You are explaining to a friend why the crew of the International Space Station feels weightless

while orbiting the Earth. Your friend responds that he thought gravity was just a lot weaker up
there. Show that it isn’t so by calculating the acceleration due to gravity 350 km above the
Earth’s surface in terms of g

Well, my friend, while it may seem like gravity is weaker up there, the truth is that it's actually pretty strong. Let me explain.

The acceleration due to gravity, denoted as "g," is a constant value on Earth's surface, which is approximately 9.8 m/s². However, as you move away from the Earth's surface, the force of gravity decreases, but it never really goes away completely.

To calculate the acceleration due to gravity 350 km above the Earth's surface, we can use an equation derived from Newton's law of universal gravitation. The equation is:

g' = g * (r / (r + h))²

Where:
g' is the acceleration due to gravity at a height h above the Earth's surface,
g is the acceleration due to gravity on the Earth's surface, and
r is the radius of the Earth.

Now, let's plug in the values. The radius of the Earth is approximately 6,371 km (or 6,371,000 meters), and the height h is 350 km (or 350,000 meters).

g' = (9.8 m/s²) * ((6,371,000 m) / (6,371,000 m + 350,000 m))²

Calculating this equation, we find that the acceleration due to gravity 350 km above the Earth's surface is approximately 8.68 m/s².

So you see, my friend, while the force of gravity does decrease as you move further away from the Earth's surface, it's definitely still strong enough to make things feel weightless for the astronauts aboard the International Space Station. It's all about that free-falling feeling!

To calculate the acceleration due to gravity at a certain distance above the Earth's surface, we can use the formula for gravitational acceleration:

g' = g * (R / (R + h))^2

where:
g' is the acceleration due to gravity at the given distance above the Earth's surface,
g is the acceleration due to gravity at the Earth's surface (9.8 m/s^2),
R is the radius of the Earth (6,371 km), and
h is the height above the Earth's surface.

First, let's convert the given distance of 350 km to meters:
h = 350 km = 350,000 m

Now, we can calculate the acceleration due to gravity at this height:
g' = (9.8 m/s^2) * ((6,371 km) / ((6,371 km) + (350 km)))^2

g' = (9.8) * (6,371 / (6,721))^2

g' = 9.8 * (6,371 / 6,721)^2

Simplifying further:
g' = 9.8 * (0.948)^2

g' = 9.8 * 0.899

g' ≈ 8.808 m/s^2

So, the acceleration due to gravity 350 km above the Earth's surface is approximately 8.808 m/s^2. This means that the crew on the International Space Station does not experience a weaker gravity but feels weightless due to constantly falling towards the Earth while simultaneously maintaining their forward momentum, resulting in an apparent lack of weight.

To calculate the acceleration due to gravity at a certain distance above the Earth's surface, we can use the formula for gravitational acceleration:

g' = (G * M) / (R + h)^2

Where g' is the acceleration due to gravity at that height, G is the gravitational constant (approximately 6.674 * 10^-11 m^3 kg^-1 s^-2), M is the mass of the Earth (approximately 5.972 * 10^24 kg), R is the radius of the Earth (approximately 6,371 km), and h is the height above the Earth's surface.

In this case, we want to calculate the acceleration due to gravity at a height of 350 km above the Earth's surface, so h = 350 km = 350,000 m.

Plugging in the values into the formula, we get:

g' = (6.674 * 10^-11 * 5.972 * 10^24) / (6,371,000 + 350,000)^2

Simplifying the calculation, we have:

g' = (3.9792 * 10^14) / (6,721,000)^2

g' ≈ 3.9792 * 10^14 / 4.51744 * 10^13

g' ≈ 8.801 * m/s^2

To compare this acceleration to the acceleration due to gravity at the Earth's surface, we can divide g' by the standard acceleration due to gravity, g.

g = 9.8 m/s^2 (approximately)

So:

g' / g ≈ 8.801 / 9.8

g' / g ≈ 0.899

Therefore, the acceleration due to gravity at a height of 350 km above the Earth's surface is approximately 0.899 times the acceleration due to gravity at the Earth's surface. This means that gravity is only slightly weaker, not significantly less. Hence, the feeling of weightlessness experienced by the crew of the International Space Station is not due to a significantly weaker gravity, but rather due to the continuous free fall they are in as they orbit the Earth.