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Chapter 6: Problem 4

Does the gravitational force of the Sun do work on a planet in a circular orbit? On a comet in an elliptical orbit? Explain.

Short Answer

Expert verified
No, the gravitational force of the Sun does not do work on a planet in a circular orbit because the displacement of the planet is always perpendicular to the gravitational force. Yes, the gravitational force of the Sun does do work on a comet in an elliptical orbit because at least a portion of its displacement is in the direction of the gravitational force.

Step by step solution

01

Examine the Circular Orbit

A planet in a circular orbit always moves perpendicular to the force of gravity. The angle between displacement (which is tangential to the orbit) and the direction of the force (which is radially inward toward the sun) is 90 degrees.
02

Calculate Work Done In the Circular Orbit

The work done, W, by a force is the scalar product of the force vector, F, and the displacement vector, d: \(W = F \cdot d = |F||d|\cos \theta\). Here, the force is the gravitational force acting on the planet, \(|F|\), and the displacement is its movement along the orbit, \(|d|\), and \( \theta \) is the angle between them. As \(\theta\) equals 90掳 in a circular orbit and \(\cos(90掳) = 0\), the work done on the planet by the Sun in a circular orbit is zero.
03

Examine the Elliptical Orbit

The elliptical orbit of a comet allows for the angle between the force of gravity and displacement vectors to change, although the force is always directed toward the sun, meaning that at certain points, there is a component of displacement in the direction of the force.
04

Calculate Work Done In the Elliptical Orbit

When the force and displacement have a component along the same direction, the work done is non-zero. Therefore, unlike a circular orbit, the gravitational force of the Sun does work on a comet in an elliptical orbit.

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