Why is the work done by the gravitational force on a satellite moving in a circular orbit around the Earth considered to be zero?

Correct Option

The correct option is

The force of gravity acts perpendicular to the direction of displacement.

Explanation

In physics, Work is defined as the product of the component of force in the direction of the displacement and the magnitude of this displacement. Mathematically, it is expressed as:

W = F · s · cos θ

Where:

• F is the magnitude of the force.
• s is the magnitude of the displacement.
• θ is the angle between the force vector and the displacement vector.

Detailed Analysis

To determine why the work done is zero, one must analyze the relationship between the direction of the gravitational force and the motion of the satellite:

• Direction of Force: The gravitational force exerted by the Earth on the satellite acts as the centripetal force. It is always directed radially inward towards the center of the Earth.
• Direction of Displacement: At any instant, the satellite moves tangentially to its circular orbit. Therefore, the instantaneous displacement is perpendicular to the radius.
• Angle of Interaction: The angle (θ) between the gravitational force (radial) and the displacement (tangential) is 90°.
• Calculation: Since cos 90° = 0, the work done is:
  W = F · s · 0 = 0.

Option Analysis

• The speed of the satellite is constant. The speed of the satellite is constant: While true for a circular orbit, constant speed implies zero change in kinetic energy (Work-Energy Theorem). However, the fundamental reason for zero work is the angle between force and displacement, not the speed itself.
• The gravitational force is zero in space. The gravitational force is zero in space: Incorrect. Gravitational force exists and provides the necessary centripetal force to keep the satellite in orbit.
• The force of gravity acts perpendicular to the direction of displacement. The force of gravity acts perpendicular to the direction of displacement: Correct. This geometric relationship ensures that the scalar product of force and displacement is zero.
• The displacement of the satellite is zero after one revolution. The displacement of the satellite is zero after one revolution: While true for a complete cycle, the work done by gravity is zero at every instant of the motion, not merely because the net displacement over a full loop is zero.

Key Takeaway: Work done by a force is zero if the force acts perpendicular (at 90°) to the direction of the body's displacement.