- 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 [3]
Write the Kepler’s laws.
- Kepler’s first law: The orbit of a planet is an ellipse with the sun at one of the foci.
- Kepler’s second law: The line joining the planet and the sun sweeps equal areas in equal intervals of time.
- Kepler’s third law: The square of the orbital period of revolution of a planet around the Sun is directly proportional to the cube of the mean distance of the planet from the Sun.
Write the answer of the question with reference to laws of gravitation.
State the universal law of gravitation.
The universal law of gravitation states that every object in the universe attracts every other object with a force called the gravitational force. The force acting between two objects is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centres.
F ∝ m M (masses of two bodies m and M)
F ∝ `1/"d"^2` (d is the distance between the two bodies.)
A and B are two bodies, the distance between which is given as d.
F = force with which one body is attracting the other.
G = proportionality constant.
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 the 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 the following:
- 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 follows:
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 × 10⁻¹¹ Nm2/kg2
Key Points
Key Points: Newton's Universal Law of Gravitation
