Question

Triple Choice The International Space Station orbits the Earth in an approximately circular orbit at a height of \(h=375 \mathrm{~km}\) above the Earth's surface. In one complete orbit, is the work done by Earth's gravity on the space station positive, negative, or zero? Explain.

Short answer

The work done by Earth's gravity on the space station is zero.

Step-by-step solution

Understanding the Concept of Work

Work is defined as the force applied on an object times the displacement in the direction of the force. Mathematically, it is expressed as \( W = F \cdot d \cdot \cos(\theta) \), where \( F \) is the magnitude of the force, \( d \) is the displacement, and \( \theta \) is the angle between the force and displacement vectors.

Analyzing the Force Applied by Earth's Gravity

Earth's gravity exerts a centripetal force on the International Space Station, directed towards the center of the Earth. This force is always perpendicular to the displacement of the station along its circular orbit.

Evaluating the Work Done by Earth's Gravity

Since the force of gravity is perpendicular to the displacement of the space station, the angle \( \theta \) between the force and displacement vectors is \( 90^\circ \). Thus, \( \cos(90^\circ) = 0 \), which means the work done \( W = F \cdot d \cdot \cos(\theta) = 0 \).

Conclusion

In the case of circular motion, when the force is perpendicular to the motion, the work done is zero. Hence, the work done by Earth's gravity on the space station in one complete orbit is zero.

Key concepts

Work Done by Gravity

When we talk about work done by gravity, we need to understand how work is calculated. Work is done when a force causes an object to move. The amount of work done can be calculated using the formula: \[ W = F \cdot d \cdot \cos(\theta) \] Here,

• \( W \) is the work done,
• \( F \) is the force applied,
• \( d \) is the displacement of the object, and
• \( \theta \) is the angle between the force and the displacement vectors.

In the context of the International Space Station, gravity acts as a centripetal force that keeps it in orbit. Importantly, this force is always directed toward the center of the Earth and is perpendicular to the path (displacement) of the space station in a circular orbit. When the angle \( \theta \) is 90 degrees, as it is in this circular motion, \( \cos(90^\circ) = 0 \). Consequently, the work done by gravity is zero because the displacement is perpendicular to the force of gravity.
This is a key feature in any circular orbit where gravity acts as a centripetal force: no energy is added or taken away by gravity, maintaining constant motion.

Centripetal Force

Centripetal force is crucial for maintaining circular motion. In any circular orbit, such as that of the International Space Station, centripetal force acts towards the center of the circle, which in this case is towards the Earth. This force is essential for keeping the object moving in a circular path instead of flying off in a straight line.

• A centripetal force does not do work on the object moving in a circle because:
  • It is always perpendicular to the direction of the object’s displacement.
  • The force changes the direction of the object, not its speed.

Using the formula for work, we see that this perpendicular relationship ensures \( \cos(\theta) \) is zero. As a result, despite the force’s presence, the work done remains zero.
This characteristic explains why satellites and space stations maintain their orbits without the need for additional energy input to counteract gravity during orbital motion.

International Space Station

The International Space Station (ISS) is a marvel of human engineering afloat in the cosmos. It serves as both a research laboratory and a habitat, orbiting Earth approximately 375 kilometers above the planet's surface. The ISS travels at a speed of about 28,000 kilometers per hour, completing an orbit roughly every 90 minutes.
The orbit of the ISS is nearly circular, which means it experiences the continuous yet balanced force of Earth's gravity serving as the centripetal force needed to sustain its orbit.

• Despite gravity's constant pull, it does not slow down or stop the ISS because:
  • This gravitational pull acts perpendicular to the movement of the ISS.
  • The exact balance of velocity and gravitational pull allows it to remain in orbit without falling back to Earth.

The energies involved in maintaining its orbit are fascinatingly efficient because no work is required by gravity, as we discussed earlier. The station, therefore, conserves energy wisely while providing a platform for invaluable scientific experiments and international cooperation.