A student of weight 990 N was sent to another planet called Nerdland. If Nerdland has a mass that is three times the mass of the earth and a radius that is half the radius of the earth, what would be the mass in kg of the student when on Nerdland?
To calculate the mass of the student on Nerdland, we need to use the law of universal gravitation, which states:
F = (G * (m1 * m2))/r²
where F is the gravitational force between two objects, G is the gravitational constant, m1 and m2 are the masses of the objects, and r is the distance between the centers of the objects.
First, let's calculate the mass of Nerdland. We're given that Nerdland's mass is three times the mass of Earth. Let's assume Earth's mass is denoted as mE. So, the mass of Nerdland, mN, would be:
mN = 3 * mE
Next, we also know that the radius of Nerdland is half the radius of the Earth, denoted as RE. Therefore, the radius of Nerdland, rN, would be:
rN = RE / 2
Now, let's find the mass of the student on Nerdland, denoted as mS. We can rearrange the equation for gravitational force as follows:
F = (G * (mE * mS))/rE²
Since the gravitational force on Nerdland would be different than that on Earth, we can set up a ratio of gravitational forces:
F(on Nerdland) / F(on Earth) = (G * (mN * mS))/(rN²) / (G * (mE * mS))/(rE²)
The gravitational constant, G, and the mass of the student, mS, are present in both ratios, so we can cancel them out:
F(on Nerdland) / F(on Earth) = (mN * rE²) / (mE * rN²)
Now, let's plug in the given values:
mN = 3 * mE
rN = RE / 2
F(on Nerdland) / F(on Earth) = ((3 * mE) * rE²) / (mE * (RE/2)²)
Simplifying the equation further:
F(on Nerdland) / F(on Earth) = (3 * rE²) / ((RE/2)²)
F(on Nerdland) / F(on Earth) = (3 * rE²) / ((RE²/4))
F(on Nerdland) / F(on Earth) = (3 * 4) / 1
F(on Nerdland) / F(on Earth) = 12
So, the gravitational force on Nerdland is twelve times stronger than on Earth.
Now, let's find the mass of the student on Nerdland. We know that the weight of an object on Earth is given by the equation:
Weight = mass * acceleration due to gravity
On Earth, the acceleration due to gravity is approximately 9.8 m/s². So we have:
990 N = mS * 9.8 m/s²
Solving for mS:
mS = 990 N / 9.8 m/s²
mS = 100.81 kg
Therefore, the mass of the student on Nerdland would also be 100.81 kg since mass is independent of gravitational force.