Class 9th || Science || Notes || Chapter 8: Gravitation

 Introduction

Gravitation is the force of attraction between any two objects in the universe. It was first described by Sir Isaac Newton, who formulated the law of universal gravitation, which explains how all objects with mass attract each other.

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Universal Law of Gravitation

The universal law of gravitation states that every object in the universe attracts every other object with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers.

Mathematical Expression:

$$F = G \frac{m_1 m_2}{r^2}$$

Where:

• $F$ = Force of attraction between two objects.
• $G$ = Universal Gravitational Constant = $6.674 \times 10^{-11} \, \text{Nm}^2/\text{kg}^2$.
• $m_1$ and $m_2$ = Masses of the two objects.
• $r$ = Distance between the centers of the two objects.

Important Points:

1. The gravitational force acts along the line joining the centers of the two objects.
2. The value of the gravitational constant $G$ is the same throughout the universe.
3. The force is always attractive.

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Importance of Gravitation

• Gravitation is the force that holds the planets in orbit around the Sun and governs the motion of celestial bodies.
• It is responsible for keeping objects anchored to the Earth.
• Gravitation causes phenomena such as the tides, which are influenced by the gravitational pull of the Moon.

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Free Fall

When an object falls under the influence of gravitational force alone, it is said to be in free fall. In free fall, all objects (regardless of their masses) accelerate towards the Earth with the same acceleration, known as acceleration due to gravity.

Acceleration Due to Gravity (g):

• The acceleration produced in a freely falling object due to the gravitational force of the Earth is called the acceleration due to gravity.
• Denoted by $g$.
• Near the surface of the Earth, the value of $g$ is approximately $9.8 \, \text{m/s}^2$.

Formula for $g$:

The acceleration due to gravity is related to the mass of the Earth and the distance from the Earth's center. It is given by:

$$g = G \frac{M}{R^2}$$

Where:

• $M$ = Mass of the Earth.
• $R$ = Radius of the Earth.
• $G$ = Universal gravitational constant.

Thus, $g$ depends on the mass of the Earth and the distance from its center.

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Difference Between Mass and Weight

1. Mass:

   • Mass is the amount of matter contained in an object.
   • It is a scalar quantity and remains constant regardless of location.
   • SI Unit: Kilogram (kg).

2. Weight:

   • Weight is the force exerted by gravity on an object.
   • It is a vector quantity.
   • Weight changes depending on the gravitational force acting on the object (for example, weight is less on the Moon due to lower gravity).
   • Formula: $$W = mg$$ Where $W$ = weight, $m$ = mass, and $g$ = acceleration due to gravity.
   • SI Unit: Newton (N).

Example:

If a person has a mass of 50 kg on Earth, their weight would be:

$$W = 50 \times 9.8 = 490 \, \text{N}$$

On the Moon, where $g \approx 1/6$th of that on Earth, the same person’s weight would be about $490 \div 6 \approx 81.7 \, \text{N}$.

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Motion of Objects Under the Influence of Gravitational Force of the Earth

When an object is dropped from a height, it accelerates towards the Earth due to gravity. Using the equations of motion, we can describe the motion of falling objects:

1. First Equation of Motion:

   $$v = u + gt$$

   Where $v$ = final velocity, $u$ = initial velocity, $g$ = acceleration due to gravity, $t$ = time taken.

2. Second Equation of Motion:

   $$s = ut + \frac{1}{2} g t^2$$

   Where $s$ = distance traveled, $u$ = initial velocity, $g$ = acceleration due to gravity, $t$ = time taken.

3. Third Equation of Motion:

   $$v^2 = u^2 + 2gs$$

   Where $v$ = final velocity, $u$ = initial velocity, $g$ = acceleration due to gravity, $s$ = distance traveled.

If the object is dropped (i.e., $u = 0$), these equations simplify accordingly.

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Mass of the Earth

The mass of the Earth can be calculated using the formula for gravitational force and the value of acceleration due to gravity.

• The formula for $g$ is:

  $$g = G \frac{M}{R^2}$$

  Where $M$ is the mass of the Earth, $R$ is the radius of the Earth, and $G$ is the gravitational constant.

• Rearranging this formula:

  $$M = \frac{gR^2}{G}$$

  Substituting known values of $g$, $R$, and $G$, we can calculate the mass of the Earth, which is approximately $5.97 \times 10^{24} \, \text{kg}$.

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Gravitational Force and Tides

Tides are caused by the gravitational pull of the Moon and the Sun on Earth's oceans. The Moon's gravitational force has a greater effect because it is much closer to the Earth than the Sun. Tides result from the differences in the Moon's gravitational pull on different parts of the Earth.

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Buoyancy

When an object is placed in a fluid (liquid or gas), it experiences an upward force known as the buoyant force. This force is responsible for making objects float or sink depending on their density relative to the fluid.

Archimedes’ Principle:

Archimedes' principle states that when an object is immersed in a fluid, it experiences an upward force equal to the weight of the fluid displaced by the object.

• Buoyant Force Formula: $$F_{\text{buoyant}} = \text{Weight of the displaced fluid}$$

If the buoyant force is greater than the weight of the object, the object floats. If it is less, the object sinks.

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Why Objects Float or Sink?

1. Objects Float:

   • When the density of the object is less than the density of the fluid.
   • Example: A piece of wood floats in water because its density is lower than that of water.

2. Objects Sink:

   • When the density of the object is greater than the density of the fluid.
   • Example: A stone sinks in water because its density is higher than that of water.

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Relative Density

Relative density is the ratio of the density of a substance to the density of water. It is a dimensionless quantity and is often used to determine whether an object will float or sink.

• Formula: $$\text{Relative Density} = \frac{\text{Density of the Substance}}{\text{Density of Water}}$$ Since the density of water is $1 \, \text{g/cm}^3$, the relative density of a substance gives a direct comparison with water.

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