To calculate the gravitational force between two objects, including a planet, you use Newton's Law of Universal Gravitation.
Newton's Law of Universal Gravitation
The formula for calculating gravitational force is:
F = G (m1 m2) / r²
Where:
- F is the gravitational force between the two objects (measured in Newtons, N).
- G is the gravitational constant (approximately 6.674 × 10⁻¹¹ N⋅m²/kg²).
- m1 is the mass of the first object (e.g., the planet) (measured in kilograms, kg).
- m2 is the mass of the second object (e.g., a person or satellite) (measured in kilograms, kg).
- r is the distance between the centers of the two objects (measured in meters, m).
Steps to Calculate Gravitational Force:
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Determine the Masses: Identify the mass of the planet (m1) and the mass of the other object (m2). These values should be in kilograms.
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Measure the Distance: Determine the distance (r) between the center of the planet and the center of the other object. This value should be in meters. If the object is on the surface of the planet, then 'r' would be the planet's radius. If it is a satellite orbiting the planet, 'r' would be the radius of the orbit (distance from the center of the planet to the satellite).
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Plug the Values into the Formula: Substitute the values of G, m1, m2, and r into the formula: F = G (m1 m2) / r².
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Calculate: Perform the calculation to find the gravitational force (F).
Example:
Let's say you want to calculate the gravitational force between Earth and a 100 kg satellite orbiting at an altitude of 200 km above Earth's surface.
- G = 6.674 × 10⁻¹¹ N⋅m²/kg²
- m1 (Earth's mass) = 5.972 × 10²⁴ kg
- m2 (Satellite's mass) = 100 kg
- Earth's Radius = 6371 km = 6,371,000 m
- Altitude of Satellite = 200 km = 200,000 m
- r (Distance between centers) = Earth's Radius + Altitude = 6,371,000 m + 200,000 m = 6,571,000 m
F = (6.674 × 10⁻¹¹ N⋅m²/kg²) (5.972 × 10²⁴ kg 100 kg) / (6,571,000 m)²
F ≈ 918 N
Therefore, the gravitational force between the Earth and the satellite is approximately 918 Newtons.
Important Considerations:
- Units: Ensure that all values are in the correct units (kilograms for mass, meters for distance, and N⋅m²/kg² for G).
- Distance: Remember that 'r' is the distance between the centers of the two objects.
- Vector Quantity: Gravitational force is a vector quantity, meaning it has both magnitude and direction. The direction is always towards the center of the planet.
