Problem 15 - University Physics with Modern Physics

Question

Calculate the earth's gravity force on a 75 -kg astronaut who is repairing the Hubble Space Telescope 600 \(\mathrm{km}\) above the earth's surface, and then compare this value with his weight at the earth's surface. In view of your result, explain why we say astronauts are weightless when they orbit the earth in a satellite such as a space shuttle. Is it because the gravitational pull of the earth is negligibly small?

Step-by-Step Solution

Answer

The gravitational force at 600 km is less but not negligible. Astronauts feel weightless because they are in free fall, not because gravity is negligible.

Step 1: Identify Given Values and Constants

The astronaut's mass (\( m \)) is 75 kg. The distance above Earth's surface is 600 km, which is 600,000 meters. The radius of Earth (\( R \)) is approximately 6,371,000 meters. The gravitational constant (\( G \)) is \( 6.674 \times 10^{-11} \, \text{N m}^2/\text{kg}^2 \). The Earth's mass (\( M \)) is \( 5.972 \times 10^{24} \, \text{kg} \). With these, we can calculate the gravitational force.

Step 2: Calculate Gravitational Force at 600 km Above Earth's Surface

First, find the total distance from the center of Earth to the astronaut, which is \( R + 600,000 \) meters. Then use the formula for gravitational force: \[ F = \frac{G M m}{(R + 600,000)^2} \].Substitute the given values:\[ F = \frac{6.674 \times 10^{-11} \times 5.972 \times 10^{24} \times 75}{(6,371,000 + 600,000)^2} \].Calculate this to find \( F \).

Step 3: Calculate Weight on Earth's Surface

The weight of the astronaut on Earth's surface is calculated using: \[ W = mg \], where the acceleration due to gravity on the Earth's surface \( g \) is \( 9.81 \, \text{m/s}^2 \). This gives:\[ W = 75 \times 9.81 \].Compute this to find the weight on the surface.

Step 4: Compare the Two Forces

Calculate the ratio of gravitational force at 600 km above the surface to the weight on Earth's surface:\[ \frac{F}{W} \].Evaluate this ratio to understand the relative strength of gravity at 600 km compared to Earth's surface gravity.

Step 5: Explain the Concept of Weightlessness

Astronauts feel weightless not because the gravitational pull is negligibly small but because they are in free fall, orbiting the Earth. In orbit, they are constantly falling towards Earth but also moving forward, creating a sensation of weightlessness.

Key Concepts

Astronaut Weight

When we talk about an astronaut's weight, we're referring to the force exerted by gravity on their mass. On Earth, this force is straightforward: it's simply their mass multiplied by the acceleration due to Earth's gravity, or

• Weight = mass × gravity (which is about 9.81 m/s²).

However, when an astronaut is in space, especially at a height of 600 km above the Earth's surface while repairing something like the Hubble Space Telescope, calculating this becomes a bit more complex.

At this altitude, the force of gravity is still there but is weaker than on the ground.
The astronaut's "weight" as felt by them isn't the same as it would be on Earth.
This might seem counterintuitive since we often assume they are weightless in space, but we'll get into why this is the case later.

Earth's Gravity

Earth's gravity is the force that keeps everything anchored to our planet. It pulls objects towards the Earth's center. The strength of this force diminishes with distance from the Earth's surface.

• Near the surface, gravity is strong enough to keep everything, from people to oceans, firmly in place.

When you are still relatively close to Earth, like a few hundred kilometers up, gravity still has a strong hold.
The gravitational force can be calculated using the equation: \[ F = \frac{G M m}{R^2} \]where

• \( G \) is the universal gravitational constant, approximately \( 6.674 \times 10^{-11} \, \text{N m}^2/\text{kg}^2 \),
• \( M \) is Earth's mass,
• \( m \) is the object's mass, and
• \( R \) is the distance from the center of the Earth.

Therefore, even at 600 km above the Earth's surface, an astronaut is still under the influence of gravity, albeit slightly less than at ground level.

Orbital Mechanics

Orbital mechanics are the rules governing movements of objects in space under the influence of gravitational forces. When an astronaut is orbiting Earth, they are subject to these principles. Imagine throwing a ball so fast that rather than falling back to the ground, it keeps missing the Earth, following a curved path around it.

• This is essentially what happens in orbit – the astronaut is in constant free fall towards Earth, but their forward motion keeps them in a continuous path around the planet.

In this state, the astronaut and their spacecraft are moving at such speeds that their downward motion due to gravity is perfectly matched by their forward motion.

This combination keeps them floating along a stable path, like dancing with destiny, around the Earth.

Weightlessness in Space

Despite gravity still acting on astronauts, they experience a sensation of weightlessness. This isn't because the gravitational force is negligible. Instead, it's due to their condition of "free fall", which is the same state achieved when skydiving briefly before a parachute deploys.

• As both the astronaut and their ship are falling, everything inside moves at the same rate, effectively "floating" relative to one another.

This is why astronauts appear to be weightless while working in space despite Earth's gravity still pulling on them. They are in a state of continuous free fall, causing them to feel as though they carry no weight at all.

Weightlessness thus is more accurately a term for the synchronized free-fall of an astronaut and their environment, rather than an absence of gravitational force.