4.11 Weightlessness in Orbit

Weightlessness is the sensation of having no weight, experienced by objects and people in a state of free fall. It is a common misconception that there is no gravity in space. At the altitude of a typical orbiting spacecraft such as the International Space Station (ISS), Earth's gravity is still about 90% as strong as it is on the surface. The feeling of weightlessness occurs because the spacecraft, and everything inside it, are continuously falling towards the Earth.

Mass vs. Weight

• Mass (m) is an intrinsic property of an object, measuring its inertia. It is a scalar quantity and remains constant everywhere.
• Weight (W) is the force of gravity acting on an object's mass ($W=mg$). It is a vector quantity.

Apparent Weight and Free Fall

What we perceive as our "weight" is not the force of gravity itself, but the normal contact force exerted on us by a surface pushing back against gravity. This contact force is called apparent weight.

Free fall is the motion of an object where gravity is the only force acting on it. In this state, there are no contact forces, and the apparent weight becomes zero.

Weightlessness in a Falling Elevator

A classic thought experiment to understand weightlessness is an elevator whose cable has snapped. The elevator and the person inside are both in free fall, accelerating downwards at the same rate $g$. Because the person is falling at the same rate as the elevator floor, the floor cannot push up on them. The normal contact force becomes zero, and the person experiences the sensation of weightlessness, floating inside the elevator.

Mathematically, the apparent weight ($T$ or Normal Force) in an accelerating elevator is:

$T=W-F_{net}=mg-ma$

When the elevator is in free fall, its acceleration $a$ is equal to $g$:

$T=mg-mg=0$

Weightlessness in an Orbiting Spacecraft

An orbiting spacecraft is in a perpetual state of free fall. It is constantly falling towards the Earth, but it also has a very high sideways velocity. This high tangential speed means that as it falls, the Earth's surface curves away from it at the same rate. The spacecraft "misses" the ground and continuously falls "around" the Earth.

Since the spacecraft and the astronauts inside are falling together at the same rate, there is no contact force between them. The astronauts float relative to the spacecraft, creating the experience of weightlessness.

Orbital Velocity (Critical Velocity)

For a satellite or spacecraft to maintain a stable orbit, it must travel at a specific horizontal speed, known as orbital velocity or critical velocity. At this speed, the force of gravity provides the exact amount of centripetal force needed to keep the satellite in its circular path.

For more details on circular motion and centripetal force, refer to Centripetal Force and Acceleration→.

Derivation

For a satellite of mass $m$ orbiting at a radius $r$ from the center of the Earth:

• Gravitational Force ($F_g$): $F_g=mg$ (where $g$ is the acceleration due to gravity at that altitude)
• Centripetal Force ($F_c$): $F_c=\frac{m{v}^2}{r}$

For a stable orbit, these two forces must be equal:

$F_g=F_c$

$mg=\frac{m{v}^2}{r}$

The mass of the satellite ($m$) cancels out on both sides:

$g=\frac{v^2}{r}$

Solving for the orbital velocity $v$:

$v^2=gr$

$v=\sqrt{gr}$

The radius of the orbit $r$ is the sum of the Earth's radius ($R$) and the satellite's altitude ($h$): $r=R+h$.

$v=\sqrt{g(R+h)}$

Example: Near-Earth Orbit

For a satellite in a low Earth orbit, the altitude $h$ is small compared to the Earth's radius $R$, so we can approximate $r\approx R$.

• $g\approx 9.8{\text{ m/s}}^2$
• $R\approx 6.4\times 10^6\text{ m}$

$v=\sqrt{(9.8{\text{ m/s}}^2)(6.4\times 10^6\text{ m})}\approx \sqrt{62.72\times 10^6{\text{ m}}^2/{\text{s}}^2}$

$v\approx 7920\text{ m/s}\approx 7.9\text{ km/s}$

This is the critical velocity required to maintain a stable orbit close to the Earth's surface.

Artificial Gravity

Prolonged weightlessness causes serious health problems including muscle atrophy, bone density loss, and cardiovascular deconditioning. To counter these effects during long-duration space missions, artificial gravity can be created.

How Artificial Gravity Works

Artificial gravity is produced by rotating the spacecraft or space station. When a cylindrical or ring-shaped station rotates, the outer rim (floor) pushes inward on the inhabitants, providing a centripetal force. The inhabitants perceive this inward push as a "weight" directed outward toward the floor — simulating the sensation of gravity.

The centripetal acceleration experienced by an inhabitant at radius $R$ from the axis of rotation is:

$a_c={\omega }^{2}R$

To simulate Earth's gravity, this is set equal to $g$:

${\omega }^{2}R=g⟹\omega =\sqrt{\frac{g}{R}}$

In terms of rotation frequency $f$ (where $\omega =2\pi f$):

$4{\pi }^2{f}^{2}R=g⟹f=\frac{1}{2\pi }\sqrt{\frac{g}{R}}$

Key point: The direction of artificial gravity is outward (away from the axis of rotation), so inhabitants stand with their heads toward the center and feet toward the outer rim.

Summary Table

Possible Questions and Answers

Q: Is there gravity on the International Space Station (ISS)?

A: Yes. The ISS orbits at an altitude of about 400 km, where Earth's gravity is still about 90% as strong as on the surface. The astronauts feel weightless because they are in a constant state of free fall around the Earth.

Q: Does the required orbital velocity depend on the mass of the satellite?

A: No. As shown in the derivation, the mass of the satellite ($m$) cancels out. The orbital velocity depends only on the gravitational acceleration and the radius of the orbit.

Q: How is artificial gravity created in a space station?

A: By rotating the space station. The centripetal acceleration $a_c={\omega }^{2}R$ acts as artificial gravity. Setting ${\omega }^{2}R=g$ gives the required angular velocity $\omega =\sqrt{g/R}$.