Suppose a person weighs 160 pounds on Earth. How much would that person weigh on a planet that was three times as massive as Earth and had a radius that was twice that of Earth's?
W on earth is GmM/r^2
W elsewhere = Gm(3M)/(2r)^2 = GmM/r^2 * 3/4 = 160*3/4 = 120lbs
So, even though the planet is more massive (linear), its radius has a larger (quadratic) effect on the weight.
To calculate the weight of a person on a different planet, we need to use the formula for gravitational force:
F = (G * m1 * m2) / r^2
Where:
F is the gravitational force between two objects,
G is the gravitational constant (approximately 6.67430 x 10^-11 N·(m/kg)^2),
m1 and m2 are the masses of the two objects, and
r is the distance between the centers of the two objects.
In this case, we have the mass of the person on Earth (160 pounds), and we want to know their weight on a different planet.
First, we need to convert the weight of the person to kilograms, as the gravitational constant is given in SI units:
1 pound is approximately 0.4536 kilograms
So, the weight of the person on Earth is:
160 pounds * 0.4536 kilograms/pound = 72.576 kilograms
Now, we can calculate the weight of the person on the other planet.
Let's call the mass of the person m1, the mass of the other planet m2, and the radius of the other planet r.
Given that the other planet is three times as massive as Earth and has a radius that is twice Earth's, we have:
m2 = 3 * mass of Earth
r = 2 * radius of Earth
Now, let's substitute the values into the formula for gravitational force:
F = (G * m1 * m2) / r^2
Weight on the other planet = F = (G * m1 * m2) / r^2
Weight on the other planet = (G * m1 * (3 * mass of Earth)) / (2 * radius of Earth)^2
Weight on the other planet = (G * m1 * 3 * mass of Earth) / (4 * radius of Earth^2)
Weight on the other planet = (G * m1 * 3 * mass of Earth) / (4 * (radius of Earth * radius of Earth))
Weight on the other planet = (G * m1 * 3 * mass of Earth) / (4 * Earth's surface area)
Now, we can calculate the weight on the other planet by substituting the values into the equation and solving for F.