Question

The weight of spaceman Speff at the surface of planet X, solely due to its gravitational pull, is 389 N. If he moves to a distance of 1.86 × 104 km above the planet's surface, his weight changes to 24.31 N. What is the mass of planet X, if Speff's mass is 75.0 kg? (G = 6.67 × 10 -11 N ∙ m2/kg2)

Answer 1

To find the mass of planet X, we can use Newton's universal law of gravitation and the weight of spaceman Speff at different distances from the planet's surface. By setting up and solving a system of equations, we can determine the mass of planet X.

Step-by-step explanation:

The weight of an object depends on its mass and the force of gravity acting on it. The force of gravity can be calculated using Newton's universal law of gravitation, which states that the force of gravity is directly proportional to the product of the masses of the two objects and inversely proportional to the square of the distance between their centers. In this case, we can use the given information to solve for the mass of planet X:

Let's denote the mass of planet X as MX.

At the surface of planet X, Speff's weight solely due to its gravitational pull is 389 N. Using the formula for the force of gravity, we have:

F1 = (G * MX * Speff's mass) / (radius of planet X)^2

Substituting the given values, we get:

389 N = (6.67 × 10^-11 N ∙ m^2/kg^2 * MX * 75.0 kg) / (radius of planet X)^2

Similarly, when Speff moves to a distance of 1.86 × 10^4 km above the planet's surface, his weight changes to 24.31 N:

F2 = (G * MX * Speff's mass) / (radius of planet X + distance above the surface)^2

Substituting the given values, we get:

24.31 N = (6.67 × 10^-11 N ∙ m^2/kg^2 * MX * 75.0 kg) / (radius of planet X + 1.86 × 10^4 km)^2

Now, we can solve this system of equations to find the mass of planet X.

Answer 2

mass of the planet X = 5.6 × 10²³ kg.

Step-by-step explanation:

According to Newtons law of universal gravitation,

F = GM₁M₂/r²

Where F = gravitational force, M₁ = mass of the speff, M₂ = mass of the planet X, G = gravitational constant r = distance between the speff and the planet X

making M₂ The subject of the equation above,

M₂ = Fr²/GM₁ .......................... equation 2

Where F = 24.31 N, r = 1.08×18⁴km ⇒( convert to m ) =1.08 × 10⁴ × 1000 m

r = 1.08 × 10⁷ m, G = 6.67 × 10 ⁻¹¹ Nm²/kg², M₁ = 75 kg

Substituting this values in equation 2,

M₂ = 24.13(1.08 × 10⁷ )²/75( 6.67 × 10 ⁻¹¹)

M₂ = 24.13 × 1.17 × 10¹⁴/500.25 × 10⁻¹¹

M₂ = (28.23 × 10¹⁴)/(500.25 × 10⁻¹¹)

M₂ = 0.056 × 10²⁵

M₂ = 5.6 × 10²³ kg.

Therefore mass of the planet X = 5.6 × 10²³ kg.