Chapter 14, Problem 20A

The dashed lines show three circular orbits about Earth.

Rank the following quantities for these orbits from greatest to least.

a. orbital speed

b. time for orbiting Earth

To determine

To rank:The three circular orbits from greatest to least on the basis of orbital speeds.

Answer to Problem 20A

The rank of the orbital speed for the three circular orbits from greatest to least is $\text{A}>\text{B}>\text{C}$ .

Explanation of Solution

Given:

The given circular orbits are,

  

Formula used:

The speed at which a body revolves around the other body is the orbital speed.

The expression for orbital speed is,

  $v=\sqrt{\frac{GM}{R}}$

Here, $G$ is the universal gravitational constant, $M$ is the mass of the earth, and $R$ is the radial distance.

Calculation:

The orbital speed of the circular orbits about Earth is related to the mass of the Earth and the radial distance of the orbit.

For higher altitudes the orbital speed is less, consider the given figure. The orbital radial distance of the circular orbit is more for the case C and less for the case A.

Less the radial distance will increase the orbital speed. So, the orbital speed of case A will be more compared to the other orbits.

Hence, the rank of the orbital speed for the three circular orbits from greatest to least is $\text{A}>\text{B}>\text{C}$ .

Conclusion:

Thus, the rank of the orbital speed for the three circular orbits from greatest to least is $\text{A}>\text{B}>\text{C}$ .

To determine

To rank:The three circular orbits from greatest to least on the basis of time for orbiting the Earth.

Answer to Problem 20A

The rank of the time for orbiting Earth for the three circular orbits from greatest to least is $\text{C}>\text{B}>\text{A}$ .

Explanation of Solution

Given:

The given circular orbits are,

  

Formula used:

The time takes a satellite to make one full orbit is known as the time for orbiting Earth.

The expression for time for orbiting Earth is,

  $T=2\pi \sqrt{\frac{R^3}{GM}}$

Here, $G$ is the universal gravitational constant, $M$ is the mass of the earth, and $R$ is the radial distance.

Calculation:

The time for orbiting Earth of the circular orbits about Earth is related to the mass of the planet and the radial distance of the orbit.

For higher altitudes the time for orbiting earth is more.

Consider given figure.The orbital radial distance of the circular orbit is more for the case C and less for the case A.Increase in radial distance of the orbit will increase the orbital period. So, the time for orbiting Earth of case C will be more compared to the other orbits.

Hence, the rank of the time for orbiting Earth for the three circular orbits from greatest to least is $\text{C}>\text{B}>\text{A}$ .

Conclusion:

Thus, the rank of the time for orbiting Earth for the three circular orbits from greatest to least is $\text{C}>\text{B}>\text{A}$ .