Gravity
Solar system.
Newton's law of universal gravitation.
Gravity of earth.
Weight.
Motion and gravity.
Escape velocity.
Orbit.
International space station.
Geo stationary satellite.
Elliptical orbit.
Moon's orbit.
Earth's orbit.
Planet's orbit.
Planetary missions.
Universal gravity.
Solar system.
About 5 centuries ago, we believed that Earth, was the centre of the universe.
We believed that the Sun went around the Earth.
Astronomers like Copernicus, Galileo and Kepler,
broke this illusion, from which we suffered.
When we have a pattern of thinking, for generations, sometimes for centuries,
it is difficult to break the belief, that we hold on to.
But we eventually realised that the Earth, is not the centre of the Universe.
The Sun is at the centre of the solar system.
The Earth and the other planets, revolve around the Sun.
These Astronomers, with remarkable accuracy, for that period,
worked out the orbit path of the planets, around the sun.
At this time, we had no explanation of why planets orbit the Earth.
Newton’s law of universal gravitation.
Newton, was one of the first scientist, who thought about,
the forces of attraction, between matter.
The story goes, that Newton thought about gravity,
when he saw an apple falling to the ground.
He speculated, that the Earth, was attracting the apple,
causing it to move down.
He extensively studied, the works of earlier astronomers.
He speculated that the planets, orbited the Sun,
due to a force of attraction, that existed between the Sun and the planets.
Based on his studies and research, he deduced that this force of attraction,
is proportional to the product of the masses.
He also deduced that the force of attraction,
is inversely proportional to the square of distance, between the objects.
He proposed the law of universal gravitation, which states that,
any two bodies in the universe,
attract each other with a force,
that is directly proportional to the product of their masses,
and inversely proportional, to the square of the distance, between them.
This law can be stated as,
’F’ is equal to ‘G’ into ‘m1’ into ‘m2’, divided by ‘r’ squared.
’F’ is the force of attraction, between the masses, in Newtons.
”m1” and ‘m2’ are mass of the two objects, in ‘kg’.
’r’ is the distance between them, in meters.
Capital ‘G’ is the gravitational constant.
Newton calculated the value of ‘G’.
The value of the gravitational constant capital ‘G’, is equal to,
6.673 into 10 to the power of minus 11,
Newton, meter squared by kg squared.
This can be expressed, as a formula,
’F’, is equal to ‘G’, ‘m1’, ‘m2’, divided by ‘r’ squared.
The motions of all the planets, mapped by previous astronomers,
conformed to Newtons law of universal gravitation.
Now it was possible to explain, and mathematically calculate the orbit path,
of each planet.
The force of attraction, between the Sun and the planet,
was directly proportional, to the mass of the Sun, multiplied by mass of the planet,
and inversely proportional, to the square of the distance,
between the planet and the Sun.
The planets orbit the Sun, in a elliptical path.
Each planet had a specific mass,
and was at a specific distance, from the Sun.
Relating the force of attraction, to the laws of elliptical motion, it was possible,
to mathematically calculate the path of each planet
Gravity on Earth.
The Earth also attracts, bodies with mass.
Without gravity, we will all be floating around.
The gravitational force of Earth, is present, all around Earth.
We can derive, the Earth’s gravity, using Newton’s universal law of gravitation.
The mass of the Earth, represented by ‘M’, is equal to,
6 into 10 to the power of 24 squared, kg.
The radius of the Earth, ‘r’, is equal to 6.4 into 10 to the power of 6, meters.
Gravitational constant, capital ’G’, is equal to,
6.673 into 10 to the power of minus 11,
Newton, meter squared by kg squared.
Let the mass of an object, on the surface of the Earth, be equal to ‘m1’.
Force is equal to mass into acceleration.
The gravitational force acting on the body, is equal to,
”m1”, into small ’g’.
Small ‘g’, is the acceleration, due to Earth’s gravity.
Let the force of attraction, between the Earth, and the body, be equal to ‘F’.
”F”, is equal to ‘m1’ into ‘g’.
Applying Newton’s law of universal gravitation,
”F”, is also equal to ‘G’, into ‘M’, into ‘m1’, divided by ‘r’ squared.
”m1”, into small ’g’ = ‘G’, into ‘M’, into ‘m1’, divided by ‘r’ squared
”m1”, is present on both sides, and cancels out.
Small ‘g’, is equal to capital ‘G’, into ‘M’, divided by ‘r’ squared.
Substituting the values for capital ‘G’, ‘M’, and ‘r’,
small ‘g’, is equal to 6.7 into 10 to the power of minus 11,
into 6 to the power of 24,
divided by 6.4 into 10 to the power of 6, squared.
If we work this out, the value of ‘g’, is equal to 9.81 m/sec, squared.
This is the force of acceleration, that the Earth’s gravity imparts,
to all the objects, on its surface.
Small ‘g’, is widely used, because, Earth’s gravity is present,
all over the Earth.
Whichever country or place, we might be in, Earth’s gravitational acceleration,
small ‘g’, is always present.
Value of small ‘g’, is sometimes rounded off, to 9.8 m/sec, squared.
Weight.
Let us take a human being, with a mass of 62 kg.
What will be his weight, on planet Earth?
Weight is a force.
His weight will be equal to, his mass, multiplied by the acceleration.
Acceleration due to gravity, is equal to small ‘g’,
which is equal to 9.8 m/sec, squared.
Weight is equal to 62 into 9.8,
is equal to 607.6 Newtons.
In a strictly scientific sense, we should express our weight, in Newtons.
Since the value of small ‘g’, is almost the same,
through out the surface of the Earth,
we conveniently, refer to our mass as weight.
So, a person with mass of 62 kg, refers to his weight, incorrectly as 62 kg.
Mass refers to the amount of substance.
Force is mass into acceleration.
Weight is equal to mass into acceleration, due to Earth’s gravity.
Weight on the surface of the Earth, is equal to ‘m’, ‘g’.
This equation is commonly used, since most of our calculations,
pertains to mass on Earth.
What happens if this human being, goes to the moon?
His mass will still be the same, that is 62 kg.
The moon’s gravity is much less than Earth.
The acceleration due to moon’s gravity, is one sixth, of that in Earth.
So, the same person, with a weight of 607.6 Newtons, on earth,
will weigh about 60 Newtons, in the moon.
The same person, will have different weights, in different planets.
Jupiter is the biggest planet, in the solar system.
The same person, will weigh much more on Jupiter, than on Earth.
There is another interesting aspect, about gravity.
Gravitational force, is a contact force.
When we stand on Earth, Earth exerts a attractive force, to our mass.
When in contact, with the Earth, Newtons third law applies.
For every action, there is an equal, and opposite reaction.
This is the force, which we feel as weight.
When we stand on a weighing scale, this is the force, that it measures.
Let us take the case, of a sky diver.
When he dives, he is in a state of free fall.
He does not feel his weight, he feels weightless.
This is because he is not in contact, with the Earth.
There is no equal and opposite force, acting on him.
When he opens the parachute, the parachute exerts a resistive force,
due to the Earth’s atmosphere.
This slows down his velocity, of descent.
He is able to land gently on Earth.
Once he comes into contact with the Earth,
he feels his weight, because now the Earth exerts, an equal and opposite force.
The same person who felt weightless, while in free fall, now feels his weight.
In an amusement park, we can also experience, difference sensations of weight.
This can happen in a roller coaster, or a giant wheel.
When going up, we can feel a extra sense of weight.
We feel pressed down, to our seat.
When coming down, we feel a little weightless.
We can sense, that the seat belt, is what is preventing us, from flying off.
The sense of weight, is dependent on the net force, acting on us.
While going up, the net force is more, we feel heavier.
While coming down, the net force is lower, we feel lighter.
When we get back to the ground, we feel our normal weight.
Motion and gravity.
The equations of motion, can be applied to the special case,
where acceleration is due to gravity.
Let ‘u’, be the initial velocity, of a body.
Let ‘v’, be the final velocity.
Let ‘a’, be the acceleration.
”v”, is equal to ‘u’, plus ‘a’,’t’.
When acceleration is due to gravity, ‘a’, is equal to small ‘g’,
equal to 9.8 m/sec, squared.
Let ’t’, be the time taken.
Let ’s’ be the distance travelled.
With gravity as acceleration, the equations of motions are:
”v”, is equal to ‘u’, plus ‘g’,’t’.
”s”, is equal to ‘u’, ’t’, plus half ‘g’,’t’ squared.
”v”, squared is equal to ‘u’, squared, plus 2 ‘g’,’s’.
Let us take a hypothetical case, of Newton’s apple.
Newton is said to have thought about gravity, when he observed the apple,
falling to the ground.
Let us say the apple, fell from a height of 19.6 meters.
”s”, is equal to 19.6 meters.
”u”, is equal to initial velocity, is equal to 0.
”s”, is equal to ‘u’, ’t’, plus half ‘g’, ’t’, squared.
”s”, is equal to 0 plus, half into 9.8 into ’t’, squared.
19.6 = 0 + 4.9 into ’t’, squared.
”t”, squared is equal to 4.
”t”, is equal to 2 sec.
The apple will fall to the ground, in 2 sec.
The final velocity of the apple,
”v”, is equal to ‘u’, plus ‘g’, ’t’.
”v”, is equal to 0 + 9.8 into 2.
”v” is equal to 19.6 m/sec.
The apple will hit the ground, with a velocity of 19.6 m/sec.
Now, let us throw the apple up vertically.
We will throw it with an initial velocity of 19.6 m/sec.
Now gravity is acting against the velocity of the apple.
Gravity will slow down, the apple.
At what time, will the velocity, become 0?
”u”, is equal to 19.6 m/sec.
”v”, is equal to 0.
We can say, ’v’, is equal to ‘u’ minus ‘g’,’t’.
Minus ‘g’, is used, because gravity, is acting against the direction of the velocity.
0 is equal to 19.6 minus 9.8 into ’t’.
”t”, is equal to 2 sec.
After 2 sec, the apple will have 0 velocity.
How far up, will the apple travel?
We can calculate this using,
”v”, squared is equal to ‘u’, squared, plus 2, a, ’s’.
”v”, is equal to 0.
”u”, is equal to 19.6 m/sec.
”g”, is equal to 9.8 m/sec, squared.
0 is equal to 19.6 squared, minus, 2 into 9.8, into ’s’.
”s”, is equal to 19.6, into 19.6, divided by 19.6.
”s”, is equal to 19.6 m.
An apple thrown up with a velocity, of 19.6 m/sec,
will reach a height of 19.6 m, and then return to Earth.
Let us try, to launch the apple, at higher speeds.
If we launch at 196 m/sec,
it will reach a height of 1960 m, and then return to Earth.
If we launch it at 1960 m/sec,
it will reach a height of 196000 m,
that is, it will reach a height of 196 km.
The higher the initial velocity, higher will be the altitude, that it will reach.
We need to pause, to think here.
We made the assumption that the accelerating force of gravity,
is a constant 9.8 m/sec.
For all practical purposes, this is true, on the surface of the Earth.
Even on top of Mount Everest, the value, is not significantly different.
Newton’s law of universal gravitational force,
”F”, is equal to capital “G”, ‘m1’, ‘m2’, divided by ‘d’, squared.
”d” is the distance between the object, ‘m1’ and ‘m2’.
In our example, ‘m1’, and ‘m2’, are the masses, of the Earth and the apple.
When the distance, between them increases, the force of attraction decreases.
Force of attraction, is proportional to 1 by ‘d’, squared.
So, when the apple travels to a very high altitude, the force of attraction,
significantly decreases.
The gravitational acceleration of Earth, decreases as we move further and further,
from the Earth.
So, when we are dealing with larger distances, we should take into account,
the decrease in the gravitational force.
Escape velocity.
The higher the apple goes, lower is the gravity.
At some point, the velocity of the apple, is sufficient to overcome,
Earth’s gravitational pull.
We will now call the apple as a projectile.
The velocity at which a projectile, should be launched, so as to escape,
Earth’s gravity, is called ‘Escape velocity’.
Since, the equations for all the forces, are known, we can work out this velocity.
The formula for escape velocity is,
square root of, Open bracket, 2 into capital ‘G’, into ‘M’, divided by ‘r’,Close bracket
Capital ‘G’, is the gravitational constant, equal to,
6.673 into 10 to the power of minus 11,
Newton, meter squared by kg squared.
”M’, is the mass of the Earth, equal to,
6 into 10 to the power of 24 squared, kg.
When we work this out, escape velocity ‘v’,
becomes equal to 11.2 km/sec.
If a projectile is launched from the surface of the Earth,
at a velocity of 11.2 km/sec, it will escape Earth’s gravity,
and never return to Earth.
11.2 km/sec is a very large velocity.
A bullet from a gun, for example, travels at 1.7 km/sec.
Even with powerful rockets, it is impractical, to launch a projectile,
at a velocity of 11.2 km/sec.
But we have successfully sent many missions to the moon, to mars,
to a comet, and to outer space.
How did these projectiles escape, the gravity of Earth?
Scientists have found a ingenious method to overcome,
the practical difficulty.
The projectile is first launched into, low Earth orbit.
The force of gravity, is less at this altitude.
In orbit the projectile, has a orbital velocity.
Taking advantage of these two factors,
additional thrust, is provided, to the projectile, while in, low Earth orbit.
In this manner, the projectile, is able to escape, the Earth’s gravity.
Once, it has escaped the Earth’s gravity, Newton’s first law, comes into effect.
Newton’s first law states,
An object remains at rest, or moves at constant velocity,
unless acted upon, by an external force.
We can state this law, in two parts, as follows.
1. An object, which is at rest, will stay at rest, unless an external force,
acts upon it.
2. An object, which is in motion, will not change its velocity,
unless an external force, acts on it.
Part 2 of Newton’s first law, now comes into effect.
A projectile which escapes the Earth’s gravity, will travel at a constant velocity.
There is no need, to provide any thrust, from a rocket engine.
This is how, we are able to send a mission to the moon, and other planets.
Orbit.
Many of Newton’s theories could not be tested, in a laboratory.
The Earth and the solar system, were his laboratory.
He used to indulge, in what he called as ‘Thought Experiments’ .
He used his intellect, and his imagination, to propound his theories.
One of the theories, related to objects orbiting the Earth.
He conducted a thought experiment.
He imagined, that a projectile is launched, from the top of a mountain,
in a horizontal direction.
We can imagine, that it will travel for a certain distance,
and then fall to Earth.
This is because, Earth’s gravity is pulling it down.
The path of the projectile, will be a curve.
If the projectile is launched, with the higher velocity,
it will travel a longer distance, and then fall to Earth.
This is easy to understand.
With the known equations for motion, we can actually calculate,
the distance it will travel.
Now let us think about, another interesting factor.
The Earth is a sphere.
The surface of the Earth, is curved.
This curvature, is more obvious, over longer distances.
If we stand at a sea shore, near a port, and gaze into the horizon,
we might not see any object.
A ship might be actually approaching the port.
But, we cannot see it, because of the Earth’s curvature.
If we wait patiently, we can see it emerge from the horizon.
For every 8 km of horizontal distance, the Earth curves in, by about 5 m.
Let us assume, that the ship is 5 m tall.
When the ship is 8 km from the shore, the ship cannot be seen.
This is because, at a distance of 8 km, the Earth has curved down, by 5 m.
As the ship approaches closer, it appears to rise above the horizon,
and comes into our line of vision.
In Newton’s thought experiment, the imaginary mountain,
from which he launched, an imaginary projectile,
is sometimes called as “Newton’s Mountain”.
Let us launch a projectile, in a horizontal direction,
from the top of Newton’s mountain.
Let the initial velocity of launch be 8 km/sec.
In one second, this projectile, will travel 8 km, in the horizontal direction.
The force of gravity, is acting on this projectile.
The force of gravity, acts in a vertical direction.
We can imagine, that this force, will be perpendicular, to the direction of motion,
of the projectile.
For convenience, let us approximate, the acceleration, due to gravity,
has 10 m/sec squared.
How much distance, will the projectile, fall vertically in one second?
”s”, is equal to ‘u’, ’t’, plus half ‘g’, ’t’, squared.
Initial velocity in vertical direction, ‘u’, is equal to 0.
”g”, is equal to 10 m/sec.
”s”, is equal to 0, plus half, into 10, into 1 squared.
”s” is equal to 5 m.
The projectile, will fall by 5 m, in one second.
The projectile, has travelled 8 km horizontally,
and 5 m down, vertically.
After 8 km, how far will the projectile, be from the ground?
After 8 km, the Earth has curved down, by 5 m.
So, the projectile, will be at the same altitude, from the Earth.
The projectile, is in orbit.
We can understand this by extending our thought experiment.
For every 8 km, that the projectile travels horizontally,
the Earth’s gravity will pull it down by 5 m.
This will happen continuously.
The projectile will circle the Earth, without ever losing it’s altitude.
This is the gravitational mechanics, of an orbit.
The speed of the orbit, of 8 km/sec, is = 28,800 km/hr.
Orbit is an important function, of gravitational force.
Hundreds of satellites, orbit the Earth, using this concept.
The moon orbits, the Earth, using this concept.
Earth orbits the Sun, using this concept.
All the planets, orbit the Earth, using this concept.
Using a thought experiment, Newton was able to explain,
how the solar system works.
Newton lived many centuries before the first artificial satellite was launched.
But his thought experiment, predicted, how it could be done.
Such is the power of imagination and intellect, if we care to use it.
International space station.
Many countries have joined together, and launched,
a international space laboratory.
It is called the international space station, and was launched, in the year 2000.
It is called as I S S in short.
Many astronauts have travelled to I S S, conducted space experiments,
and returned to Earth.
The I S S orbits, the Earth, at an altitude of about 400 km.
It’s average speed is about 27,700 km/hr.
This is close to the 8 km/sec, or 28,800 km/hr, that Newton theorised.
It orbits the Earth, once in 93 minutes.
Astronauts working in the space station, experience a feeling of weightlessness.
Is it because, the Earth’s gravity is not present at a height of 400 km?
The answer is “No”. Earth’s gravity is very much present, in the I S S.
The acceleration due to gravity, at a height of 400 km,
is about 8.7 m/sec, squared.
The astronauts in the I S S, are in a virtual state of free fall.
This is very much similar to the sky diver, who feels weightless.
The I S S which is in orbit, is in a state of free fall.
It falls about 5 m/sec, but it retains it’s altitude.
The astronauts are also in a state of free fall.
This is why, they do not feel their weight.
Geo stationary satellite.
Geo stationary satellites, orbit the Earth, at a height of about 36,000 kilo meters.
It orbits the Earth, once in 24 hr.
But, when observed from a point in Earth,
it appears to be in the same spot, in the sky.
This is because the Earth rotates on its axis, once in 24 hr.
So, relative to the Earth, the satellite appears to be stationary.
This is very useful for many applications.
We are able to position a satellite, which is always in the same position,
above us, in the sky.
This is widely used, in communication and other applications.
If we transmit a signal, to a geo stationary satellite,
it can transmit it back to Earth.
This return transmission can cover a large area in Earth.
For example, a T V station, any where, in the line of sight,
of the satellite, can upload a program.
A geo stationary satellite, can transmit it back.
This transmission can be received through out the country,
and even in neighbouring countries.
Television, mobile communications, GPS are some of the applications,
which makes use of geo stationary satellites.
Elliptical Orbit.
A Ellipse is like a compressed circle.
Unlike a circle, a ellipse does not have uniform radius.
When a satellite is in elliptical orbit, around the Earth,
it’s nearest point to the Earth, is called as the Perigee.
The furthest point from the Earth, is called as the Apogee.
Objects in elliptical orbit, also follow Newton’s law of universal gravitation.
The calculations are adapted, for elliptical orbits.
Many orbits in nature, are elliptical.
The Moon, has an elliptical orbit, around the Earth.
The Earth and the planets, are in elliptical orbit, around the Sun.
Moon’s Orbit.
The Moon orbits the Earth.
It has an elliptical orbit.
It’s perigee, is about 360000 km.
It apogee, is about 400000 km.
We can observe the phases of the moon, from full moon,
to no moon, to full moon.
This happens in cycles.
The cycle time period from full moon to full moon, is 29.5 days.
This is called the synodic month.
Early calendars were based on this lunar month.
We now however, follow a solar calendar.
Usually, in a solar calendar month, there will be one full moon.
The number of synodic month cycles in a year, is about 12.4.
In certain months, we can observe, two full moons in a month.
This understandably happens, very infrequently.
Colloquially, we call the second full moon occurrence,in a month, as a blue moon.
Since, the moon is an elliptical orbit, some full moons will be closer to Earth.
The full moon, when the moon is closest to Earth, is called as the super moon.
During the super moon, the moon appears much bigger and brighter.
The orbit of the moon, is also governed by the gravitational force,
between the Earth and the Moon.
The moon’s gravity, can be felt on Earth.
High tides and low tides, in the sea, is due to moon’s gravity.
Earth’s Orbit.
The Earth orbits the Sun, once in 365.25 days.
Our solar calendar is based on this.
The solar calendar, has 365 days in a year.
To make up for the extra quarter day, we add 1 day every four years.
This year, is called as a leap year.
The distance of the Earth, from the sun, is about 150,000,000 km.
This distance is called one astronomical unit.
Planet’s Orbit.
The solar system has 8 planets.
Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus and Neptune.
Mercury is the planet closest to the Sun.
It orbits the sun, once in about 88 Earth days.
Jupiter is the biggest planet.
It orbits the sun, once in about 12 Earth years.
If some one is 60 years old, on Earth, he will be only 5 Jupiter years old.
Neptune is the furthest planet.
It orbits the sun, once in about 165 years.
All these planetary motions are governed by,
Newton’s law of universal gravitation.
Planetary missions.
We have sent many missions, to the moon.
We have also sent, missions to Mars, and other planets.
We have sent voyager satellites, which has travelled,
throughout the solar system, and beyond.
To achieve this, we have to escape, the Earth’s gravity.
This can be done by achieving, a escape velocity, of 11.2 km/sec.
This can be done in a practical, two step process.
First the satellite, is launched into a low Earth orbit.
In this orbit, it already has a speed of about 8 km/sec.
By giving it additional thrust, we can increase this to 11.2 km/sec.
This will cause the satellite to escape the gravity of Earth.
If we point it in the right direction, it can travel to the moon.
Once it has escaped the Earth’s gravity, it does not require any more thrust,
from rocket engines.
Obeying Newton’s first law, It will continue to travel,
at the same speed, towards the moon.
Same way, missions are launched to Mars.
In practice, since the planets are in motion, some course corrections,
are made during the voyage.
Once the satellite, comes close to mars, it is attracted,
by the gravity of Mars.
By suitable manoeuvring, we can place the satellite, in an orbit around Mars.
The principles of orbit of Mars, is the same as the principles, of orbit of Earth.
India has sent a satellite, called Mars Orbiting Mission, or MOM in short.
It is orbiting, the planet Mars.
There are man made satellites, orbiting Mercury, Venus, and Saturn.
We have launched a space craft called voyager, which has travelled more than,
ten thousand million kilo meters, into outer space.
It has no intention of returning back to Earth.
But during its voyage, it has provided valuable images and data,
of other planets and outer space.
Many more exploratory planetary missions, are being planned.
Universal gravity.
We are able to stand on Earth, because of gravity.
Earth and all other planets, orbit the Sun, because of gravity.
Gravity is truly universal.
We have not reached the end, of fully grasping the nature of gravity.
Research is still going on, to understand gravity, even better.