Chapter 7, Problem 35A

To determine

The orbital period of Pluto around the Sun in years.

Answer to Problem 35A

  $245\text{years}$ approximately

Explanation of Solution

Given:

Pluto has a mean distance of $5.87\times {10}^{12}\text{m}$ from the Sun.

Formula used:

Period of a Planet orbiting the Sun is given by,

  $T=2\pi \sqrt{\frac{r^3}{G{m}_S}}$

Where $G$ is the gravitational constant, $r$ is the orbital radius, and $m_S$ is the mass of the Sun given as $1.99\times {10}^{30}\text{kg}$ .

Calculation:

It is known that 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 and known values, the orbital period of Pluto around the Sun will be,

  $\begin{array}{c}T=2\pi \sqrt{\frac{r^3}{G{m}_S}}\\ =2\pi \sqrt{\frac{{\left(5.87\times {10}^{12}\right)}^3}{6.67\times {10}^{-11}\times 1.99\times {10}^{30}}}\\ =2\pi \left(1.23\times {10}^9\right)\\ =7.72\times {10}^9\text{s}\times \frac{1\text{}\text{hr}}{3600\text{s}}\times \frac{1\text{}\text{day}}{24\text{hr}}\times \frac{1\text{}\text{year}}{365\text{days}}\\ \approx 245\text{years}\end{array}$

Conclusion:

Thus, the orbital period of Pluto is approx $245\text{years}$ .