Gravity and Acceleration due to Gravity


Gravity

Everyone knows that the earth is spherical. But do you know that why people staying just opposite to us in the surface of the earth don’t fall? It is because of the Gravity of the earth. Gravity of the earth pulls every objects towards its centre.

Earth keeps us on its surface due to its gravitational force

In the Newton’s law of gravitation, if one of the two involved objects is a heavenly body, then the gravitational force is called as Gravity. Gravity is the gravitational force in which there is involvement of a heavenly body. The gravitational force between the earth and the objects near or on its surface is the gravity of the earth.

The gravitational force between the Jupiter and the objects near or on its surface is the gravity of the Jupiter.
​So, we can say that the force by which the earth attracts every objects towards its centre is known as the gravity of the earth. The force by which a body is attracted towards the centre of the planet is known as its weight.

Gravity is affected by the mass and the radius of the planet or satellite

Gravity is affected by the mass and the radius of the planet or satellite

Let the mass of the heavenly body be $M$ and the mass of the object placed on the surface of the heavenly body be $m$. Here, the distance between the heavenly body and the object is the radius of the heavenly body $R$. Then, the weight of the object is,
\[W = \frac{GMm}{R² }\text{ ___(1) (F = W)}\]
By keeping the $m$ constant,
\[W ∝\frac{ M}{R²}\text{ ___(2)}\]

From equation $(2)$, the weight of the object $(W)$ on the surface of the heavenly body is directly proportional to the mass of the heavenly body $(M)$. Thus, If the mass of the heavenly body is more, then gravity will be more or the weight of the object will be more.

The weight of the object $(W)$ is inversely proportional to the square of the radius of the heavenly body $(R)$. If the radius of the heavenly body is long then the weight of the object on the heavenly body will be less.

​Because of the less mass of the moon than the earth, the gravity of the moon is $1/6$ of the gravity of the earth. Because of this, the person who can jump $1\text{ m}$ high on the surface of the earth, that person can jump $6\text{ m}$ on the surface of the moon. Also, If a person can lift an object of mass $20\text{ kg}$ on earth then he/she can lift an object of mass $120\text{ kg}$ on moon.

​Here, Jupiter is about $319$ times greater than the earth. But due to its longer radius, its gravity is only $2.5$ times of the earth’s gravity as $F ∝ \frac{1}{R^2}$. It means that if a person can lift a object of mass $100\text{ kg}$ on earth then that person can lift an object of mass $20\text{ kg}$ on Jupiter.

Remember that the gravitational force is the force of attraction between any two objects in the universe but the gravity is the force of attraction between a heavenly body and an object.

Acceleration due to Gravity

When a body falls freely towards the surface of the earth, acceleration is produced on the body which is equal to the acceleration due to gravity of the earth. Thus, acceleration due to gravity is defined as the acceleration produced on a freely falling body due to the gravity of the earth.
Acceleration due to gravity is denoted by $g$. Its SI unit is $\text{m/s}^2$.

​The average value of the acceleration due to gravity near the surface of the earth is $9.8\text{ m/s}^2$. It means that if the air resistance is neglected, the velocity of body increases by $9.8\text{ m/s}$ every second.

For example, Let an object such as a ball is kept at some height, then it will have zero initial velocity. When the ball is allowed to fall freely, after one second, its velocity becomes $9.8\text{ m/s}$. After further one second, its velocity becomes $19.6\text{ m/s}$. After one more second, its velocity becomes $29.4\text{ m/s}$. The velocity of the object increases like this.

Acceleration due to Gravity

​But what will happen to its velocity when the object is thrown upward?
– The velocity of the body decreases by $9.8\text{ m/s}$ by every second.

Acceleration due to gravity does not depend upon the mass of the object

Galileo Galilee did an experiment in 1590 AD to show that acceleration due to gravity doesn’t depend upon the mass of the object. Before him, it was thought that the speed by which an object falls depends upon its mass. If an object has high mass, then it falls faster. This was Aristotle’s thought and was prove to be wrong.

To demonstrate it, Galileo dropped two iron balls of different masses simultaneously from the top of the leaning tower of Pisa. Both the iron balls reached the ground at the same time which shows that all bodies of different masses fall with the same acceleration.

But if a feather and a stone are allowed to fall from same height then what happens? We will find that the stone reaches the ground faster than the feather. If all bodies fall with the same acceleration, then why did the stone falls faster than the feather?

It is because of the air resistance. The speed of falling of a body is not only determined by gravity, air resistance is also responsible for it. The air resistance on the surface area of the feather is very large than the force of gravity applied on it.

​What does it mean? It means that every body whether they are heavier or lighter, fall with the same acceleration if there is no air resistance. Without air resistance, the feather falls with the same acceleration as the stone. It was at first demonstrated by Robert Boyle after Galileo’s death.

Coin and Feather Experiment

Coin and Feather Experiment Figure (1)

Sir Isaac Newton took a long and wide glass tube and put a coin and a feather into it. He inverted the tube and the coin fell much more faster than the feather. He again inverted the tube to get the coin and the feather at the bottom of the tube.

Coin and Feather Experiment Figure (2)

In second time, He evacuated the tube by using a vacuum pump. When he inverted the evacuated tube, it was found that the coin and the feather fell together and reached the bottom at same Time. It demonstrated that they fall at the same rate when there is no air resistance.

This experiment demonstrated that the acceleration due to gravity does not depend upon the mass of the body and the body will fall at the same acceleration in the absence of air resistance.

If a body has a larger surface area then it will experience more air resistance. That’s why, a paper ball falls more faster than a sheet of a paper.

It was again experimented at the moon by the astronauts who were travelling on Apollo-12. They dropped a mallet and a feather from the same height at the same time. It was found that both the mallet and the feather fall at the same rate and reached the surface at the same time. This is because there is no air in moon so the air resistance is not applied on the feather to slow down it while it was falling.

Mathematically,

Acceleration due to gravity does not depend upon the mass of the object

Let the mass of the earth be $M$ and its radius be $R$. And the mass of a body on the surface of the earth be $m$.

According to Newton’s Law of Gravitation,
\[F = \frac{GMm}{R²} \text{ ____(i)}\]
The force by which the body is attracted towards the centre of the earth is given by,
\[F = mg \text{ ____(ii)}\]
From equation $(i)$ and $(ii)$,
\[mg = \frac{GMm}{R²}\]
Cancelling $m$ on both sides, we get,
\[g = \frac{GM}{R²} \text{ ____(iii)}\]

This equation $(iii)$ does not contain the mass of the body $(m)$ which shows that $(g)$ does not depend upon the mass of the freely falling body $(m)$.

​Here, the value of $G$ and $M$ are constant. But due to the orange shape of the earth, the value of $R$ is variable then,
\[F ∝\frac {1}{R² }\text{ ____(iv)}\]
From equation $(iv)$, we will find that $(g)$ depends upon the radius of the heavenly body.

Effects

We all use gravity in our daily life. Without gravity, we are not able to do many of our activities. Some effects of gravity are-

  • We can perform the activities like walking, running, swimming, standing up, etc due to the gravity of the earth.
  • Without the gravity of the earth, there will be no atmosphere. Without atmosphere, Noone can survive and there will be no change in season.
  • When a body is thrown upwards, gravity opposes its motion and it comes back to the earth’s surface.
  • A body experiences acceleration due to gravity when it falls on the earth’s surface.
  • We can construct buildings and bridges on the earth’s surface due to its gravity.
  • There is more force of gravity between the moon and the ocean due to the greater mass of the ocean water. So, the gravity of the moon pulls the ocean water by which ocean level increases and tides occur.
  • Rivers and streams flow due to the gravity of the earth.

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