FORCE
Why does the moon revolve around the earth? Why does the earth revolve around the sun? Why is there no collision between the stars and planets in the universe? What is the difference between the mass and weight? The answer to such questions and other related questions take us into subject of gravitation.
Gravitation
In the universe, one object attracts another object with a certain force. The force of attraction that exists between them is termed as gravitational force. It is simply called gravitation. Such a force is attractive in nature. Its unit is N (Newton) in SI system and dyne in CGS system.
Experimentally, it is found that the gravitational force between any two objects depends upon their masses and the distance between their centers. After different observations, the great scientist Sir Isaac Newton formulates the experiment in the experiment in the form of law which is known as Newton's Universal Law of Gravitation.
Newton's Universal Law of Gravitation
It states that the gravitational force between any two objects directly proportional to the products of their masses and inversely proportional to the square of distance between their centers. This law is applicable in the masses of two objects on the universe. So. it is also called the Universal Law of Gravitation.
Let consider two objects of masses 'm1' and 'm2' which are separated by a distance 'd' as shown in the figure. If 'F' is force of attraction between the two objects, then according to Newton's Universal Law:
i. The force is directly proportional to the product of their masses, F∝ m1m2............(i)
ii. The force is inversely proportional to the square of the distance between them, F∝1/d^2..........(ii)
Combining (i) and (ii), we get;
F∝ m1m2/d^2,
F= G (m1m2/d^2)..............(iii)
Where G is a proportionality constant called universal gravitation constant. G is called universal gravitational constant because its value remains same throughout the universe.
Definition and unit of the universal gravitational constant (G)
We know that; F=G(m1m2/d^2) [By Newton Universal Law]
In SI system, If m1=1kg, m2=1kg and d=1m
then F=G.
The universal gravitational constant is numerically equal to the force of attraction between two masses of 1kg each at a distance of 1m from their centers.
Similarly in CGS system, the universal gravitational constant is the force of attraction between two masses of 1g each kept at a distance of 1cm from their centers.
So in general, the force of attraction between any two objects of unit masses when they are separated by unit distance is called universal gravitational constant.
The value of G could not be found when Newton gave the la of gravitation. It was determined by Henry Cavendish in 1798 AD by using an instrument called torsion balance and he found the value equal to 6.67x10^-11 Nm^2/kg^2 in SI system.
The value of G in CGS is 6.67x10^-8 dyne cm^2/g^2.
Why is the gravitational force between two objects usually unnoticeable?
The value of G is very small and hence the gravitational force between two smaller objects becomes negligibly small. Therefore, the gravitational force between any two objects is usually unnoticeable. It becomes appreciable only when at least one body is huge like earth, the sun, etc.
Some important points which should be remembered regarding gravitational force are:
i. The gravitational force does not change with the change in medium.
ii. It is the weakest force in nature and is always attractive.
iii. It obeys the inverse square law.
iv. Gravitational force is a central force. This means that this force acts along the line joining the particles towards their center.
v. Gravitational force in the long range force. It is applicable between stars and galaxies.
vi. Newton's law of gravitation is universal. This means that it is applicable anywhere in the universe.
Points to know:
i. Gravitational force is increased by two times when mass of an object is doubled keeping the distance between them constant.
ii. Gravitational force is four times when mass of each object is doubled keeping the distance between them constant.
iii. Gravitational force is decreased by four times when distance between them is made double keeping their masses constant.
iv. Gravitational force is increased by four times when distance between them is made half keeping their masses constant.
Gravitation
In the universe, one object attracts another object with a certain force. The force of attraction that exists between them is termed as gravitational force. It is simply called gravitation. Such a force is attractive in nature. Its unit is N (Newton) in SI system and dyne in CGS system.
Experimentally, it is found that the gravitational force between any two objects depends upon their masses and the distance between their centers. After different observations, the great scientist Sir Isaac Newton formulates the experiment in the experiment in the form of law which is known as Newton's Universal Law of Gravitation.
Newton's Universal Law of Gravitation
It states that the gravitational force between any two objects directly proportional to the products of their masses and inversely proportional to the square of distance between their centers. This law is applicable in the masses of two objects on the universe. So. it is also called the Universal Law of Gravitation.
Let consider two objects of masses 'm1' and 'm2' which are separated by a distance 'd' as shown in the figure. If 'F' is force of attraction between the two objects, then according to Newton's Universal Law: i. The force is directly proportional to the product of their masses, F∝ m1m2............(i)
ii. The force is inversely proportional to the square of the distance between them, F∝1/d^2..........(ii)
Combining (i) and (ii), we get;
F∝ m1m2/d^2,
F= G (m1m2/d^2)..............(iii)
Where G is a proportionality constant called universal gravitation constant. G is called universal gravitational constant because its value remains same throughout the universe.
Definition and unit of the universal gravitational constant (G)
We know that; F=G(m1m2/d^2) [By Newton Universal Law]
In SI system, If m1=1kg, m2=1kg and d=1m
then F=G.
The universal gravitational constant is numerically equal to the force of attraction between two masses of 1kg each at a distance of 1m from their centers.
So in general, the force of attraction between any two objects of unit masses when they are separated by unit distance is called universal gravitational constant.
The value of G could not be found when Newton gave the la of gravitation. It was determined by Henry Cavendish in 1798 AD by using an instrument called torsion balance and he found the value equal to 6.67x10^-11 Nm^2/kg^2 in SI system.
The value of G in CGS is 6.67x10^-8 dyne cm^2/g^2.
Why is the gravitational force between two objects usually unnoticeable?
The value of G is very small and hence the gravitational force between two smaller objects becomes negligibly small. Therefore, the gravitational force between any two objects is usually unnoticeable. It becomes appreciable only when at least one body is huge like earth, the sun, etc.
Some important points which should be remembered regarding gravitational force are:
i. The gravitational force does not change with the change in medium.
ii. It is the weakest force in nature and is always attractive.
iii. It obeys the inverse square law.
iv. Gravitational force is a central force. This means that this force acts along the line joining the particles towards their center.
v. Gravitational force in the long range force. It is applicable between stars and galaxies.
vi. Newton's law of gravitation is universal. This means that it is applicable anywhere in the universe.
Points to know:
i. Gravitational force is increased by two times when mass of an object is doubled keeping the distance between them constant.
ii. Gravitational force is four times when mass of each object is doubled keeping the distance between them constant.
iii. Gravitational force is decreased by four times when distance between them is made double keeping their masses constant.
iv. Gravitational force is increased by four times when distance between them is made half keeping their masses constant.
Click here to see some Numerical related to Gravitation
Gravity
According to the law of gravitation. the earth attracts each object around it towards its center. The object also attracts the earth by an equal force. Since the object is free to move, it moves toward the earth, but the earth because of its large inertia does not move toward the object. The force with which the earth attracts the objects is called Force of gravity on the body. It always acts vertically downwards at the center of the earth. Such a force is produced by other planets, satellites and other heavenly bodies too.
The force of gravity on a body of mass m kept on the surface of the earth of mass M and the radius R is equal to the force of attraction between the earth and that of the body. It is given as :
F=GMm/R^2
Taking the mass of the earth (M) = 6 x 10 ^24 kg and the radius of the earth (R) = 6.4 x 10^6 m, the force of gravity on the body of the mass 1kg on the surface of the earth will be;
F=GMm/R^2
={(6.67 x 10^-11) x (6 x 10^24) x 1} / (6.4 x 10^6)^2
={6.67 x 6 x 10^(24-11)} / (40.96 x 10^12)
=(40.02 x 10^13) / (40.96 x 10^12)
={40.02 x 10^(13-12)} / (40.96)
=40.02 x 10 / 40.96
=400.2 / 40.96
=9.8 N
∴ The force of gravity on the object of mass 1kg is 9.8N.
Variation of acceleration due to gravity on the earth surface
1. Due to shape of the earth
Earth is not a perfect sphere. It is flattened at the poles and bulges out at equator. Therefore, radius of equator (RE) is greater than the radius of the poles (RP). We know that g= GM / R^2
Here,
Mass of Planet 'M' and 'G' are constant.
∴g ∝ 1/R^2
So, the acceleration due to gravity 'g' at the equator (gE) is less than the acceleration due to gravity 'g' at poles (gP) of the earth.
The value of acceleration due to gravity at equatorial region of the earth (gE) is 9.78m/s^2 and that of polar region (gP) is 9.83 m/s^2. The average value of the acceleration due to gravity is taken as 9.8 m/s^2.
2. Due to altitude from the earth's surface
If we go up a hill at a height 'h' above the surface of the earth, height 'h' from the surface of the earth increases, i.e if the body moves to the higher altitude, the acceleration due to gravity 'g' decreases. As we know that when 'h' is considered, then the force of attraction between the body of mass (m) and the earth of mass (M) can be given as;
F=GMm / (R+h)^2.........(i)
Let 'gh' be the acceleration due to gravity at height 'h'. Then, the force acting on the body at height 'h' is :
F= mass ✖ acceleration
F=m ✖ gh.........(ii)
From (i) and (ii), we get :
mgh= GMm / (R+h)^2
gh=GM / (R+h)^2
gh∝ 1/ (R+h)^2
Thus, it is clear that as height increases, the acceleration due to gravity decreases.
Gravity
According to the law of gravitation. the earth attracts each object around it towards its center. The object also attracts the earth by an equal force. Since the object is free to move, it moves toward the earth, but the earth because of its large inertia does not move toward the object. The force with which the earth attracts the objects is called Force of gravity on the body. It always acts vertically downwards at the center of the earth. Such a force is produced by other planets, satellites and other heavenly bodies too.
The force of gravity on a body of mass m kept on the surface of the earth of mass M and the radius R is equal to the force of attraction between the earth and that of the body. It is given as :
F=GMm/R^2
Taking the mass of the earth (M) = 6 x 10 ^24 kg and the radius of the earth (R) = 6.4 x 10^6 m, the force of gravity on the body of the mass 1kg on the surface of the earth will be;
F=GMm/R^2
={(6.67 x 10^-11) x (6 x 10^24) x 1} / (6.4 x 10^6)^2
={6.67 x 6 x 10^(24-11)} / (40.96 x 10^12)
=(40.02 x 10^13) / (40.96 x 10^12)
={40.02 x 10^(13-12)} / (40.96)
=40.02 x 10 / 40.96
=400.2 / 40.96
=9.8 N
∴ The force of gravity on the object of mass 1kg is 9.8N.
Variation of acceleration due to gravity on the earth surface
1. Due to shape of the earth
Earth is not a perfect sphere. It is flattened at the poles and bulges out at equator. Therefore, radius of equator (RE) is greater than the radius of the poles (RP). We know that g= GM / R^2Here,
Mass of Planet 'M' and 'G' are constant.
∴g ∝ 1/R^2
So, the acceleration due to gravity 'g' at the equator (gE) is less than the acceleration due to gravity 'g' at poles (gP) of the earth.
The value of acceleration due to gravity at equatorial region of the earth (gE) is 9.78m/s^2 and that of polar region (gP) is 9.83 m/s^2. The average value of the acceleration due to gravity is taken as 9.8 m/s^2.
2. Due to altitude from the earth's surface
If we go up a hill at a height 'h' above the surface of the earth, height 'h' from the surface of the earth increases, i.e if the body moves to the higher altitude, the acceleration due to gravity 'g' decreases. As we know that when 'h' is considered, then the force of attraction between the body of mass (m) and the earth of mass (M) can be given as;
F=GMm / (R+h)^2.........(i)
Let 'gh' be the acceleration due to gravity at height 'h'. Then, the force acting on the body at height 'h' is :
F= mass ✖ acceleration
F=m ✖ gh.........(ii)
From (i) and (ii), we get :
mgh= GMm / (R+h)^2
gh=GM / (R+h)^2
gh∝ 1/ (R+h)^2
Thus, it is clear that as height increases, the acceleration due to gravity decreases.
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