Mars (6.42 × 10²³ kg) pulls a 500 kg probe whose center is 2.0 × 10⁷ m from Mars's center. Enter the force in newtons.
No one has ever put a planet on a scale. Yet planet masses fill the data books. Where do they come from? They come from Newton's law, run backward.
Given the force and all but one quantity, the law hands you the missing one. Two steps join the routine: make the unknown the subject, and, for r, take a square root.
Step 1 and Step 2 are unchanged: write down the values, mark the unknown, and select F = G m₁m₂ / r².
A moon of mass 5.0 × 10²² kg orbits planet P with centers 2.0 × 10⁸ m apart. The force on the moon is 5.0 × 10²⁰ N. Calculate planet P's mass.
Write down the values in the question:
m₂ = 5.0 × 10²² kg
r = 2.0 × 10⁸ m
F = 5.0 × 10²⁰ N
G = 6.67 × 10⁻¹¹ N·m²/kg²
m₁ = ?
Write down the equation:
F = G m₁m₂ / r²
Make m₁ the subject:
m₁ = F r² / (G m₂)
Substitute in the values:
m₁ = (5.0 × 10²⁰)(2.0 × 10⁸)² / ((6.67 × 10⁻¹¹)(5.0 × 10²²))
Simplify:
m₁ = 6.0 × 10²⁴ kg (2 significant figures)
Starting from F = G m₁m₂ / r², which line is m₁ made the subject?
Planet Q's moon (2.0 × 10²¹ kg) orbits at r = 5.0 × 10⁸ m and feels a pull of 3.2 × 10¹⁹ N. Enter planet Q's mass in kilograms.
Planet P (mass 6.0 × 10²⁴ kg) attracts a second moon, of mass 3.0 × 10²¹ kg, with a force of 7.5 × 10¹⁸ N. Calculate the distance between their centers.
Write down the values in the question:
m₁ = 6.0 × 10²⁴ kg
m₂ = 3.0 × 10²¹ kg
F = 7.5 × 10¹⁸ N
G = 6.67 × 10⁻¹¹ N·m²/kg²
r = ?
Write down the equation:
F = G m₁m₂ / r²
Make r² the subject:
r² = G m₁m₂ / F
Substitute in the values:
r² = (6.67 × 10⁻¹¹)(6.0 × 10²⁴)(3.0 × 10²¹) / (7.5 × 10¹⁸)
Simplify:
r² = 1.6 × 10¹⁷ m²
Find r — square root both sides:
r = 4.0 × 10⁸ m (2 significant figures)
Earth (6.0 × 10²⁴ kg) pulls a 750 kg probe with 120 N. Enter the distance between their centers, in meters.
Planet Y pulls its 4.0 × 10²¹ kg moon, centers 4.0 × 10⁸ m apart, with 6.67 × 10¹⁸ N. Enter planet Y's mass in kilograms.
A student solves F = G m₁m₂ / r² for the separation and reaches r² = G m₁m₂ / F. Which step finishes the solve?
Planet Z (8.0 × 10²³ kg) pulls a 3,000 kg spacecraft with 4.0 N. Enter the distance between their centers, in meters.