Law of Universal Gravitation

Sir Isaac Newton proposed that every mass in the universe attracts every other mass with a force. This gravitational force is described by the Law of Universal Gravitation: \[ F = G \frac{m_1 m_2}{r^2} \] where:

\( F \): gravitational force (in newtons)
\( G \): gravitational constant, \( 6.674 \times 10^{-11} \, \text{Nm}^2/\text{kg}^2 \)
\( m_1 \) and \( m_2 \): masses of the two objects (in kilograms)
\( r \): distance between the centers of the two masses (in meters)

Characteristics of Gravitational Force

It is always attractive, never repulsive.
It acts along the line joining the centers of the two masses.
It is a mutual force: both objects exert equal and opposite forces on each other.
The force decreases rapidly with distance (inversely proportional to the square of the distance).

Mass vs. Weight

- Mass is a measure of how much matter is in an object. It does not change with location.
- Weight is the gravitational force acting on a mass. It is given by: \[ W = mg \] where \( g \) is the local acceleration due to gravity.

Calculate the gravitational force between two objects of mass \( 10 \, \text{kg} \) and \( 20 \, \text{kg} \) separated by \( 2 \, \text{m} \).

Solution:

\( G = 6.674 \times 10^{-11} \, \text{Nm}^2/\text{kg}^2 \)
\[ F = \frac{6.674 \times 10^{-11} \times 10 \times 20}{2^2} = \frac{1.3348 \times 10^{-8}}{4} = 3.337 \times 10^{-9} \, \text{N} \]

The gravitational force is very small: \( 3.337 \times 10^{-9} \, \text{N} \).

Two masses are \( 0.5 \, \text{m} \) apart and attract each other with a force of \( 2 \times 10^{-6} \, \text{N} \). If one mass is \( 4 \, \text{kg} \), find the other mass.

Solution:

Use: \( F = G \frac{m_1 m_2}{r^2} \)
\[ 2 \times 10^{-6} = \frac{6.674 \times 10^{-11} \times 4 \times m_2}{0.5^2} \Rightarrow m_2 = \frac{2 \times 10^{-6} \times 0.25}{6.674 \times 10^{-11} \times 4} \approx 1.874 \times 10^3 \, \text{kg} \]

The second mass is approximately \( 1874 \, \text{kg} \).

Written by Thenura Dilruk