Universal Gravitation
The fundamental law that describes the gravitational attraction between masses in the universe.
Introduction
Universal Gravitation, formulated by Sir Isaac Newton, is one of the cornerstones of physics. It explains how every object in the universe attracts every other object with a force that depends on their masses and the distance between them. This principle governs the motion of planets, stars, and even objects on Earth.
Variables & Units
The law is expressed by the equation F = G × (m₁ × m₂) / r², where:
- F (Force): The gravitational force between two masses, measured in newtons (N).
- G (Gravitational Constant): Approximately 6.674×10⁻¹¹ N·m²/kg².
- m₁ and m₂ (Masses): The masses of the two objects, measured in kilograms (kg).
- r (Distance): The distance between the centers of the two masses, measured in meters (m).
Using consistent SI units is essential for accurate calculations.
The Fundamental Equation
Newton’s Law of Universal Gravitation is given by:
F = G × (m₁ × m₂) / r²
This equation can be rearranged to solve for any variable. For example:
- To calculate a mass: m₂ = (F × r²) / (G × m₁)
- To calculate the distance: r = √(G × m₁ × m₂ / F)
How to Use Universal Gravitation
Applying Universal Gravitation is straightforward:
- Identify the Known Values: Determine the masses (m₁ and m₂) and the distance (r) between their centers.
- Select the Correct Formula: Use F = G × (m₁ × m₂) / r² to calculate the gravitational force.
- Substitute and Solve: Ensure all values are in SI units, substitute them into the equation, and solve for the unknown variable.
Example Problems
Example 1: Calculating Gravitational Force
Problem: Calculate the gravitational force between two 5 kg masses that are 2 meters apart. (G ≈ 6.674×10⁻¹¹ N·m²/kg²)
F = 6.674×10⁻¹¹ × (5 × 5) / 2²
F = 6.674×10⁻¹¹ × 25 / 4
F ≈ 4.17×10⁻¹⁰ N
Explanation: The gravitational force is very small because of the low masses and the distance between them.
Example 2: Calculating Mass
Problem: If the gravitational force between two objects is 1×10⁻⁵ N, they are 1 meter apart, and one mass is 1000 kg, what is the other mass? (G ≈ 6.674×10⁻¹¹ N·m²/kg²)
m₂ = (F × r²) / (G × m₁) = (1×10⁻⁵ × 1²) / (6.674×10⁻¹¹ × 1000)
m₂ ≈ 1.50×10² kg
Explanation: Rearranging the equation allows you to solve for the unknown mass.
Example 3: Calculating Distance
Problem: Two objects, each of mass 2000 kg, attract each other with a force of 0.1 N. What is the distance between them? (G ≈ 6.674×10⁻¹¹ N·m²/kg²)
r = √(G × m₁ × m₂ / F) = √(6.674×10⁻¹¹ × 2000 × 2000 / 0.1)
r ≈ √(6.674×10⁻¹¹ × 4×10⁶ / 0.1)
r ≈ √(2.67×10⁻⁴) ≈ 0.0163 m
Explanation: This example shows how to rearrange the formula to find the distance between two masses.
Practical Applications
Universal Gravitation is crucial in many fields, such as:
- Astronomy & Astrophysics: Calculating the gravitational forces between planets, stars, and galaxies.
- Orbital Mechanics: Designing satellite orbits and understanding planetary motion.
- Space Exploration: Planning spacecraft trajectories and gravitational assists.
- Geophysics: Studying Earth’s gravitational field and its variations.
Advanced Concepts
Beyond the basic equation, Universal Gravitation leads to more advanced topics:
- Gravitational Potential Energy: The energy due to the gravitational attraction between masses, given by U = -G × (m₁ × m₂) / r.
- General Relativity: Einstein’s theory refines our understanding of gravitation in strong fields and at high speeds.
- Celestial Mechanics: The principles of gravitation underpin the motion of celestial bodies and the dynamics of orbits.
Frequently Asked Questions
What is Universal Gravitation?
It is Newton’s law stating that every mass attracts every other mass with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between them.
What is the formula for Universal Gravitation?
The formula is F = G × (m₁ × m₂) / r².
What units are used in this formula?
Force is measured in newtons (N), mass in kilograms (kg), distance in meters (m), and the gravitational constant in N·m²/kg².
How do I solve problems using Universal Gravitation?
Identify the known values, substitute them into the formula, and solve for the unknown variable. You can also rearrange the equation to find mass or distance as needed.
Conclusion
Universal Gravitation is a foundational principle in physics that explains the gravitational attraction between masses. Mastering this law provides deep insights into planetary motion, celestial mechanics, and the dynamics of our universe.