would’ve followed if there were no forces acting on it.
• It falls “around” the Earth.
  • Reason → Tangential velocity
• The Moon’s distance of fall person second is comparable to the distance the apple falls in one second.
  • This didn’t align with Newton’s calculations.
• Years later, Newton corrected his experimental data and got results.
  • He then published the Law of Universal Gravitation.
• According to Newton, everything pulls on everything else in a way that involves only mass and distance.
• Every body attracts every other body with a force that is directly proportional to the product of their mass and inversely proportional to the square of the distance between them.
• Force ~ (mass1 x mass2)/(distance)^2

9.2

The Universal Gravitational Constant, G:

• The proportionality in the universal law of gravitation can be expressed as G.

• F = G((m1 x m2)/d^2)

• Magnitude of G = gravitational force between two 1 kg masses 1m apart.

  • G = 6.67 x 10^Nm^2/kg^2

• We feel gravitational force as weight.

  • We feel it due to the massive number of atoms on Earth pulling at us.

• Newton could calculate the product of G and Earth’s mass, but couldn’t calculate them individually.

• G was first calculated by Henry Cavendish in 1798.

  • He measured forces between lead masses with a sensitive torsion balance.

• Philip von Jolly developed a simpler method.

  • He attached a spherical flask of mercury to one arm of a sensitive balance.
  • When the balance was put in equilibrium, a 6-ton lead sphere was rolled beneath the mercury flask.
  • Force of gravity between the masses = weight needed to restore equilibrium on the other end of the balance.

• G = F/((m1 x m2)/d) = 6.67x 10^-11 Nm^2/kg^2

• Value of G shows that gravity is the smallest of the four fundamental forces.

  • Gravity is sensed only when masses like Earth’s are involved.

• Force of attraction between you and Earth → your weight

  • Weight depends on your mass and your distance from the center of the Earth
  • Your weight on a mountain would be slightly less than at ground level.
  • Reason → distance from Earth’s center lesser on ground

• Mass of Earth was easily calculated after value of G was found.

• Earth exerts a force of 9.8N (rounded off to 10N) on a mass of 1kg at its surface.

• F = G((m1 x m2)/d^2)

  • 9.8N = 6.67 X 10^-11 Nm^2/kg^2 ((1kg x m1)/(6.4 x 10^6m)^2)
  • m1 = 6 x 10^24 kg

9.3

Gravity and Distance: The Inverse Square Law:

• Gravity weakens with distance.
• Compare to: paint from a spray can spreading as the distance increases.
  • Position a spray paint can at the center of a radius 1m sphere.
  • Let a burst of spray paint travel 1m to produce a 1mm thick patch.
  • If the same is done with a 2m radius sphere, the height and width of the patch will be twice the initial value.
• Thickness of the paint decreases as square of the distance increases. This is the inverse square law.
• Inverse square law holds for gravity as well.
  • It applies to all phenomena where the effect from a localized source spreads uniformly through the surrounding space.
• Newton’s law of gravity applies to particles, spherical bodies, and non-spherical bodies sufficiently far apart.

In Newton’s equation, d → distance between the centers of masses of the objects

• For objects on Earth:
  • As the distance between the object and Earth’s surface increases, Earth’s gravitational force approaches zero
  • The force never actually reaches zero.
• Gravitational field of every material object extends through all of space.

9.4

Weight and Weightlessness:

• Gravity can produce acceleration.
• Objects influenced by gravity accelerate towards each other.
• Since we’re always in contact with Earth, we feel gravity as something that presses against us instead of accelerating us.
  • This sensation is weight.
• Consider a weighing scale on the floor.
  • Earth’s gravitational force pulls you towards the scale and the floor.
  • At the same time, the scale and the floor push up towards you. (Newton’s third law)
  • Inside the scale, there’s a spring like device to calibrate weight.
• Consider weighing yourself on a scale in a moving elevator.
  • Elevator accelerates upwards → scale and floor push up harder against your feet.
  • Spring compression in the scale is greater.
  • Scale shows increase in weight.
  • Elevator accelerates downwards → opposite happens
  • Scale shows a decrease in your weight.
• Weight experienced by an object is the force it exerts against a supporting surface.
• If the elevator were in free fall, the scale would read zero.
  • In this case, you are weightless.
  • Even when weightless, gravity acts on a body.
  • Since there is no supporting force, the gravity is not felt.
• Astronauts in orbit don’t have a supporting force.
  • They are always in a state of weightlessness.
  • Until they become accustomed to the weightlessness, they experience space sickness.
  • They are in continuous free fall.
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9.5

Ocean Tides:

• Ocean tides are caused by differences in the gravitational pull between the Moon and the Earth on opposite sides of Earth.
  • On the side of Earth nearer to the Moon → the force between Moon and Earth is stronger
  • On the side farther from the Moon → the force is weaker
  • Reason → gravitational force is weaker with increased distance
• Consider a ball of Jell-O:
  • If the same force is applied on every part of the ball, it stays perfectly round, as if accelerated.
  • If one side is pulled harder than the other, different pulls stretch at it, it isn’t uniform.
• The ball is Earth.
• Different pulls of the Moon stretch Earth. We see this in ocean tides.
  • Ocean bulges happen on opposite sides of Earth.
  • This is why there are two sets of tides: high tides, and low tides.
• Ocean bulges are 1m above surface level.
• Earth spins once a day.
• Conclusion: A fixed point on Earth passes both bulges each day.
  • This produces two sets of ocean tides.
• When a part of Earth passes beneath one of the bulges, the part has a high tide.
• 6 hours later (quarter turn completed), water level of the same part of the ocean is 1m below sea level.
  • This is low tide.
• After another quarter turn, another tidal bulge occurs.
• This is how there are two high tides and low tides a day.
• Tides don’t occur at the same time every day.
  • Reason → When Earth spins, the Moon moves in orbit and appears at the same position in the sky every 24 hours and 50 minutes.
  • High tide cycles are thus 24h 50min.
• The Sun contributes to ocean tides.
  • Sun’s pull on Earth is 180 times the Moon’s pull.
  • Still, the Sun is less than half as effective as the Moon in raising tides.
  • Reason → Sun’s great distance from the Earth.
  • The difference in gravitational pull on opposite sides of the Earth becomes very small.
• Tides due to the Sun and Moon coincide when the Sun, Moon, and Earth are aligned.
  • This leads to high and low tides that are higher and lower than average respectively.
  • These are called spring tides.
  • They aren’t related to spring as a season.
  • They occur during a full Moon or new Moon.
  • Full Moon → Earth is between Sun and Moon. (perfect alignment: lunar eclipse)
  • New Moon → Moon is between Sun and Earth. (perfect alignment: solar eclipse)
• All spring tides aren’t equally high.
  • Reason → distance between the Earth and the Moon and between the Earth and the Sun vary.
  • Orbital paths of Earth and Moon are elliptical, not circular.
  • Moon’s distance from Earth varies by 10%, effect in raising tides varies by 30%.
• Highest spring tides occur when the Moon and the Sun are closest to Earth.
• Neap tides: when the high tides are lower than average, and the low tides aren’t as low as they usually are.
  • Occur when the Moon is halfway between a new Moon and a full Moon, in either direction.
  • Tides due to the Sun and the Moon partially cancel each other.
• Tides are affected by the tilt of the Earth’s axis.
• Opposite tidal bulges are theoretically equal.
• Earth’s tilt makes the two daily high tides unequal.
• Tides don’t occur in ponds.
  • Reason → no part of a pond is significantly closer than the rest of the pond to the Sun/Moon.
• The same logic applies to the fluids in your body.
• Humans aren’t tall enough for tides.
  • The Moon produces micro-tides in our bodies.
  • These are 1/200th of the tides produced by a 1kg melon held 1m above your head.
• Landmasses and friction with the ocean floor complicate tidal motions.
• Eg: Tides break up into smaller “basins of circulation” in many places.
  • What that means → tidal bulge travels like a circulating wave in a small tilted basin of water
  • These cause high tides to occur hours after the Moon is overhead.
• In mid ocean, the range between high and low tides is about 1m.
  • Variations in this range occur in different parts of the world.
  • Range is greatest in Alaskan fjords.
  • Range is most noticeable in the basin of the Bay of Fundy.

Tides in the Earth and Atmosphere:

• Earth is a semi-molten liquid covered by a thin, solid, pliable crust.
• So, Moon-Sun tidal forces produce Earth tides and ocean tides.
• Surface of Earth rises and falls by 1/4m twice a day.
• Volcanic eruptions and earthquakes have a higher chance of occurring when Earth is experiencing a spring tide.
• We live at the bottom of an ocean of air.
  • This ocean of air also experiences tides.
  • We don’t notice them.
• Tidal effects occur in the ionosphere.
• These produce electric currents that alter the magnetic field that surrounds Earth.
  • These are magnetic tides.
  • They regulate the degree to which cosmic rays penetrate into the lower atmosphere.
  • Highs and lows of magnetic tides are greatest when atmosphere is experiencing spring tides.

Ionosphere → upper part of the atmosphere, called so because it consists of ions

Ions → electrically charged atoms that are the result of UV light and cosmic ray bombardment.

Tidal Bulges on the Moon:

• There are two tidal bulges on the Moon.
  • Reason → near and far sides of each body are pulled differently.
• The Moon is pulled into a football shape with the long axis pointing to Earth.
• Unlike on Earth, the tidal bulges are in fixed locations.
  • There’s no daily rising and falling of tides on the Moon.
• Moon takes 27.3 days to revolve on its axis.
  • So, the same lunar hemisphere faces Earth constantly.
  • Reason → center of gravity of the elongated moon is slightly displaced from its center of mass.
  • Earth exerts a small torque on the Moon when the Moon’s axis isn’t lined up toward Earth.
  • This twists the Moon and aligns it with Earth’s gravitational field.
  • This is why the Moon always shows us the same face.
• This tidal lock works on Earth too.
• Our days are getting longer.
  • Increase of 2ms per century.

9.6

Gravitational Fields:

• Earth and the Moon pull on each other.
  • Action at a distance → interact without being in contact
• Moon is in contact with and interacts with the gravitational field of the Earth.
• Properties of space surrounding any massive body alter so the body in this region experiences a force.
  • This alteration of space is a gravitational field.

Rockets and space probes are influenced by the gravitational fields at their locations in space, not Earth’s.

• Gravitational field is a force field.
  • Reason → any body with mass experiences force in the field.
• A magnetic field is another force field.
  • Iron filings line up in patterns around a magnet.
  • Pattern of filings at different points shows strength and direction of the field.
  • Where filings closest together, field is the strongest.
  • Direction of filings shows direction of field.
• Pattern of Earth’s gravitational field can be represented by field lines.
  • Where the lines are closer together, the field is stronger.
  • At each point on the line, direction of field is along the line.
  • Arrows show the direction of the field.
  • Any body/particle in the vicinity of Earth will be accelerated in the direction of the field line at that location.
• Strength of Earth’s gravitational field follows inverse-square law.
• Gravitational field at Earth’s surface varies slightly from location to location.
  • Above large caverns → field weaker
  • Above large subterranean lead deposits → field stronger

Gravitational Field Inside a Planet:

• Gravitational field exists inside Earth too.

• Imagine a hole drilled through Earth from the North Pole to the South Pole.

  • If you started to fall at the North Pole, you’d gain speed down the way to the center, then lose speed “up” to the South Pole.
  • If you didn’t grab onto something at the South Pole, you’d fall back to the North Pole.

• Acceleration reduces on moving towards Earth’s center.

  • Reason → less mass pulling you towards the center, more mass pulling you back up

• Net force at Earth’s center is zero.

  • Reason → Pull down balanced by the pull up

• Maximum velocity, minimum acceleration at Earth’s center.

• Gravitational field at Earth’s center is zero.

• Earth is most dense at its core and least dense at its surface.

• Consider a hypothetical planet with uniform density.

  • The field inside increases linearly.
  • 0 at the center to g at the surface.
  • 

• Imagine a spherical cavern at the center of a planet:

  • Cavern would be cavity free.
  • Reason → gravitational forces in all directions cancel each other out.
  • This doesn’t depend on the size of the planet.

• A hollow planet wouldn’t have any gravitational field inside it.

  • All the gravitational forces inside would cancel out.

• Consider this figure:

  

• According to inverse square law, particle P should only be attracted to the left side by 1/4th the times it’s attracted to the left side.

• This isn’t the case in reality.

  • Reason → Gravity doesn’t depend only on distance. It also depends on mass.

• Region A has 4 times the area and thus 4 times the mass of B.

  • P is 4 times closer to B than A.
  • A is 4 times heavier than B.
  • Thus, P is equally attracted to both A and B with the same force.
  • The forces cancel out.

• Cancellation happens anywhere inside a planetary shell with uniform density and thickness.

• A gravitational field exists within the shell and outside it.

• The gravitational field of a planet acts like all the mass of a planet is concentrated at its center.

  • This applies to the field at the outer surface and beyond that.

• Anyone inside the hollow part would be weightless.

  • Reason → zero gravitational field.

• Gravity can be cancelled inside a body or between bodies.

• Gravity can’t be shielded.

  • Electric forces can be.
  • Reason → they repel and attract.

• Gravity only attracts. Thus, no shielding.

• Evidence of this: Eclipses.

  • The Moon is in the gravitational field of the Sun and the Earth.
  • Shielding of the Sun’s field by Earth would deviate the Moon’s orbit during a lunar eclipse.
  • Even the slightest shielding effect would have accumulated over the years and disrupted the timing of eclipses.
  • No such discrepancies have been found thus far.

Einstein’s Theory of Gravitation:

• Einstein presented a model for gravitation in the early 20th century.
• He thought of gravitational fields as geometrical warpings of 4-dimensional spacetime.
• According to him, bodies dent space and time.
  • The more the mass, the bigger the dent.
• Imagine rolling a marble across a bed with a ball on it.
  • The marble rolls in a straight line when it’s away from the ball.
  • The marble rolls in a curve when it’s rolled closer to the ball, because the surface dents.
  • The closer the marble gets to the ball, the more circular the path gets.
  • Eventually, the marble ends up circling the ball in an orbit.
• By Newton’s theory, the marble curves because it’s attracted to the ball.
• By Einstein’s theory, the marble curves because the surface it moves on is curved.

9.7

Black Holes:

• Imagine being indestructible and traveling to the center of a star in a spaceship:

  • Your weight on the star would depend on your mass, the star’s mass, and the distance between the center of the star and your center of mass.
  • The more the star shrinks, the stronger the gravitational field becomes.
  • Eg: If the star collapsed to 1/10th its radius, your weight would go up 100 times.
  • The velocity needed to escape the star, escape velocity, increases.
  • (It becomes harder to leave the star.)

• The Sun is a star.

• If it’s radius reduced to less than 3km, the escape velocity of its surface would become greater than the speed of light.

  • Not even light would be able to escape.

• The Sun would be invisible. It would become a black hole.

• The Sun itself has too little mass to experience this collapse.

  • Some stars with greater mass than the Sun collapse like this when they reach the end of their nuclear resources.
  • If the stars don’t rotate fast enough, the collapse continues until they reach infinite densities.
  • As the stars keep collapsing, the gravitational force on their surface keeps increasing until it becomes so massive that light can’t escape.
  • Black holes get formed.

• Black holes are completely invisible.

• Black holes are as big as the stars they’re formed from.

• The gravitational field is enormous in the vicinity of a blackhole.

  • Anything which passes too close is drawn into it.
  • Any object that falls into a blackhole will be torn into pieces.
  • Any object in a black hole disappears from the observable universe.

• Black holes are detected by their gravitational influence on nearby matter and stars.

Wormholes are a speculative notion. They open up the possibility of time travel.

• According to evidence, there are binary star systems that have a luminous star and a companion similar to a black hole.

• According to stronger evidence, there are more massive black holes at the centers of many galaxies.

• In a young galaxy (called a quasar), the central black hole sucks in matter than emits a large amount of radiation.

• In an older galaxy, stars circle in powerful gravitational fields around centers that are apparently empty.

• The center of our galaxy holds a black hole with a mass of 4 million solar masses.

9.8

Universal Gravitation:

• Earth is round because of gravity.
• Everything attracts everything.
  • Earth has attracted itself together as far as it can.
  • All the “corners” of Earth have been pulled together.
  • Every point on the surface is equidistant from the center of gravity.
  • Earth is thus a sphere.

Rotational effects make the Sun, the Moon, and the Earth slightly ellipsoidal.

• The planets are pulled by the Sun.

• They also pull each other.

• The effect of the planets pulling each other is minimal compared to the pulling of the Sun.

  • When the planets pull each other, they wobble.
  • Perturbations - the interplanetary forces that cause the wobbling.

• When Uranus was discovered, it showed deviations from its orbit that perturbations could not explain.

• Possible reasons:

  • The law of gravitation was failing at this distance from the Sun
  • (or) There was an eighth planet perturbing Uranus’ orbit.

• The efforts that came from this speculation resulted in Neptune being discovered that very night.

• Pluto was then discovered by the tracking of Uranus’ and Neptune’s orbits.

  • Pluto was discovered at the Lowell Observatory, Arizona.

• Pluto → dwarf planet

  • Dwarf planet → a category that includes certain astroids in the Kuiper belt

• Pluto takes 248 years to make a single revolution.

  • Meaning → its discovered position won’t be seen again till 2178.

• The universe is expanding.

• The universe is accelerating outwards.

  • It is pushed by an antigravity dark energy that makes up 73% of the universe.

• 23% of the universe is made of dark matter.

  • Dark matter → this is invisible matter that also pulls at stars.
  • We see this from the rate at which stars circle galaxies → it’s not just the masses of visible stars pulling on them.

• 4% of the universe is made of ordinary matter.

• Newton’s insights on the working of the universe brought in the Age of Reason.

• Newton uncovered that people could uncover the workings of the universe.

• 

• This formula given by Newton's is major reason for success in science.

• John Locke → English philosopher, argued that observation and reason should always guide humanity.

• He used Newtonian physics to model a system of government.

  • There found adherents in 13 British colonies.
  • These ideas results in the Declaration of Independence and the Constitution of the United States of America.