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. Show all work.
weight W = M g
g , and the weight, are proportional to M/R^2
M/R^2 is 4Me/(16 Re^2) = (1/4) Me/Re^2
on the new planet.
Me = earth mass
Re = earth radius
Weight will be four times less on the new planet
The "M" in the first equation is the object mass, not the earth mass. I should have used m.
The answer is the same
Well, on this planet, "weight" might become a bit of a subjective term. I like to think of it as "how much gravitational pull does the planet have on me" rather than "how heavy am I?"
So, let's calculate that gravitational pull! The gravitational force between two objects is given by the equation:
F = G * (m1 * m2) / r^2
Here, G is the gravitational constant, m1 and m2 are the masses of the two objects, and r is the distance between their centers. The weight of an object is the force of gravity acting on it.
On Earth, let's say your mass (m1) is 70 kilograms. The mass of Earth (m2) is about 5.97 * 10^24 kilograms. The radius of Earth (r) is roughly 6,371 kilometers.
So, the gravitational force on Earth would be:
F_earth = G * (70 kg * 5.97 * 10^24 kg) / (6,371 km)^2
Now let's calculate the gravitational force on the larger planet. The mass of the larger planet would be 4 times the mass of Earth, and the radius would also be 4 times the radius of Earth.
F_largerplanet = G * (70 kg * 4 * 5.97 * 10^24 kg) / (4 * 6,371 km)^2
Now, let's compare these two forces:
F_largerplanet / F_earth = (G * (70 kg * 4 * 5.97 * 10^24 kg) / (4 * 6,371 km)^2) / (G * (70 kg * 5.97 * 10^24 kg) / (6,371 km)^2)
A lot of things cancel out:
F_largerplanet / F_earth = (4 * 5.97 * 10^24 kg) / (4 * 6,371 km)^2
And after simplifying, the fraction is:
F_largerplanet / F_earth = 1
So, you would weigh the same on that planet as you would on Earth! No weight gain or loss, just a change of scenery. I guess the gravitational pull likes to keep things balanced, just like a good circus act!
To calculate how much more or less you would weigh on a planet, we can use the concept of gravity. The weight of an object is determined by the gravitational force acting on it. The formula to calculate weight is:
Weight = mass × gravitational acceleration
The gravitational acceleration on a planet can be found using the formula:
gravitational acceleration = (gravitational constant × mass of the planet) / (radius of the planet)^2
Let's first calculate the gravitational acceleration on Earth:
Mass of Earth = 5.972 × 10^24 kg
Radius of Earth = 6,371 km = 6,371,000 m
Gravitational constant = 6.674 × 10^-11 m^3/(kg s^2)
Plugging these values into the formula, we can calculate the gravitational acceleration on Earth:
Gravitational acceleration on Earth = (6.674 × 10^-11 × 5.972 × 10^24) / (6,371,000)^2 m/s^2
Now let's calculate the gravitational acceleration on the new planet:
Mass of the new planet = 4 × Mass of Earth = 4 × 5.972 × 10^24 kg
Radius of the new planet = 4 × Radius of Earth = 4 × 6,371,000 m
Gravitational acceleration on the new planet = (6.674 × 10^-11 × 4 × 5.972 × 10^24) / (4 × 6,371,000)^2 m/s^2
Now we can calculate the ratio of the weight on the new planet compared to Earth. Since weight is directly proportional to gravitational acceleration, we can write:
Weight on the new planet / Weight on Earth = (Gravitational acceleration on the new planet) / (Gravitational acceleration on Earth)
Substituting the values we calculated, we get:
Weight on the new planet / Weight on Earth = [(6.674 × 10^-11 × 4 × 5.972 × 10^24) / (4 × 6,371,000)^2] / [(6.674 × 10^-11 × 5.972 × 10^24) / (6,371,000)^2]
Simplifying the expression, we find:
Weight on the new planet / Weight on Earth = 4
Therefore, you would weigh 4 times more on this planet compared to your weight on Earth.