Planet Z is 9000 km in diameter. The free-fall acceleration on Planet Z is 10.0 m/s^2.What is the mass of Planet Z?
part two
What is the free-fall acceleration 1.00×10^4 km above Planet Z's north pole?
To find the mass of Planet Z, we can use Newton's law of gravitation, which states that the force of gravity between two objects is equal to the product of their masses divided by the square of the distance between them.
The equation for Newton's law of gravitation is:
F = (G * M * m) / r^2
Where:
F is the gravitational force,
G is the gravitational constant (approximately 6.67430 x 10^-11 m^3 kg^-1 s^-2),
M is the mass of Planet Z,
m is the mass of an object,
r is the distance between the center of Planet Z and the object.
In this case, we can assume the object's mass (m) to be negligible compared to the mass of Planet Z. Therefore, we can denote m as zero.
The acceleration due to gravity (g) can be calculated using the formula:
g = G * M / r^2
From the problem statement, we know that the free-fall acceleration on Planet Z is 10.0 m/s^2, thus:
g = 10.0 m/s^2
The radius (r) of Planet Z is half of the diameter, which is 9000 km. So:
r = 9000 km / 2 = 4500 km = 4,500,000 m
Now, we can plug in the values (g and r) into the equation and solve for M:
10.0 m/s^2 = (6.67430 x 10^-11 m^3 kg^-1 s^-2) * M / (4,500,000 m)^2
Simplifying the equation:
10.0 = (6.67430 x 10^-11) * M / (4,500,000)^2
Cross-multiplying and rearranging the equation:
M = (10.0 * (4,500,000)^2) / (6.67430 x 10^-11)
Calculating the mass (M):
M = (10.0 * 20,250,000,000,000) / 6.67430 x 10^-11
M ≈ 3.040 x 10^24 kg
Therefore, the mass of Planet Z is approximately 3.040 x 10^24 kg.
To find the mass of Planet Z, we can use the relationship between gravitational acceleration, mass, and radius.
The formula that relates these quantities is:
g = G * (M / r^2)
Where:
- g is the acceleration due to gravity (10.0 m/s^2)
- G is the gravitational constant (approximately 6.67430 x 10^-11 N*m^2/kg^2)
- M is the mass of the planet
- r is the radius of the planet (half of the diameter, which is 4500 km or 4,500,000 m)
To solve for M, we rearrange the formula as follows:
M = (g * r^2) / G
Let's calculate the mass of Planet Z using this equation:
Substituting the values:
M = (10.0 m/s^2 * (4,500,000 m)^2) / (6.67430 x 10^-11 N*m^2/kg^2)
Now, let's plug it into a calculator:
M ≈ 1.348 x 10^24 kg
Therefore, the mass of Planet Z is approximately 1.348 x 10^24 kg.
The value of surface acceleration, gs, for Earth is 9.81 m/s^2. For planet Z, is is nearly the same, just 1.9% higher.
gs is proportional to M/R^2, where M is the planet's mass and R is its radius
[M/R^2]z = 1.019 [M/R^2]earth
Mz/Mearth = 1.019 [Rz/Rearth]^2
The radius of the Earth is 6371 km. This equaiton will let you solve for the mass of planet Z divided by that of the earth