Acceleration Due to Gravity on Earth (g): Derivation, Formula, Factors and Value
Acceleration due to gravity (g) is one of the most important concepts in Gravitation. Every object near the Earth's surface experiences a force of attraction due to the Earth's gravity. The acceleration produced by this gravitational force is called acceleration due to gravity (g).
What is Acceleration Due to Gravity?
Acceleration due to gravity is the acceleration experienced by a body due to the gravitational force of the Earth.
Derivation of Acceleration Due to Gravity
Consider a body of mass m lying on the surface of the Earth of mass M and radius R.
According to the shell theorem, the gravitational force acting on the body of mass m due to the Earth is equal to the gravitational force between mass m and the mass of Earth concentrated at the centre of the Earth.
Therefore, the gravitational force acting on the body is:
F = GMm / R²
Where:
- G = Universal Gravitational Constant
- M = Mass of the Earth
- m = Mass of the body
- R = Radius of the Earth
The weight of the body is:
\[ F=mg \]
The acceleration experienced by the body of mass m is:
Equating the two expressions:
\[ mg=\frac{GMm}{R^2} \]
Cancelling \(m\) from both sides:
\[ g=\frac{GM}{R^2} \]
This is the expression for acceleration due to gravity.
Factors on Which the Value of g Depends
From the equation:
g = GM / R²
The value of g depends upon:
1. Mass of the Earth (or Planet)
The greater the mass of the planet, the greater the value of g.
2. Radius of the Earth (or Planet)
The greater the radius of the planet, the smaller the value of g.
Important Point
The value of g does not depend on the mass of the body.
Important Facts About g
- As all heavenly bodies like planets, the Sun and the Moon have different masses and different radii, the value of g is different on different heavenly bodies.
- g is a vector quantity.
Value of g on the Surface of the Earth
We know:
g = GM / R²
Mass of the Earth:
M ≈ 6 × 1024 kg
Radius of the Earth:
R = 6.4 × 106 m
Substituting the values:
g = (6.67 × 10-11 × 6 × 1024) / (6.4 × 106)²
g = 9.8 m s-2
Thus, the value of acceleration due to gravity on the Earth's surface is:
g = 9.8 m s-2
Value of g on the Moon
The value of g on the surface of the Moon is one-sixth of its value on the Earth's surface.
gmoon = 1/6 × gearth
Key Formula
g = GM / R²
Frequently Asked Questions (FAQs)
Q1. What is acceleration due to gravity?
Acceleration due to gravity is the acceleration experienced by a body due to the gravitational force of the Earth.
Q2. What is the formula for acceleration due to gravity?
g = GM / R²
Q3. On which factors does g depend?
The value of g depends on the mass and radius of the Earth (or planet).
Q4. Does g depend on the mass of the body?
No, the value of g does not depend on the mass of the body.
Q5. What is the value of g on the Earth's surface?
9.8 m s-2
Q6. What is the value of g on the Moon?
The value of g on the Moon is one-sixth of the value of g on the Earth's surface.
Q7. Is g a scalar or vector quantity?
g is a vector quantity.
Quiz: Acceleration Due to Gravity (g)
1. What is acceleration due to gravity?
A) Force acting on a body
B) Velocity of a falling body
C) Acceleration experienced by a body due to Earth's gravitational force
D) Momentum of a body
Answer: C) Acceleration experienced by a body due to Earth's gravitational force
2. The gravitational force acting on a body of mass m on the Earth's surface is:
A) F = GM/R²
B) F = GMm/R²
C) F = mgR
D) F = GMmR²
Answer: B) F = GMm/R²
3. The expression for acceleration due to gravity is:
A) g = GM/R
B) g = GM/R²
C) g = GMR/2
D) g = GMR²
Answer: B) g = GM/R²
4. The value of g depends upon:
A) Mass of the body only
B) Radius of the body only
C) Mass and radius of the Earth (or planet)
D) Shape of the body
Answer: C) Mass and radius of the Earth (or planet)
5. The value of g does not depend upon:
A) Mass of the Earth
B) Radius of the Earth
C) Mass of the body
D) Planet's size
Answer: C) Mass of the body
6. The value of acceleration due to gravity on the Earth's surface is:
A) 8.9 m/s²
B) 9.8 m/s²
C) 10.8 m/s²
D) 6.8 m/s²
Answer: B) 9.8 m/s²
7. The value of g on the Moon is:
A) Equal to Earth
B) Double that of Earth
C) Half that of Earth
D) One-sixth that of Earth
Answer: D) One-sixth that of Earth
8. 'g' is a:
A) Scalar quantity
B) Vector quantity
C) Tensor quantity
D) Dimensionless quantity
Answer: B) Vector quantity
9. The SI unit of acceleration due to gravity is:
A) N
B) kg
C) m/s²
D) J
Answer: C) m/s²
10. Different heavenly bodies have different values of g because they have different:
A) Colors
B) Temperatures
C) Masses and radii
D) Shapes only
Answer: C) Masses and radii
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