Gravitational field of a point mass
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State Newton's Law of Gravitation.
The gravitational force between two point masses is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers. Mathematically: F = Gm₁m₂ / r².
Derive the formula for gravitational field strength (g) due to a point mass.
Starting with Newton's Law of Gravitation (F = GMm/r²) and the definition of gravitational field strength (g = F/m), substitute F to get g = (GMm/r²)/m. Simplifying, g = GM/r².
What is the equation for gravitational field strength (g) due to a point mass?
g = GM/r², where G is the gravitational constant, M is the mass of the point mass, and r is the distance from the center of the point mass.
Calculate the gravitational field strength on the surface of a planet with mass M = 6 x 10^24 kg and radius r = 6.4 x 10^6 m. (G = 6.67 x 10^-11 Nm²/kg²)
Using g = GM/r², g = (6.67 x 10^-11 Nm²/kg² * 6 x 10^24 kg) / (6.4 x 10^6 m)² = 9.77 N/kg (or m/s²).
Explain why 'g' is approximately constant for small changes in height near the Earth's surface.
Near the Earth's surface, small changes in height (Δr) result in negligible changes to the overall distance 'r' from the Earth's center. Since g is inversely proportional to r², g remains approximately constant (g ≈ GM/r²).
Describe the relationship between gravitational field strength and distance from a point mass.
Gravitational field strength (g) is inversely proportional to the square of the distance (r) from the point mass (g ∝ 1/r²). As distance increases, field strength decreases rapidly.
If the distance from a planet's center doubles, how does the gravitational field strength change?
If the distance doubles, the gravitational field strength is reduced to one-quarter of its original value. Since g ∝ 1/r², if r becomes 2r, then g becomes g/4.
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