Universal Law of Gravitation - Physics

1. Fundamental Concepts

Definition: The Universal Law of Gravitation states that every particle attracts every other particle in the universe with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers.
Gravitational Constant: The gravitational constant $G$ is approximately $6.674 \times 10^{-11} \, \text{Nm}^2/\text{kg}^2$.
Force Equation: The force $F$ between two masses $m_1$ and $m_2$ separated by a distance $r$ is given by $F = G \frac{m_1 m_2}{r^2}$.

2. Key Concepts

Basic Rule: $$F = G \frac{{m_1 {m_2}}}{{r^2}}$$

Distance Dependence: The force decreases as the square of the distance between the objects increases.

Application: Used to calculate the gravitational forces between planets, moons, and other celestial bodies.

3. Examples

Example 1 (Basic)

Problem: Calculate the gravitational force between two masses of $5 \, \text{kg}$ and $10 \, \text{kg}$ separated by a distance of $2 \, \text{m}$.

Step-by-Step Solution:

Substitute the values into the formula: $F = 6.674 \times 10^{-11} \frac{{5 \cdot 10}}{{2^2}}$
Simplify: $F = 6.674 \times 10^{-11} \frac{{50}}{{4}} = 8.3425 \times 10^{-11} \, \text{N}$

Validation: Substitute the values back into the formula to ensure consistency.

Example 2 (Intermediate)

Problem: Determine the distance between two masses of $100 \, \text{kg}$ each if the gravitational force between them is $1 \times 10^{-9} \, \text{N}$.

Step-by-Step Solution:

Rearrange the formula to solve for $r$: $r^2 = G \frac{{m_1 {m_2}}}{{F}}$
Substitute the values: $r^2 = 6.674 \times 10^{-11} \frac{{100 \cdot 100}}{{1 \times 10^{-9}}}$
Simplify: $r^2 = 6.674 \times 10^{-11} \frac{{10000}}{{1 \times 10^{-9}}} = 667.4$
Take the square root: $r = \sqrt{667.4} \approx 25.8 \, \text{m}$

Validation: Substitute the calculated distance back into the original formula to check the force.

4. Problem-Solving Techniques

Visual Strategy: Draw diagrams to visualize the positions and distances between masses.
Error-Proofing: Always check units and ensure they are consistent throughout the calculation.
Concept Reinforcement: Practice problems involving different scenarios to reinforce understanding of the law.