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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 mightâve 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 wouldâve followed if there were no forces acting on it.
It falls âaroundâ the Earth.
Reason â Tangential velocity
The Moonâs distance of fall person second is comparable to the distance the apple falls in one second.
This didnât align with Newtonâs 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 Earthâs mass, but couldnât 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 Earthâs are involved.
Force of attraction between you and Earth â 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 â distance from Earthâs 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.
Newtonâs law of gravity applies to particles, spherical bodies, and non-spherical bodies sufficiently far apart.
In Newtonâs equation, d â distance between the centers of masses of the objects
For objects on Earth:
As the distance between the object and Earthâs surface increases, Earthâs 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 weâre 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.
Earthâs gravitational force pulls you towards the scale and the floor.
At the same time, the scale and the floor push up towards you. (Newtonâs third law)
Inside the scale, thereâs a spring like device to calibrate weight.
Consider weighing yourself on a scale in a moving elevator.
Elevator accelerates upwards â scale and floor push up harder against your feet.
Spring compression in the scale is greater.
Scale shows increase in weight.
Elevator accelerates downwards â 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 donât 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 â the force between Moon and Earth is stronger
On the side farther from the Moon â the force is weaker
Reason â 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 isnât 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 donât occur at the same time every day.
Reason â 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.
Sunâs pull on Earth is 180 times the Moonâs pull.
Still, the Sun is less than half as effective as the Moon in raising tides.
Reason â Sunâs 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 arenât related to spring as a season.
They occur during a full Moon or new Moon.
Full Moon â Earth is between Sun and Moon. (perfect alignment: lunar eclipse)
New Moon â Moon is between Sun and Earth. (perfect alignment: solar eclipse)
All spring tides arenât equally high.
Reason â 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.
Moonâs 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 arenât 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 Earthâs axis.
Opposite tidal bulges are theoretically equal.
Earthâs tilt makes the two daily high tides unequal.
Tides donât occur in ponds.
Reason â 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 arenât 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 âbasins of circulationâ in many places.
What that means â 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 donât 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 â upper part of the atmosphere, called so because it consists of ions
Ions â 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 â 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.
Thereâs 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 â 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 Moonâs axis isnât lined up toward Earth.
This twists the Moon and aligns it with Earthâs 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 â 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 Earthâs.
Gravitational field is a force field.
Reason â 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 Earthâs 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 Earthâs gravitational field follows inverse-square law.
Gravitational field at Earthâs surface varies slightly from location to location.
Above large caverns â field weaker
Above large subterranean lead deposits â 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, youâd gain speed down the way to the center, then lose speed âupâ to the South Pole.
If you didnât grab onto something at the South Pole, youâd fall back to the North Pole.
Acceleration reduces on moving towards Earthâs center.
Reason â less mass pulling you towards the center, more mass pulling you back up
Net force at Earthâs center is zero.
Reason â Pull down balanced by the pull up
Maximum velocity, minimum acceleration at Earthâs center.
Gravitational field at Earthâs 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 â gravitational forces in all directions cancel each other out.
This doesnât depend on the size of the planet.
A hollow planet wouldnât 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 itâs attracted to the left side.
This isnât the case in reality.
Reason â Gravity doesnât 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 â zero gravitational field.
Gravity can be cancelled inside a body or between bodies.
Gravity canât be shielded.
Electric forces can be.
Reason â 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 Sunâs field by Earth would deviate the Moonâs 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.
Einsteinâs 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 itâs away from the ball.
The marble rolls in a curve when itâs 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 Newtonâs theory, the marble curves because itâs attracted to the ball.
By Einsteinâs 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 starâs 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 itâs 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 donât 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 canât escape.
Black holes get formed.
Black holes are completely invisible.
Black holes are as big as the stars theyâre 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 âcornersâ 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 Uranusâ orbit.
The efforts that came from this speculation resulted in Neptune being discovered that very night.
Pluto was then discovered by the tracking of Uranusâ and Neptuneâs orbits.
Pluto was discovered at the Lowell Observatory, Arizona.
Pluto â dwarf planet
Dwarf planet â a category that includes certain astroids in the Kuiper belt
Pluto takes 248 years to make a single revolution.
Meaning â its discovered position wonât 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 â this is invisible matter that also pulls at stars.
We see this from the rate at which stars circle galaxies â itâs not just the masses of visible stars pulling on them.
4% of the universe is made of ordinary matter.
Newtonâs 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 â 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.
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