What is the Earth's gravitational force on you when you are standing on the Earth and when you are riding in the Space Shuttle 380 km above the Earth's surface? Answer as a fraction of the surface gravity of Earth. (The Earth's radius is 6400 km.)
F = G Mearth Moject / r^2
F1 = G Mearth m/6400^2
F2 = G Mearth m/6780^2
so
F2/F1 = (6400/6780)^2 = 0.891
Calculate the acceleration due to gravity when a spaceship is at a distance equal to twice the earth radius from the centre of the earth
Well, did you know that the Earth's gravitational force is a real crowd-pleaser? When you're standing on the Earth's surface, the gravitational force pulls you in with a whopping 1 g, which is equal to one surface gravity. It's like getting a big bear hug from Mother Nature herself!
Now, when you're soaring high in the Space Shuttle, things get a bit wacky. At a distance of 380 km above the Earth's surface, the gravitational force weakens a tad. You see, gravity doesn't like traveling long distances. So up there, you'll only experience about 0.86 g of Earth's surface gravity.
In simple terms, it's like gravity telling you, "Hey buddy, I'm here, but I'm taking a little break." So just remember, whether you're on Earth or floating above, gravity always finds a way to make its presence felt, even if it likes to goof around sometimes!
To calculate the gravitational force on an object, we can use Newton's law of universal gravitation, which states that the force between two objects is directly proportional to the product of their masses and inversely proportional to the square of the distance between them.
Let's analyze the two scenarios separately:
1. Standing on Earth:
When you are standing on the Earth's surface, the distance between you and the center of the Earth is the Earth's radius, which is 6400 km. The mass of the Earth and your mass are constant in both scenarios.
Using the formula F = (G * m1 * m2) / 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)
- m1 and m2 are the masses of the two objects
- r is the distance between the centers of the two objects
Let's denote the gravitational force on you when standing on the Earth as F1.
2. Riding in the Space Shuttle 380 km above the Earth's surface:
In this scenario, the distance between you and the center of the Earth is the sum of the Earth's radius (6400 km) and the altitude of the Space Shuttle (380 km).
Let's denote the gravitational force on you when riding in the Space Shuttle as F2.
To compare the two gravitational forces, we can express F2 as a fraction of F1.
F2/F1 = (G * m1 * m2) / ((R + h)^2) / (G * m1 * m2) / R^2
Cancelling out common terms, we get:
F2/F1 = (R^2) / ((R + h)^2)
Plugging in the values:
F2/F1 = (6400^2) / ((6400 + 380)^2)
Now, we can calculate the fraction:
F2/F1 = (6400^2) / (6780^2)
F2/F1 ≈ 0.79
Therefore, the Earth's gravitational force on you when you are riding in the Space Shuttle 380 km above the Earth's surface is approximately 0.79 of the gravitational force when standing on the Earth's surface.