Chapter 7, Problem 1RQ

Review Question 7.1 Assuming that Earth’s orbit around the Sun is circular, what information do you need to estimate the work done by the Sun on Earth while Earth travels 1000 km? Explain your answer.

To determine

The information needed to estimate the work done by the Sun on the Earth while Earth travels $1000\text{ km}$ in an assumed circular orbit.

Answer to Problem 1RQ

Solution:

No information is needed because work done by the Sun on the Earth is zero in this case.

Explanation of Solution

Introduction:

Work done by any force is defined as the product of force and the displacement in the direction of the force.

In this question, it is assumed that Earth’s orbit is circular, i.e., Earth’s displacement in the radial direction is zero at every point of the path.

Explanation:

When the orbit is assumed to be circular radius, it is constant at every point of the path, i.e., there is no displacement in the direction of the radius. Since, the Sun is at the center of this circular orbit, so the gravitational force between the Sun and Earth is always towards the center and along the radius.

The equation for the work done is given below.

$W=Fd\cos \theta$

Here, $W$ is the work done by the applied force (gravitational force between the Sun and Earth), $F$ is the gravitational force between Sun and Earth, $d$ is the displacement, and $\theta$ is the angle between the applied force and displacement.

In the case of a circular path, the angle between displacement and the force is $90°$.

When $90°$ is substituted for $\theta$ in the expression for work done, $\cos 90°$ will be zero, resulting in zero work done.

Conclusion:

Work done by centripetal force in case of circular motion is always zero. In this case, the Earth’s orbit is assumed to be circular, the gravitational force’s direction is always along the radius towards the center, i.e., the Sun. But, displacement along the radius is zero at every point of the path. Hence, the work done is zero.