FORCE
Gravitational Force
The force that attracts any two objects with mass is called gravitational force.
It is also defined as the mutual force of attraction between any two heavenly bodies due to their masses which are acting equal and opposite.
It is the weakest force between any two masses. So, it is not noticeable.
The SI unit of gravitational force is Newton [N].
We call the gravitational force attractive because it always tries to pull masses together, it never pushes them apart.
The factors affecting gravitational force are product of masses and distance between the centers of masses.
The effect of gravitation can be seen more on liquids than in solid due to more intermolecular space of liquids than that of solids.
Consequences / examples/ effects of gravitational force
Existence of solar system due to the gravitational force between the sun and other heavenly bodies.
Occurrences of sea tides due to the gravitational force of the sun and the moon on the sea water.
The satellite orbiting around the planet due to their gravitational force.
Newton’s law of Gravitation [ Propounded by Sir Isaac Newton in 1687AD]
Newton’s universal law of gravitation states that “Every object in this universe attract each other with a force called gravitation which is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers.”
i.e. F = GMm / d2
The direction of force is along the line joining the center of two masses.
Suppose two bodies A and B having their masses m1 and m2 respectively are separated by a distance 'd' from their centers. If the force of attraction between them is F, then according to Newton's universal law of gravitation:
F ∝ m1 x m2 ……………… (i) [Keeping the distance constant]
And
F ∝ 1/d2 ……………. (ii) [Keeping the masses m1 and m2 constant]
Combining Equation (i) and (ii), we have,
F ∝ m1 x m2 / d2
[Where G is a constant of proportionality which is known as universal gravitational constant and its value is 6.67 x 10 -11Nm2 kg -2]
Hence, the Equation (iii) gives the measure of the gravitational force between two bodies.
Gravitational law is also called the Universal law because this law is applicable/holds true for all the bodies terrestrial or celestial.
Effect on gravitational force with its factors
Gravitation force depends upon the product of the masses and distance between their centers. This relationship can be explained by using following examples,
The masses of both objects are doubled keeping distance constant.
Let m1 and m2 be the masses of two bodies separated by distance d, If F is the force of attraction between them then,
F1 = G m1 m1 /d2……….eqn (i)
From the above statement,
If the masses of both objects are doubled by keeping distance constant.
m1 = 2m1
m2 = 2m2
d = d
∴ New force of attraction (F2)
F2=G 2m1 x 2m2/d2
=4 G m1m2/d2
Therefore, the new force will be four times more when the masses of each bodies increased by two times.
The mass of one body is doubled with distance reduced to half
Let m1 and m2 be the masses of two bodies separated by distance d, If F is the force of attraction between them then,
F1 = G m1 m1 /d2………..eqn (i)
From the above statement,
If the mass of one object is doubled by with distance reduced to half.
m1= 2m1
m2 = m2
D = d/2
∴ New force of attraction F2
F2=G2m1m2/(d/2)2
=2 G m1m2/d2/4
=2x4 G m1m2/d2
Hence the new force will be 8 times more than the original force when one body is doubled with distance reduced to half
Universal gravitational constant (G)
The universal gravitational constant, denoted by letter G, can be defined as the force of gravitation produced between two bodies of unit mass each, separated by unit distance between their centers
. It was first calculated by Henry Cavendish by using sensitive balance [ called torsion balance].
It is called universal gravitational constant since the value of G is independent to any two masses of the universe, their shape or size or structure.
It is expressed as:
G = F x d2 / m1 x m2
Condition of F = G
We have,
F = G (m1 x m2 / d2)
If m1 = m2 = 1 and d = 1m. Putting the value of m1 and m2 and d in the given relation,
F = G (1 x 1 / 12)
∴ F = G
Thus, gravitational force is equal to gravitational constant when any two bodies are having unit masses and separated by unit distance apart from their centers.
Unit of G
We have,
G = F x d2 / m1 x m2
In the SI system, the unit of force (F) is Newton(N), distance(d) is meter(m) and mass(m) is kilogram(k). Then the unit of G is:
G = newton x (meter)2 / kg x kg = Nm2 / kg2
∴ The SI unit of G is Nm2 kg -2.
Gravity
The force of attraction with which the earth pulls a body towards its center is called gravity.
The factors that affect gravity are mass and radius of the planets and satellites.
The SI unit of gravity is N.
Formula to calculate Gravity
Gravity (F) = GMmR2
M = Mass of the heavenly body
m= mass of an object on the heavenly body
R = Radius of the heavenly body
Gravity of a planet or satellite or weight of objects of mass "m" depends upon:
Mass of planets or satellite( M) [ i.e. FαM or WαM]
Radius (R) of planets or satellite (R) [ i.e. Fα1/R2 or Wα1/R2]
Mass of the bodies (m) i.e. Fαm or Wαm.
Effects of Gravity
We are able to stand up, run and perform other activities freely due to gravity of the earth.
Construction of buildings and bridges is possible due to the gravity of the earth.
The existence of atmosphere around the earth is due to the force of gravity of the earth. The atmosphere makes life possible on the earth. As the gravity of moon is very small, it does not have its atmosphere.
If a body is thrown upwards, its motion is opposed by the gravity of the earth. So, it comes back to the earth.
The flowing property of liquid is also one of the effects of gravity. For example: flowing of water and streams etc.
Acceleration due to gravity:
Acceleration due to gravity is the acceleration produced in a freely falling body due to the force of the gravity.
It is denoted by g and its SI unit is m/s2.
The average value g is 9.8 m/s2, g at poles is 9.83m/s2 and g at equator is 9.78m/s2
The value of g at moon is 1.66 [1.67]m/s2 and that of Jupiter is 25m/s2.
Acceleration due to gravity of the earth is 9.8m/s2 means that every freely falling object towards the earth increases its velocity by 9.8m/s in every second.
Prove g=GMR2 or gα1R2or [Acceleration due to gravity is independent to mass of the object]
Let ‘M’ be the mass of the earth and ‘R’ its radius; then the force of a body of mass ‘m’ in its surface.
According to Newton’s law of gravitation, the force of attraction between them is given by,
F= GMmR2……… (i)
From the second law of motion,
F = mg……….(ii)
Combining equation (i) and (ii)
We get,
Or, mg=GMmR2 (since, F=mg)
(Since, cancelling out m on both sides)
As GM and are constant
gα1R2
It means that acceleration due to gravity is inversely proportional to the square of radius. (if radius is more, acceleration due to gravity becomes less and vice versa)
The shape of earth is not completely spherical. Somewhere it is flat and somewhere it has hills and mountains. Its polar regions are flattened a little and it bulges out at the equator. Due to this, radius of different place is different. As radius changes from place to place, value of g also changes from place to place.
Questions
The value of g is more at poles than at the equator. Why?
As we know gα1R2. The radius is less at the poles due to which the value of g is also more at the poles. But radius is more at the equator due to which the value of g also becomes less.
Weight of an object is more at terai that at the mountain. Why?
As we know W = mg. Acceleration due to gravity is more at terai than at the mountain. So, the value of g is also more at teria than at the mountain.
Acceleration due to gravity at a certain height g= GMR+h2
Variation of g due to height
Consider an object of mass m at a height h from the surface of the Earth. The acceleration due to gravity at the surface of the earth is g=GMR2…. ( i)
and acceleration due to gravity due to height is
gh= GMR+h2….(ii)
Dividing equation (ii) by equation (i)
ghg=GM(R+h)2GMR2
ghg=R2(R+h)2
Thus, value of acceleration due to gravity decreases with the height from the earth’s surface
Note:
g is zero [i.e.gd=0] at the center of the earth since gd=1-dRg, as d=R.
For an object placed at a height h, the acceleration due to gravity is less as compared to that placed on the surface.
As depth increases, the value of acceleration due to gravity (g) falls.
The value of g is more at poles and less at equator.
Formulae related to Acceleration due to gravity
g=GMR2 m/s2
gh= GMR+h2 m/s2
W = mg (N)
Falling bodies and acceleration due to gravity
Before Galileo Galilei, it was believed that the speed with which a body falls to the ground depends on its mass. Higher the mass more will be its velocity.
But, Galileo Galilei did an experiment to demonstrate it in 1950 and proved that all the bodies of different masses fall with the same acceleration.
He released several iron balls of different masses from the top of leaning tower of Pisa.
All the balls reached the ground at the same time.
Air resistance plays an important role during the fall of the bodies.
If a stone and a feather are allowed to fall from a height, the stone will fall on the ground much quickly than the feather.
This is because more air resistance will act on the larger surface area than on the smaller surface area which ultimately affects their acceleration.
It proves that acceleration due to gravity is same for all falling bodies if air resistance is neglected.
Coin and feather experiment [performed by Sir Isaac Newton
The shown experimental fact is the fall of an object does not depend on its mass. The acceleration of freely falling bodies is found to be equal for all bodies. If the object falls through air, then air resistance must also be taken into account.
In the first figure: - Since the feather has a large surface area than the coin, the air resistance on it very quickly builds up to equal the pull of gravity. After that, the feather gains no more speed, but just drifts slowly downward. Whereas, the coin has a small surface area so it encounters less air resistance and falls faster with less disturbance.
In the second figure: - The tube is evacuated using a vacuum pump so with no presence of air there is no air resistance. So, in the absence of air resistance the coin and feather fall simultaneously due having the same acceleration due to gravity.
In conclusion: Larger the surface area of an object it experiences more air resistance than a smaller surfaced object. If air resistance can resist the force of gravity in greater amount or balance it than the object will fall slowly hovering through the air but if it cannot resist the force of gravity much than it will fall much faster. With no air resistance all objects fall at the same rate since acceleration due to gravity is independent to mass.
Acceleration due to gravity of freely falling bodies, whatever their masses, is the same in the absence of air resistance. Larger the surface area, more will be air resistance.
Free fall
If a body falls freely under the influence of gravity only neglecting air resistance, then the fall of the body is said to be free fall.
Free fall is any motion of a body where gravity is the only force acting upon it.
In other words, if a body is falling with acceleration equal to the acceleration due to gravity then the body is said to have a free fall
A body falling in vacuum or space have free fall.
Since moon does not have the atmosphere, the fall of the bodies in it becomes free fall.
On earth, there is no perfect free fall of any falling body due to air resistance.
During free fall, the body is in weightlessness state as the body feels zero reaction force.
Weightlessness
Weightlessness is the state of the body in which it feels that the body is not influenced by any force
Conditions of weightlessness
When a body falls freely under the influence of force of gravity only or when the acceleration of a falling body is equal to the acceleration due to gravity.
When a body in at the earth’s center or in space at null point (i.e. g=0)
The body in the artificial satellite becomes weightlessness when the artificial satellite is orbiting around a heavenly body.
Weightlessness in a lift
A person standing in a lift moving down along with it near the surface of the earth. The person feels weightless. When the lift is falling freely with acceleration, the floor of the lift cannot exert any upward force to prevent it from falling. The person is not pressing any force on the floor and hence no reaction force takes place and the person feels weightless.
Activity of measuring the weight of a stone with a spring balance. Spring balance shows the weight of the stone when we carry the stone but shows zero weight when it is released from our hand.
Falling from a parachute
A parachute is used for jumping out from a flying aero plane. It opens and expands when a para trooper jumps out from the aero plane. Due to large surface area of the parachute, the air resistance will be more. As a result, the acceleration of the parachute decreases. The decreased acceleration due to gravity causes the para trooper to fall down slowly. As a result of slow motion, the para trooper can balance their body and land safely without any hurt.
The use of parachute is not safe on the moon, why?
It is because of the absence of atmosphere on the moon, hence no air pressure is felt. Thus, the parachute gets attracted by the gravitational force of the moon and the velocity of the parachute is not controlled causing the parachute falls forcibly onto the surface of the moon.
At what condition is the acceleration due to gravity of falling parachute approximately zero while getting down from a parachute?
When air resistance is equal to the force of gravity of the earth on the falling parachute the acceleration of the falling parachute is nearly zero while getting down from the parachute.
A person in an artificial satellite of the earth feels weightless. But the same person on the moon has weight though the moon is also the satellite of the earth.
When a spacecraft or the moon orbits the earth, necessary centripetal force is provided by gravitational force between the earth and the spacecraft or the moon. The only force that the astronaut experience while orbiting the earth is due to the gravitational force exerted by the spacecraft. This force is negligibly small force and the astronaut does not experience any gravity/weight. However, when he is on the moon, he will experience the force of gravity of the moon. As the mass of the moon is much more relative to the spaceship, the person has weight there.
Differences between weightlessness and freefall
Differences between acceleration due to gravity and gravitation
Differences between mass and weight
Differences between acceleration due to gravity and gravitational constant
Differences between weightlessness due to freefall and weightlessness in space.
For numerical related to Gravitational Force
Formula
F = Gm1m2d2
Where,
F = Gravitational force/ Force of attraction---Newton (N)
M1 = Mass of first body--kilogram(kg) --if gram is given then it should be converted into kg (divide by 1000)
M2 = Mass of second body-- kilogram(kg) --if gram is given then it should be converted into kg (divide by 1000)
D = distance between the bodies--meter (m)--if kilometer is given then it should be converted into meter (multiply by 1000)
G = Gravitational constant (G= 6.67 x 10-11Nm2/kg2) ------[need to remember this value]
For Gravity
Gravity (F) = GMmR2
Important Topics
Gravitation
Gravitational law and its derivation.
Why is gravitational law called the universal law?
Gravitational constant
Why is gravitational constant called universal constant?
Value of G is 6.67 x 10-11Nm2/kg2. What does it mean?
It means that the force of attraction between two unit masses separated by a unit distance is 6.67 x 10-11N.
Acceleration due to gravity of earth is 9.8m/s2. What does it mean?
It means that the velocity of the falling bodies on earth surface increases by 9.8m/s in 1 second.
Mass and weight
G and g
Acceleration due to gravity and its derivation.
Value of g is more at poles than at the equator. Why?
Weight of an object is more at terai than at the mountain. Why?
Free fall
Weightlessness
Conditions for weightlessness
Conclusion of feather and coin experiment
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