- Every object attracts every other with a gravitational force.
- Force increases with mass — more mass means a stronger pull.
- Force decreases with distance — doubling the distance halves the force.
- A force acts along the line joining the centres (or centres of mass) of the two bodies.
Definitions [2]
Definition: Universal Law of Gravitation
"Every particle of matter attracts every other particle of matter with a force which is directly proportional to the product of their masses and inversely proportional to the square of the distance between them."
Definition: Escape velocity
"The minimum velocity with which a body should be thrown vertically upwards from the surface of the Earth so that it escapes the Earth’s gravitational field, is called the escape velocity (ve) of the body."
Formulae [4]
Formula: Universal Law of Gravitation
The gravitational force of attraction (F) between two bodies of mass m1 and m2 separated by a distance r is:
\[\mathbf{F} = \mathbf{G}\frac{m_1 m_2}{r^2}\]
- F: Gravitational Force of attraction (in Newtons, N).
- \[m_1, m_2\]: Masses of the two objects (in kilograms, kg).
- r (or d in the first part): Distance between the two objects (in meters, m).
- G: The constant of proportionality, called the Universal gravitational constant.
- Value in SI units: \[G=6.67\times10^{-11}\mathrm{N}\cdot\mathrm{m}^2/\mathrm{kg}^2\]
- Dimensions: \[[G]=[\mathrm{L}^3\mathrm{M}^{-1}\mathrm{T}^{-2}]\]
Formula: Gravity with Altitude
The formulas for acceleration due to gravity (g) are provided below:
On the Earth's Surface:
\[g = \frac{G M}{R^2}\]
At height $h$ above the Earth's Surface:
\[g_h = g \frac{R^2}{(R+h)^2} \quad \text{or} \quad g_h = g \left(I + \frac{h}{R}\right)^{-2}\]
Simplified Formula for Small Altitudes ($h \ll R$):
\[g_h = g \left(1 - \frac{2h}{R}\right)\]
Definition of Terms:
- g: Acceleration due to gravity on the Earth's surface.
- gh: Acceleration due to gravity at height h above the Earth's surface.
- G: Universal Gravitational Constant.
- M: Mass of the Earth.
- R: Radius of the Earth.
- h: Altitude or height above the Earth's surface.
Formula: Gravitational Potential Energy
U(r) = -\[\frac {GMm}{r}\]
Where:
- U(r) = Gravitational potential energy at distance r from Earth's center
- G = Universal gravitational constant (6.67 × 10⁻¹¹ N·m²/kg²)
- M = Mass of Earth (kg)
- m = Mass of the object (kg)
- r = Distance between the centers of mass of Earth and object (m)
- Negative sign = Shows that potential energy is negative (zero at infinity)
Formula: Escape velocity
\[v_e=\sqrt{\frac{2GM}{R}}\]
- ve = Escape velocity (minimum speed needed to escape Earth’s gravity)
- G = Universal gravitational constant (6.674 × 10−11 Nm2/kg2)
- M = Mass of the Earth (or celestial body)
- R = Radius of the Earth (or distance from the centre of the mass to the object)
Theorems and Laws [1]
Law: Universal Law of Gravitation
Statement:
The law which states that every particle of matter attracts every other particle in the universe with a force whose magnitude is directly proportional to the product of masses and inversely proportional to the square of distance between them is called Newton's Law of Gravitation.
Derivation:
Newton's Universal Law of Gravitation states that every particle of matter attracts every other particle of matter with a force which is:
- Directly proportional to the product of their masses: F ∝ m1 ⋅ m2
- Inversely proportional to the square of the distance between them: F ∝ \[\frac {1}{r^2}\]
Combining both, the gravitational force is expressed as:
F = G\[\frac{m_1m_2}{r^2}\]
where G is the Universal Gravitational Constant, measured by Henry Cavendish using the Cavendish balance, with the value:
G = 6.67 × 10−11Nm2/kg2
Key Points
Key Points: Newton's Universal Law of Gravitation
