Problem 40 - College Physics

Question

Apparent weightlessness in a satellite. You have probably seen films of astronauts floating freely in orbiting satellites. People often think the astronauts are weightless because they are free of the gravity of the earth. Let us see if that explanation is correct. (a) Typically, such satellites orbit around 400 \(\mathrm{km}\) above the surface of the earth. If an astronaut weighs 750 \(\mathrm{N}\) on the ground, what will he weigh if he is 400 \(\mathrm{km}\) above the surface? (b) Draw the orbit of the satellite in part (a) to scale on a sketch of the earth. (c) In light of your answers to scale on a sketch of the earth. (c) In light of your answers to parts (a) and (b), are the astronauts weightless because gravity is so weak? Why are they apparently weightless?

Step-by-Step Solution

Answer

(a) The astronaut weighs less but not zero at 400 km. (b) They orbit Earth at 6,771 km radius. (c) They're apparently weightless due to free-fall, not weak gravity.

Step 1: Find the Acceleration Due to Gravity at 400 km Altitude

The formula for gravitational acceleration is given by \( g' = \frac{GM}{(R + h)^2} \), where \( G \) is the gravitational constant \( (6.674 \times 10^{-11} \ \text{Nm}^2/\text{kg}^2) \), \( M \) is the mass of the Earth \( (5.972 \times 10^{24} \ \text{kg}) \), \( R \) is the radius of the Earth \( (6.371 \times 10^6 \ \text{m}) \), and \( h \) is the altitude (400 km = 400,000 meters). Substitute these values to find \( g' \).

Step 2: Calculate the Weight at 400 km Above Earth

The weight of the astronaut is given by \( W = mg' \), where \( m \) is the mass of the astronaut. We know the weight on Earth is 750 N, so \( m = \frac{750}{9.81} \ \text{kg} \). Use \( m \) and the newly calculated \( g' \) from Step 1 to find the weight at 400 km altitude.

Step 3: Draw a Scaled Orbit of the Satellite

Draw a circle to represent the Earth with a radius of 6,371 km. Then, draw another circle with a radius of 6,771 km (i.e., the sum of Earth's radius and the altitude of 400 km) to represent the orbital path of the satellite. This represents the orbit scale.

Step 4: Interpret Apparent Weightlessness

Astronauts are not truly weightless due to weak gravity. Their weight decreases slightly because gravity is still strong at 400 km. The apparent weightlessness is due to free-fall: astronauts are continuously falling towards Earth but moving forward fast enough that they keep missing it, maintaining orbit.

Key Concepts

Gravitational Force

Gravitational force is what keeps planets in orbit and causes objects to fall towards the Earth. It's a fundamental interaction between two masses. In the case of the Earth and an astronaut, this force depends on the mass of each and the distance between them. Newton's Law of Universal Gravitation states that the force (\( F \)) between two masses is given by: \[ F = \frac{GMm}{r^2} \]where:- \( G \) is the gravitational constant \((6.674 \times 10^{-11} \, \text{Nm}^2/\text{kg}^2)\)- \( M \) is the mass of the Earth \((5.972 \times 10^{24} \, \text{kg})\)- \( m \) is the mass of the astronaut- \( r \) is the distance from the center of the Earth to the astronaut which includes Earth's radius plus the altitudeEven at high altitudes like 400 km above the surface, gravity is still significant. This is why astronauts can still "feel" weight, albeit reduced.

Satellite Orbits

Satellite orbits are pathways that satellites follow around the Earth. These orbits result from a perfect balance of gravitational force pulling the satellite towards Earth and the satellite's linear momentum keeping it traveling forward. Understanding satellite orbits is crucial to dispel the myth of weightlessness. Even though they are high above the surface, satellites, including spacecraft with astronauts, orbit the Earth. To depict this, imagine drawing a circle to represent the Earth, then putting another circle around it—including Earth's radius and the altitude of the orbit. The satellite follows this path. The careful design of this pathway allows continuous motion around the Earth without falling back to its surface.

Acceleration Due to Gravity

Even in orbit, there is still acceleration due to gravity acting on objects. On the Earth's surface, this value is approximately 9.81 m/s². However, as one moves away from the Earth's surface, this value decreases.For instance, at 400 km above the Earth, the acceleration due to gravity can be calculated with the formula:\[ g' = \frac{GM}{(R + h)^2} \]where:- \( G \) is the gravitational constant- \( M \) is the mass of the Earth- \( R \) is the radius of the Earth- \( h \) is the altitude, here 400 kmSo, with increasing distance from Earth, the \( g' \) decreases, meaning that objects still experience gravity but with reduced intensity, causing a lighter weight sensation in orbit.

Free-Fall in Orbit

The key reason astronauts feel "weightless" in orbit is due to free-fall—a state of continuous fall towards Earth. Imagine this: an astronaut in a satellite is constantly falling towards the Earth. However, as the satellite moves forward at a high enough speed, this forward motion counteracts the fall. It's like running around a curve where you constantly miss hitting the ground. In essence, astronauts are experiencing free-fall together with their satellite. They're not losing gravitational pull but instead both the rate of descent and the forward speed synchronizes in such a way that they do not crash into the Earth. This balance gives the sensation of apparent weightlessness common in space exploration footage.