Section 3.6: Newton’s Law of Universal Gravitation
Every two masses in the universe attract each other with a force proportional to the product of their masses and inversely proportional to the square of the distance between them. This is described by Newton's Law of Universal Gravitation.
Formula:
\[
F = G \frac{m_1 m_2}{r^2}
\]
where:
- \(F\) = gravitational force
- \(m_1, m_2\) = masses
- \(r\) = distance between centers
- \(G = 6.674 \times 10^{-11} \text{ Nm}^2/\text{kg}^2\)
Key Points:
- Gravitational force acts along the line joining the centers of mass.
- It is always attractive.
- It governs planetary motion and orbital dynamics.
Example 1 — Gravitational Force Between Two Spheres
Two spheres, each of mass 500 kg, are 2 m apart. Find the gravitational force between them.
\(F = G \frac{m_1 m_2}{r^2} = 6.674 \times 10^{-11} \frac{500 \cdot 500}{2^2} \approx 4.17 \times 10^{-6} \text{ N}\)
Practice Problems
- Find the gravitational force between Earth (mass \(5.97 \times 10^{24}\) kg) and a 1000 kg satellite at a distance of 6.7 × 10^6 m from Earth’s center.
- Two planets with masses \(3 \times 10^{24}\) kg and \(4 \times 10^{24}\) kg are 1 × 10^7 m apart. Compute the gravitational force.
- A 50 kg object is 1 m from a 1000 kg mass. Determine the force.
- Double the distance between two masses. How does the gravitational force change?
- Determine the weight of a 70 kg person on the Moon (mass \(7.35 \times 10^{22}\) kg, radius \(1.74 \times 10^6\) m).