Chapter 7, Problem 2STP

To determine

To Choose: The correct option.

Answer to Problem 2STP

Option D. $4.0\times {10}^{28}\text{}\text{kg}$

Explanation of Solution

Given:

The satellite is orbiting around the planet. The satellite’s orbit radius is $6.7\times {10}^4\text{km}$ and its speed is $2.0\times {10}^5\text{m/s}$ .

Formula used:

The velocity of the object orbiting around another object of mass $m$ , is given by,

  $v=\sqrt{\frac{Gm}{r}}$

Where, $G$ is the gravitational constant, and $r$ is the distance between the masses.

Calculation:

The value of gravitational constant is,

  $G=6.67\times {10}^{-11}\text{N}{\text{.m}}^{\text{2}}{\text{/kg}}^{\text{2}}$

Then using the given values of velocity of the satellite and its orbital radius in SI units, the mass of the planet around which the satellites orbits, is given by,

  $\begin{array}{c}v=\sqrt{\frac{Gm}{r}}\\ v^2=\frac{Gm}{r}\\ {\left(2\times {10}^5\right)}^2=\frac{6.67\times {10}^{-11}\times m}{6.7\times {10}^4\times {10}^3}\\ m=\frac{4\times {10}^{10}\times 6.7\times {10}^7}{6.67\times {10}^{-11}}\\ m=4.0\times {10}^{28}\text{}\text{kg}\end{array}$

Conclusion:

Thus, the mass of the planet around which the satellites orbits, is $4.0\times {10}^{28}\text{}\text{kg}$ .