6. Suppose that you travel to a planet that has 4 times the Earth’s mass and 4 times the Earth’s radius. Calculate how much more or less you would weigh on this planet compared to your weight on Earth. Express your answer as a factor or fraction of your weight on Earth.
1/4
Oh boy, we're going on a cosmic adventure, huh? Well, on this planet with 4 times the mass and 4 times the radius of Earth, you might expect that your weight would increase, right? Well, surprise surprise! That's not exactly the case.
You see, the force of gravity on an object depends on both the mass of the planet and your distance from its center. In this scenario, even though the planet is bigger and heavier, your distance from its center has also increased proportionally.
Since the radius of the planet has also increased by a factor of 4, your distance from the center has effectively doubled. And here's the kicker: the force of gravity is inversely proportional to the square of the distance. So, in this case, the force of gravity (and hence your weight) would actually be 1/4th of what it is on Earth!
In simpler terms, if you weigh, let's say, 100 pounds on Earth, you would weigh a mere 25 pounds on this planet. So, you'd be feeling light as a feather! But hey, don't go floating away just yet – enjoy your newfound interplanetary weight loss!
To determine how much more or less you would weigh on this planet compared to Earth, you can use the concept of gravitational force.
The gravitational force between two objects can be calculated using the equation:
F = (G * m1 * m2) / r^2
Where:
F is the gravitational force
G is the gravitational constant (approximately 6.67430 × 10^-11 N(m/kg)^2)
m1 and m2 are the masses of the two objects
r is the distance between the centers of the two objects
In this case, we want to compare the gravitational force on the planet to the gravitational force on Earth. The mass of an object is directly proportional to its weight, so we can use mass as a proxy for weight.
Let's assume your mass is "m" on Earth. On the other planet:
Mass = 4 * Mass on Earth = 4m
Radius = 4 * Radius on Earth = 4r
To calculate the weight on the other planet, we need to find the gravitational force on that planet and compare it to the gravitational force on Earth.
Gravitational force on Earth = (G * m * Mass of Earth) / (Radius of Earth)^2
Gravitational force on the other planet = (G * m * 4m) / (4r)^2
Now, let's simplify and compare the two forces:
Gravitational force on the other planet = (G * m * 4m) / (16 * r^2)
= (G * 4m^2) / (16 * r^2)
= (G * m^2) / (4 * r^2)
We can see that the gravitational force on the other planet is (1/4) times the gravitational force on Earth.
Since weight is directly proportional to the gravitational force, we can conclude that you would weigh 1/4 (or 25%) of your weight on Earth when on this planet.
To calculate how much more or less you would weigh on the planet compared to Earth, we need to consider the effect of the planet's mass and radius on its gravitational field.
The formula for gravitational force is given by Newton's Law of Universal Gravitation:
F = G * (m1 * m2) / r^2
Where:
F is the gravitational force
G is the gravitational constant
m1 and m2 are the masses of the two objects
r is the distance between the centers of the two objects
Let's assume your mass remains the same on both Earth and the planet. The gravitational force acting on you will be:
F_Earth = G * (M_Earth * m) / R_Earth^2
F_planet = G * (M_planet * m) / R_planet^2
Now, we can calculate the ratio of the gravitational forces on the planet and on Earth:
(F_planet / F_Earth) = [(G * (M_planet * m) / R_planet^2) / (G * (M_Earth * m) / R_Earth^2)]
Since the masses and the gravitational constant are the same in both equations, we can cancel them out:
(F_planet / F_Earth) = (M_planet * m * R_Earth^2) / (M_Earth * m * R_planet^2)
Simplifying the equation further:
(F_planet / F_Earth) = (M_planet * R_Earth^2) / (M_Earth * R_planet^2)
Now, let's substitute the given values:
M_planet = 4 * M_Earth (mass of the planet)
R_planet = 4 * R_Earth (radius of the planet)
(F_planet / F_Earth) = (4 * M_Earth * R_Earth^2) / (M_Earth * (4 * R_Earth)^2)
Simplifying the equation again:
(F_planet / F_Earth) = (4 * M_Earth * R_Earth^2) / (M_Earth * 16 * R_Earth^2)
(F_planet / F_Earth) = 1 / 4
Therefore, you would weigh 1/4 (or 25%) of your weight on Earth on this planet, which means you would weigh less on this planet compared to Earth.