Force and Gravitation
The fundamental concept that governs the motion of celestial bodies and objects on Earth.
Heliocentric Model
The heliocentric theory argues that the Sun is the central body of the solar system and perhaps of the universe. Everything else (planets and their satellites, asteroids, comets, etc.) revolves around it.
The first evidence of the theory is found in the writings of ancient Greece. Greek philosopher-scientists deduced by the sixth century B.C. that Earth is round (nearly spherical) from observations that during eclipses of the Moon, Earth's shadow on the Moon is always a circle of about the same radius wherever the Moon is on the sky. Only a round body can always cast such a shadow.
Key Figure: Nicolaus Copernicus (1473-1543)
Geocentric Model
Geocentric model, any theory of the structure of the solar system (or the universe) in which Earth is assumed to be at the centre of it all. The most highly developed geocentric model was that of Ptolemy of Alexandria (2nd century CE).
It was generally accepted until the 16th century, after which it was superseded by heliocentric models such as that of Nicolaus Copernicus.
Key Figure: Ptolemy (100-170 CE)
Gravitational Force
It is the mutual force of attraction that exists between any two masses in the universe. Its SI unit is newton (N).
Effects of Gravitational Force
- Due to the gravitational force between the moon and the earth, tides occur in the ocean or sea.
- All the heavenly bodies are in their positions due to gravitational force.
- Keeps planets in orbit around the sun.
Newton's Universal Law of Gravitation
It states that everybody in the universe attracts each other with the force, which is:
- Directly proportional to the product of masses
- Inversely proportional to the square of distance between their centres
Formula
F = Gravitational force
G = Universal gravitational constant
m₁, m₂ = Masses of the two objects
d = Distance between their centers
Derivation
Consider a body of mass m₁ attracts another body of mass m₂ with a force F towards its centre O₁. The body of mass m₂ in turn attracts the body of mass m₁ with the same force F towards its centre O₂. Suppose the distance between their centers is 'd' then:
F ∝ m₁m₂ ...(i)
F ∝ 1/d² ...(ii)
Combining above equations:
F ∝ m₁m₂/d²
F = Gm₁m₂/d²
Where G is the universal gravitational constant. This equation gives the measure of gravitational force between two masses.
Universal Gravitational Constant 'G'
Universal gravitational constant 'G' can be defined as the force of gravitation that is exerted between two unit masses, separated by a unit distance.
Key Properties
- The SI unit of G is Nm²kg⁻²
- The approximate value of G is 6.67 × 10⁻¹¹ Nm²kg⁻²
Significance
G is a fundamental physical constant that appears in Newton's law of universal gravitation and in Einstein's general theory of relativity.
Numerical Problems
1 Basic Gravitational Force Calculation
Find the force of gravitation between two bodies of unit mass separated by unit meter distance apart.
Given:
- Mass of the first body (m₁) = 1 kg
- Mass of the second body (m₂) = 1 kg
- Distance between their centers (d) = 1 m
- Universal gravitational constant (G) = 6.67 × 10⁻¹¹ Nm²/kg²
Solution:
From Newton's universal law of gravitation:
F = (G m₁ m₂)/d²
Substituting the values:
F = (6.67 × 10⁻¹¹ × 1 × 1)/1²
F = 6.67 × 10⁻¹¹ N
Answer:
The gravitational force between them is 6.67 × 10⁻¹¹ N.
2 Gravitational Force Between Different Masses
Find the gravitational force between two bodies of masses 10kg and 20kg when separated by the distance of 10m between their centers.
Given:
- Mass of the first body (m₁) = 10 kg
- Mass of the second body (m₂) = 20 kg
- Distance between their centers (d) = 10 m
- Universal gravitational constant (G) = 6.67 × 10⁻¹¹ Nm²/kg²
Solution:
From Newton's universal law of gravitation:
F = (G m₁ m₂)/d²
Substituting the values:
F = (6.67 × 10⁻¹¹ × 10 × 20)/10²
F = (1334 × 10⁻¹¹)/100
F = 1.334 × 10⁻¹⁰ N
Answer:
The gravitational force between them is 1.334 × 10⁻¹⁰ N.
Force of Gravity
It is defined as the force by which the Earth or other heavenly bodies attract each and every object lying on their surfaces towards their centers.
Weight of the body = Force of gravity
Derivation of Force of Gravity
Let us suppose that:
- Mass of the Earth = M
- Mass of the object on its surface = m
- Radius of the Earth = R
There is a mutual force of attraction between them which is:
a) Directly proportional to the product of the masses
F ∝ Mm ...(i)
b) Inversely proportional to the square of radius of the Earth
F ∝ 1/R² ...(ii)
Combining equations (i) and (ii), we get:
F ∝ Mm/R²
F = G Mm/R²
Where G is the proportionality constant called the universal gravitational constant.
Force of Gravity Formula:
Acceleration Due to Gravity (g)
It is the acceleration acquired by a freely falling body due to force of gravity.
SI Unit
m/s² or ms⁻²
Nature
Vector quantity
Variation
Its value differs from place to place
Average Values of g
| Celestial Body | Acceleration Due to Gravity (g) |
|---|---|
| Earth | 9.8 m/s² |
| Moon | 1.67 m/s² |
| Jupiter | 25 m/s² |
Free Fall
Object falls due to gravity only (no air resistance).
Equations of motion under gravity:
- v = u + gt
- s = ut + ½gt²
- v² = u² + 2gs
Weightlessness
When a body experiences no normal reaction due to free fall or orbit.
Example: Astronauts in space feel weightless.
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