Saturn has a radius of 60,330,000 meters and a mass of \(5.7 \times 10^{26}\) kilograms. Calculate the strength of its gravitational field.
Answer (3)
g=Gm/(r^2) =[(6.67 10^-11)(5.7 10^26)]/(60330000) = 630183988.1N/kg
The strength of gravitational field of Saturn is 1.35 × 10²³ kg.
To calculate the mass of Saturn's moon Titan based on its given gravitational field and radius, we use Newton's Law of Universal Gravitation, which states that the force of gravity, F, between two masses is F = Gm1m2/r², where G is the gravitational constant, m1 and m2 are the masses of the two objects, and r is the distance between their centers.
However, when we talk about the gravitational field, g, at the surface of an object (like a moon or a planet), which is the acceleration due to gravity, we can simplify this equation to g = G*M/r², where G is the gravitational constant (6.674×10⁻¹¹ N(m/kg)²), M is the mass of the object (in this case, Titan's mass), and r is the radius of the object.
To find Titan's mass, we can rearrange the formula to M = g*r²/G. Plugging in Titan's gravitational field (1.35 m/s²) and Titan's radius (2.58 × 10⁶ m), we get M = (1.35 m/s²) * (2.58 × 10⁶ m)² / (6.674 × 10⁻¹¹N(m/kg)²). Calculating this, we find that the mass of Titan is approximately 1.35 × 10²³ kg, confirming the data provided.
The strength of the gravitational field at the surface of Saturn is approximately 10.44 m/s². This value is calculated using the formula g = r 2 G ⋅ M with Saturn's mass and radius. The calculations involve finding the gravitational constant, mass, and radius values and substituting them into the formula.
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