Law of Universal Gravitation

1. Overview of the Law of Universal Gravitation

The Law of Universal Gravitation, proposed by Sir Isaac Newton, states that every mass in the universe 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 their centers.

2. The Formula

F=Gm1m2r2F = G \frac{m_1 m_2}{r^2}F=Gr2m1​m2​​

Where:

  • F = Gravitational force between two objects (in Newtons, N)

  • G = Gravitational constant (6.674×10−11 Nm2/kg26.674 \times 10^{-11} \, \text{Nm}^2/\text{kg}^26.674×10−11Nm2/kg2)

  • m₁ and m₂ = Masses of the two objects (in kilograms, kg)

  • r = Distance between the centers of the two objects (in meters, m)

3. Key Points to Remember

  1. Directly Proportional Relationship

    • The gravitational force increases as the masses of the objects increase.

    • Example: A larger planet exerts a stronger gravitational pull than a smaller one.

  2. Inverse Square Law

    • The gravitational force decreases rapidly as the distance between the objects increases.

    • Example: Doubling the distance reduces the gravitational force to one-fourth.

  3. Gravitational Force is Universal

    • It acts on all objects, regardless of their size, mass, or location.

  4. Gravity is Weak at Large Scales

    • Gravitational force is relatively weak compared to other fundamental forces like electromagnetism.

4. Applications of the Law

  • Explains the motion of planets, moons, and satellites.

  • Governs tidal forces caused by the Moon's gravitational pull on Earth.

  • Aids in understanding the orbits of celestial bodies and spacecraft.

5. Example Problems

Problem 1: Gravitational Force Between Two Masses
Two objects, each with a mass of 5 kg5 \, \text{kg}5kg, are placed 2 m2 \, \text{m}2m apart. What is the gravitational force between them?

F=Gm1m2r2=(6.674×10−11)(5)(5)22F = G \frac{m_1 m_2}{r^2} = \left(6.674 \times 10^{-11}\right) \frac{(5)(5)}{2^2}F=Gr2m1​m2​​=(6.674×10−11)22(5)(5)​

Answer: F=4.17×10−10 NF = 4.17 \times 10^{-10} \, \text{N}F=4.17×10−10N

6. Common Misconceptions

  1. Gravitational Force Only Exists on Earth

    • Gravity exists everywhere in the universe, not just on Earth.

  2. Gravity Only Acts on Large Masses

    • All objects with mass experience gravitational force, regardless of size.

7. Quick Tips for Studying

  • Practice solving problems involving the formula.

  • Understand the relationships between force, mass, and distance.

  • Visualize how distance affects gravity using graphs or diagrams.

8. Example Question for Practice

Question:
The gravitational force between two objects is 100 N100 \, \text{N}100N. If the distance between the objects is doubled, what will the new force be?
A. 25 N25 \, \text{N}25N
B. 50 N50 \, \text{N}50N
C. 75 N75 \, \text{N}75N
D. 100 N100 \, \text{N}100N

Answer:
A. 25 N25 \, \text{N}25N
Explanation: Doubling the distance reduces the force to 14\frac{1}{4}41​ of the original.