Gravitation Solved Examples (Class 11 Physics)
Gravitation numericals apply Newton's law of universal gravitation to calculate gravitational forces, orbital mechanics, and escape velocity. These problem
TL;DR: Gravitation numericals apply Newton's law of universal gravitation to calculate gravitational forces, orbital mechanics, and escape velocity. These pr…
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Gravitation numericals apply Newton's law of universal gravitation to calculate gravitational forces, orbital mechanics, and escape velocity. These problem
Gravitation — Solved Numerical Examples (Step by Step)
Example 1: Calculate the gravitational force between two masses of 5 kg and 10 kg separated by 2 m. (G = 6.67 × 10^-11 N·m²/kg²)
Solution: m₁ = 5 kg, m₂ = 10 kg, r = 2 m, G = 6.67 × 10^-11 N·m²/kg². F = (G × m₁ × m₂) / r² = (6.67 × 10^-11 × 5 × 10) / 4 = (3.335 × 10^-9) / 4 = 8.34 × 10^-10 N.
Example 2: Calculate the acceleration due to gravity on Earth's surface. (G = 6.67 × 10^-11 N·m²/kg², M = 6 × 10^24 kg, R = 6.4 × 10^6 m)
Solution: g = (G × M) / R² = (6.67 × 10^-11 × 6 × 10^24) / (6.4 × 10^6)² = (40.02 × 10^13) / (40.96 × 10^12) = 9.77 m/s².
Example 3: A satellite orbits Earth at a height of 400 km. Calculate its orbital velocity. (M = 6 × 10^24 kg, R = 6.4 × 10^6 m, G = 6.67 × 10^-11 N·m²/kg²)
Solution: Orbital radius r = R + h = 6.4 × 10^6 + 400 × 10^3 = 6.4 × 10^6 + 0.4 × 10^6 = 6.8 × 10^6 m. Orbital velocity v = sqrt(G × M / r) = sqrt((6.67 × 10^-11 × 6 × 10^24) / 6.8 × 10^6) = sqrt(5.88 × 10^7) = 7.66 × 10^3 m/s.
Example 4: Calculate the escape velocity from Earth's surface. (G = 6.67 × 10^-11 N·m²/kg², M = 6 × 10^24 kg, R = 6.4 × 10^6 m)
Solution: Escape velocity ve = sqrt(2 × G × M / R) = sqrt((2 × 6.67 × 10^-11 × 6 × 10^24) / 6.4 × 10^6) = sqrt(1.253 × 10^8) = 1.119 × 10^4 m/s.
Example 5: The period of Moon's orbit around Earth is 27.3 days. Calculate the orbital radius using Kepler's third law. (G = 6.67 × 10^-11 N·m²/kg², M = 6 × 10^24 kg)
Solution: T = 27.3 days = 27.3 × 86400 = 2.36 × 10^6 s. Using T² = (4π²/GM) × r³, r³ = (GMT² / 4π²) = (6.67 × 10^-11 × 6 × 10^24 × (2.36 × 10^6)²) / (4π²). r³ = 5.63 × 10^24, so r = 3.83 × 10^8 m.
Tips
- Newton's law: F = G(m₁m₂)/r². Force decreases with square of distance.
- For orbits: gravitational force provides centripetal force.
- Escape velocity is independent of the object's mass being launched.
Frequently Asked Questions
Why does gravity act on all objects?
All objects with mass produce gravitational fields. The gravitational force between any two masses is directly proportional to their masses and inversely proportional to the square of the distance between them.
Why do satellites not fall to Earth?
Satellites are in constant free fall toward Earth, but their forward orbital velocity causes them to miss Earth and continuously orbit. The gravitational force provides the centripetal force for circular motion.
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