In the previous section we completed the discussion on Force and laws of motion. In this section we will see Gravitation.
■ Lift a small stone to a certain height and let go.
• We can see that it falls down.
■ Consider a small stone thrown vertically into the air.
• We can see that it’s velocity goes on decreasing.
• After a while, at a point, the velocity finally becomes zero. Then it starts to fall back.
• While falling back, it’s velocity goes on increasing.
• It’s velocity will be maximum at the time when it hits the ground.
Let us analyse the above.
• In the first case, when the stone was let go, it immediately fell to the ground. Instead of floating around in the air or moving upwards.
• In the second case, the stone went on decreasing it’s speed while ascending.
And while descending, went on increasing it’s speed. As if to reach the ground as early as possible.
• In both the above cases, it looks as if the stone does not want to leave the surface of the earth.
In fact, it is not the wish of the stone, that is being fulfilled here. It is the wish of the earth. The earth attracts every object towards it’s centre. So the stone is not able to float away or rise high up. Note that the force of attraction is towards the centre of the earth. This is shown in the fig.3.1 below:
So, if the falling stone does not meet the ground, it will continue it’s fall till it reaches the centre of the earth.
• So why doesn’t the earth move towards the stone?
The following steps give the answer:
1. Let FES be the force with which the earth attracts the stone. We have seen that Force = mass acceleration. Details here.
• Then FES = msas (Where ms is the mass of stone and as is the acceleration with which the stone moves towards the earth)
2. Let FSE be the force with which the stone attracts the earth
• Then FSE = meae (Where me is the mass of earth and ae is the acceleration with which the earth moves towards the stone)
3. These two forces are equal and opposite. So we write:
msas = -meae
• Here mass of earth me is enormous, while mass of stone ms is negligibly small.
• So, to maintain the equality, as will be large, while ae will be negligibly small.
■ Now, if the earth attracts every object towards it’s centre, what about the moon? Does the earth attract the moon towards it’s centre?
• The answer is yes. The earth attracts the moon too. And the moon in turn attracts the earth.
• This force of attraction causes the sea level to rise up. We call it: the ‘high tide’.
• But we do not see the moon falling onto the earth. We will see the reason when we learn about circular motion.
■ The sun attracts all the planets which orbits around it.
■ In fact, there is a force of attraction between every object in the universe.
■ There is a force of attraction even between two small wooden blocks placed on top of a table. It is a universal phenomenon.
■ Sir Isaac Newton studied about this phenomenon and published his findings in the form of a law. It is the Universal Law of Gravitation. Let us see the details about this law:
• All bodies in the universe attract each other
• The force of this attraction between two bodies is
♦ Directly proportional to the product of their masses
♦ Inversely proportional to the square of the distance between them
Let us put it into a mathematical form:
In the fig.3.2 below, masses of the two bodies are m1 and m2. The distance between their centres is d
Based on this fig., we can derive an equation for the force of attraction F between any two bodies:
Unit of G
Eq. 3.1 above can be written as:
Now we can obtain the unit of 'G' as follows:
• Unit of force is newton N
• Unit of distance is m
• Unit of mass is kg
• So the unit of G will be:
■ The value of G was calculated by the British scientist Henry Cavendish. The accepted value of G is 6.673 × 10-11 N m2 kg-2
A child of mass 40 kg is sitting at a distance of 1 m from another child of mass 50 kg. Calculate the gravitational force of attraction between them
Solution:
1. m1 = 40 kg • m2 = 50 kg • d = 1 m • G = 6.7 × 10-11 N m2 kg-2.
2. We have: F = G(m1m2⁄d2) = 6.7 × 10-11 × (40×50⁄12) = 13400 × 10-11 = 1.34 × 10-7 = 0. 000000134 N.
3. We can see that it is a very small force. Such a small force cannot overcome the frictional resistance and other forces. So two children sitting close to each other do not come closer to each other due to the attractive force between them.
Solved example 3.2
A body of mass 50 kg and another body of mass 60 kg are separated by a distance of 2 m. What is the force of attraction between them?
Solution:
1. m1 = 50 kg • m2 =650 kg • d = 2 m • G = 6.7 × 10-11 N m2 kg-2.
2. We have: F = G(m1m2⁄d2) = 6.7 × 10-11 × (50×60⁄22) = 5025 × 10-11 = 0.5025 × 10-7 = 0. 00000005025 N.
Solved example 3.3
The mass of the earth is 6 × 1024 kg and that of the moon is 7.4 × 1022 kg. If the distance between the earth and the moon is 3.84 × 105 km, calculate the force exerted by the earth on the moon. G = 6.7 × 10-11.
Solution:
1. Mass of the earth, m1 = 6 × 1024 kg
2. Mass of the moon, m2 = 7.4 × 1022 kg
3. Distance between earth and the moon, d = 3.84 × 105 km = 3.84 × 105 ×1000 m = 3.84 × 108 m
4. The force can be calculated as follows:
Importance of the Universal law of gravitation
The universal law of gravitation successfully explained several phenomena that we see in the universe:
• The force that binds us to the earth. If there was no gravitational force, we would not be able to walk on the surface of the earth. We will all be floating.
• The motion of the moon around the earth. If there was no gravitational force, the moon would not be orbiting around the earth in a definite orbit. It would travel away in a straight line
• The motion of the planets around the sun. If there was no gravitational force, the planets also would not be orbiting around the sun.
• The tides due to the moon and the sun. If there was no gravitational force, there would not be any high tides or low tides
In the next section, we will see Acceleration due to Gravity.
■ Lift a small stone to a certain height and let go.
• We can see that it falls down.
■ Consider a small stone thrown vertically into the air.
• We can see that it’s velocity goes on decreasing.
• After a while, at a point, the velocity finally becomes zero. Then it starts to fall back.
• While falling back, it’s velocity goes on increasing.
• It’s velocity will be maximum at the time when it hits the ground.
Let us analyse the above.
• In the first case, when the stone was let go, it immediately fell to the ground. Instead of floating around in the air or moving upwards.
• In the second case, the stone went on decreasing it’s speed while ascending.
And while descending, went on increasing it’s speed. As if to reach the ground as early as possible.
• In both the above cases, it looks as if the stone does not want to leave the surface of the earth.
In fact, it is not the wish of the stone, that is being fulfilled here. It is the wish of the earth. The earth attracts every object towards it’s centre. So the stone is not able to float away or rise high up. Note that the force of attraction is towards the centre of the earth. This is shown in the fig.3.1 below:
 |
| Fig.3.1 |
• Now we consider another aspect of this force of attraction. The earth attracts the stone. Does the stone attract the earth?
• Indeed it does. By the newton's third law, there will be equal and opposite forces. • So why doesn’t the earth move towards the stone?
The following steps give the answer:
1. Let FES be the force with which the earth attracts the stone. We have seen that Force = mass acceleration. Details here.
• Then FES = msas (Where ms is the mass of stone and as is the acceleration with which the stone moves towards the earth)
2. Let FSE be the force with which the stone attracts the earth
• Then FSE = meae (Where me is the mass of earth and ae is the acceleration with which the earth moves towards the stone)
3. These two forces are equal and opposite. So we write:
msas = -meae
• Here mass of earth me is enormous, while mass of stone ms is negligibly small.
• So, to maintain the equality, as will be large, while ae will be negligibly small.
• Thus it is clear that there is indeed a Force. A force of attraction.
• When there is a force, there will be acceleration. That is why, the stone ‘accelerates’ towards the earth. That is., it increases it’s velocity continuously.
■ Now, if the earth attracts every object towards it’s centre, what about the moon? Does the earth attract the moon towards it’s centre?
• The answer is yes. The earth attracts the moon too. And the moon in turn attracts the earth.
• This force of attraction causes the sea level to rise up. We call it: the ‘high tide’.
• But we do not see the moon falling onto the earth. We will see the reason when we learn about circular motion.
■ The sun attracts all the planets which orbits around it.
■ In fact, there is a force of attraction between every object in the universe.
■ There is a force of attraction even between two small wooden blocks placed on top of a table. It is a universal phenomenon.
■ Sir Isaac Newton studied about this phenomenon and published his findings in the form of a law. It is the Universal Law of Gravitation. Let us see the details about this law:
• All bodies in the universe attract each other
• The force of this attraction between two bodies is
♦ Directly proportional to the product of their masses
♦ Inversely proportional to the square of the distance between them
Let us put it into a mathematical form:
In the fig.3.2 below, masses of the two bodies are m1 and m2. The distance between their centres is d
 |
| Fig.3.2 |
Eq. 3.1 above can be written as:
• Unit of force is newton N
• Unit of distance is m
• Unit of mass is kg
• So the unit of G will be:
We will now see some solved examples
Solved example 3.1A child of mass 40 kg is sitting at a distance of 1 m from another child of mass 50 kg. Calculate the gravitational force of attraction between them
Solution:
1. m1 = 40 kg • m2 = 50 kg • d = 1 m • G = 6.7 × 10-11 N m2 kg-2.
2. We have: F = G(m1m2⁄d2) = 6.7 × 10-11 × (40×50⁄12) = 13400 × 10-11 = 1.34 × 10-7 = 0. 000000134 N.
3. We can see that it is a very small force. Such a small force cannot overcome the frictional resistance and other forces. So two children sitting close to each other do not come closer to each other due to the attractive force between them.
Solved example 3.2
A body of mass 50 kg and another body of mass 60 kg are separated by a distance of 2 m. What is the force of attraction between them?
Solution:
1. m1 = 50 kg • m2 =650 kg • d = 2 m • G = 6.7 × 10-11 N m2 kg-2.
2. We have: F = G(m1m2⁄d2) = 6.7 × 10-11 × (50×60⁄22) = 5025 × 10-11 = 0.5025 × 10-7 = 0. 00000005025 N.
Solved example 3.3
The mass of the earth is 6 × 1024 kg and that of the moon is 7.4 × 1022 kg. If the distance between the earth and the moon is 3.84 × 105 km, calculate the force exerted by the earth on the moon. G = 6.7 × 10-11.
Solution:
1. Mass of the earth, m1 = 6 × 1024 kg
2. Mass of the moon, m2 = 7.4 × 1022 kg
3. Distance between earth and the moon, d = 3.84 × 105 km = 3.84 × 105 ×1000 m = 3.84 × 108 m
4. The force can be calculated as follows:
The universal law of gravitation successfully explained several phenomena that we see in the universe:
• The force that binds us to the earth. If there was no gravitational force, we would not be able to walk on the surface of the earth. We will all be floating.
• The motion of the moon around the earth. If there was no gravitational force, the moon would not be orbiting around the earth in a definite orbit. It would travel away in a straight line
• The motion of the planets around the sun. If there was no gravitational force, the planets also would not be orbiting around the sun.
• The tides due to the moon and the sun. If there was no gravitational force, there would not be any high tides or low tides
In the next section, we will see Acceleration due to Gravity.
No comments:
Post a Comment