Section 2.1: Universal Gravitation
Universal gravitation is the attractive force between any two masses in the universe. Newton’s Law of Universal Gravitation states that every particle attracts every other particle with a force proportional to the product of their masses and inversely proportional to the square of the distance between them.
\( F = G \frac{m_1 m_2}{r^2} \)
- \( F \) is the gravitational force
- \( m_1, m_2 \) are the masses
- \( r \) is the distance between their centers
- \( G = 6.674 \times 10^{-11} \text{ N·m²/kg²} \)
Example 1
Calculate the gravitational force between two 1000 kg masses separated by 2 m.
\( F = G \frac{m_1 m_2}{r^2} = 6.674 \times 10^{-11} \frac{1000*1000}{2^2} = 1.6685 \times 10^{-5} \text{ N} \)
Example 2
Find the gravitational force between Earth (mass \(5.97 \times 10^{24}\) kg) and a 1000 kg satellite 400 km above the surface.
Distance from Earth's center: \( r = 6371 + 400 = 6771 \text{ km} = 6.771 \times 10^6 \text{ m} \)
\( F = G \frac{5.97\times10^{24}*1000}{(6.771\times10^6)^2} \approx 1.38 \times 10^4 \text{ N} \)
Practice Problems
- Two masses of 50 kg and 70 kg are 3 m apart. Find the gravitational force between them.
- A satellite of 500 kg orbits Earth at 500 km above surface. Calculate gravitational force.
- Two planets of mass \(3\times10^{24}\) kg each are 1×10^7 m apart. Find the gravitational force.
- A 1000 kg spacecraft is 10,000 km from a planet of mass 5×10^24 kg. Find gravitational force.
- Two objects of 2 kg each are 0.5 m apart. Compute gravitational attraction.
- A satellite of mass 800 kg is at 300 km altitude. Determine gravitational force from Earth.
- Two asteroids of mass 1×10^12 kg each, separated by 1 km. Find gravitational force.
- A 50 kg astronaut is 2×10^6 m from a small moon of mass 7×10^20 kg. Calculate force.
- Two satellites, each 1000 kg, are 5 km apart. Find gravitational attraction.
- A spacecraft of 2000 kg is 4000 km above a planet of mass 6×10^23 kg. Compute gravitational force.