Proof of Derivation: Gravitation | Science Class 9 PDF Download

Universal Law of Gravitation

The force is inversely proportional to the square of the distance (d) between the centers of the two objects:
F ∝ 1/d² - (equation 2)

Combining the two relations from Equation 1 and 2, we get
F ∝ (M x m) / d² - (equation 3)

To convert the proportionality into an equation, a constant G (the gravitational constant) is introduced:
F = G[(M x m) / d²] - (equation 4) 

Where G is the constant of proportionality and is called the gravitational constant.
To determine the value of G, we can rearrange Equation 4 as follows:
G = (F x d²) / (M x m) - (equation 5)

The SI unit of G can be derived by substituting the units of force (N), distance (m), and mass (kg) in Equation 5, which gives the unit of G as N m² kg⁻².
The value of G was experimentally determined by Henry Cavendish (1731-1810) using a sensitive balance. The accepted value of G is:
G = 6.673 x 10^-11 N m²kg^-2

Conclusion:
The law shows that there exists a gravitational force between any two objects, but due to the extremely small value of G, the force between objects like you and a friend sitting close by is negligible.

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Derivation of Acceleration Due to Gravity (g)

Where:

• m is the mass of the object,
• g is the acceleration due to gravity.

From the Universal Law of Gravitation, the force between two objects, with masses M (mass of the Earth) and m (mass of the object), at a distance d (distance between the object and the Earth) is given by:
F = G x (M x m) / d² - (equation 2)

From both the equations for force (F), we can equate the expressions from Equation 1 and Equation 2:
m x g = G x (M x m) / d²- (equation 3)

By canceling m from both sides, we get the equation for g (acceleration due to gravity):
g = (G x M) / d²- (equation  4) 

Where G is the gravitational constant, M is the mass of the Earth, and d is the distance between the object and the Earth.
For objects near the Earth's surface, the distance d is approximately equal to the radius of the Earth (R). Substituting d = R in Equation 4 , we get:
g = (G x M) / R²- (equation 5)
Note:  For objects far from Earth, the acceleration due to gravity is given by Equation 4 

Derivation of Weight of an Object

Substituting the acceleration due to gravity (g) for a, we get:
F = m x g - (equation 2)

The force acting on an object due to the gravitational attraction of the Earth is known as the weight (W) of the object. Thus, we can substitute the force (F) with weight (W) in Equation :
W = m x g
Where:

• W is the weight of the object,
• m is the mass of the object,
• g is the acceleration due to gravity.

The SI unit of weight is the same as the unit of force, which is the newton (N). Therefore, the weight of an object is expressed in newtons, and it acts vertically downwards with both magnitude and direction.

Since g(acceleration due to gravity) is constant at a given location, the weight of an object is directly proportional to its mass:
W ∝ m
This means that at any given place, the weight of an object can be used as a measure of its mass.

Variation of Weight Across Planets:
The mass of an object remains constant everywhere, but its weight depends on the value of g at different locations. Therefore, the weight of an object will vary depending on the location (such as on Earth or another planet), as g changes with location.

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Derivation of Weight of an Object on the Moon

Just as the Earth attracts objects with a gravitational force, the Moon also attracts objects, but with a weaker force because the Moon has less mass than the Earth. The weight of an object on the Moon is the force with which the Moon attracts the object.
Let the mass of the object be m. Let the weight of the object on the Moon be denoted as Wₘ, the mass of the Moon be Mₘ, and the radius of the Moon be Rₘ. By applying the universal law of gravitation, the weight of the object on the Moon is given by:
W_m = (G x M_m x m) / R²_m - (equation 1)

Let the weight of the same object on Earth be denoted as Wₑ, with the mass of the Earth being M and the radius of the Earth being R. From the universal law of gravitation, the weight of the object on Earth is given by:
W_e = (G x M x m) / R²- (equation 2)

Using the values for the mass and radius of the Earth and the Moon, we substitute them into Equations 1 and 2:
Wₘ = (7.36 × 10²² kg × m) / (1.74 × 10⁶ m)² = 2.431 × 10¹⁰ G × m - (equation 3)
Wₑ = (5.98 × 10²⁴ kg × m) / (6.37 × 10⁶ m)² = 1.474 × 10¹¹ G × m - (equation 4)

Dividing Equation 3 by Equation 4:
Wₘ / Wₑ = (2.431 × 10¹⁰) / (1.474 × 10¹¹) ≈ 1/6
Hence, the weight of an object on the Moon is approximately 1/6 of its weight on Earth.

Conclusion:
Wₘ / Wₑ = 1/6  or  Wₘ = (1/6) × Wₑ

$(Equation\; 9.18a)Wₘ\; =\; \backslash frac\{7.36\; \backslash times\; 10^\{22\}\; \backslash ,\; \backslash text\{kg\}\; \backslash times\; m\}\{(1.74\; \backslash times\; 10^6\; \backslash ,\; \backslash text\{m\})^2\}\; =\; 2.431\; \backslash times\; 10^\{10\}\; G\; \backslash times\; m\; \backslash quad\; \backslash text\{(Equation\; 9.18a)\}$

$(Equation\; 9.5)G\; =\; \backslash frac\{F\; \backslash times\; d^2\}\{M\; \backslash times\; m\}\; \backslash quad\; \backslash text\{(Equation\; 9.5)\}$