The gravitational force between an object & the Earth is inversely proportional to the square of the distance from the object to the center of the Earth. If an astronaut weighs 210 lbs on the surface of the Earth, what will he weigh 100 miles above the Earth? Assume that the radius of Earth is 4,000 miles
210/4000² = x/(4000+100)²
Solve for x:
x=210*(4000/4100)²
F = G M1Me/r^2
210 = G Me M1/4000^2
W = G Me M1 /(4100^2
W/210 = 4000^2/4100^2
W = 210 (4*10^3)^2/(4.1*10^3)^2
W = 210 (16/16.81)
To find the weight of the astronaut 100 miles above the Earth's surface, we need to use the inverse square law of gravitational force.
According to the inverse square law, the gravitational force (F) between two objects is inversely proportional to the square of the distance (d) between their centers. The equation can be written as:
F = k / d^2
where k is the constant of proportionality.
Let's denote the weight of the astronaut on the surface of the Earth as W1 and the weight of the astronaut at a height of 100 miles above the Earth as W2.
First, let's calculate the value of k. We know that the astronaut weighs 210 lbs on the surface of the Earth, so we can substitute the values of W1 = 210 lbs and d = 4000 miles into the equation:
210 = k / (4000^2)
Next, we can solve for k:
k = 210 * (4000^2)
Now, we can use the value of k to find the weight (W2) of the astronaut at a height of 100 miles above the Earth's surface. We substitute the values of k and d = 4000 + 100 = 4100 miles into the equation:
W2 = k / (4100^2)
Now we can calculate W2:
W2 = (210 * (4000^2)) / (4100^2)
Calculating the above expression will give us the weight of the astronaut 100 miles above the Earth's surface.
To find the weight of an astronaut 100 miles above the Earth's surface, we need to understand that the gravitational force decreases as the distance from the object to the center of the Earth increases.
Given that the gravitational force is inversely proportional to the square of the distance, we can use the formula:
F = k / r^2
where:
F = gravitational force
k = constant of proportionality
r = distance from the object to the center of the Earth
Now, let's find the value of k using the information given. We know that the astronaut weighs 210 lbs on the surface of the Earth, which means the weight is the gravitational force acting on the astronaut at sea level. At the surface of the Earth, the distance from the object to the center of the Earth is equal to the radius of the Earth, which is 4,000 miles.
Plugging these values into the formula:
210 = k / (4000^2)
To find k, we can rearrange this equation:
k = 210 * (4000^2)
Now that we have the value of k, we can calculate the weight of the astronaut 100 miles above the Earth's surface. At this distance, the new distance from the object to the center of the Earth is the radius of the Earth plus 100 miles (4,000 + 100).
Using the formula again:
Weight = k / (r^2)
Weight = k / (4100^2)
Plugging in the value of k we calculated earlier:
Weight = (210 * (4000^2)) / (4100^2)
Simplifying the equation:
Weight = (210 * 16000000) / 16810000
Weight = 200 lbs
Therefore, the astronaut would weigh 200 lbs 100 miles above the Earth's surface.