Chapter 12, Problem 56QAP

To determine

The time period of each of the planets in our solar system.

Answer to Problem 56QAP

Time period for Venus: $T=2.11\sec$

Time period for Mercury: $T=3.31\sec$

Time period for Earth: $T=2.00\sec$

Time period for Mars: $T=3.23\sec$

Time period for Jupiter: $T=1.23\sec$

Time period for Saturn: $T=1.82\sec$

Time period for Uranus: $T=1.92\sec$

Time period for Neptune: $T=1.67\sec$

Explanation of Solution

Given:

Length = l
g= $9.8{\text{m/s}}^{\text{2}}$

Formula used:

  $T=2\pi \sqrt{\frac{l}{g}}$

g is gravity

Calculation:

  $T=2\pi \sqrt{\frac{l}{g}}$

  $\begin{array}{l}T=2\pi \sqrt{\frac{l}{g}}\\ T=2\pi \sqrt{\frac{1}{g}}\\ T=\frac{2\pi }{\sqrt{g}}\end{array}$

Time period for Venus:

  $\begin{array}{l}T=\frac{2\pi }{\sqrt{g_{venus}}}\\ g_{venus}=8.87\\ T=\frac{2\pi }{\sqrt{8.87}}\\ T=\frac{2\pi }{2.976}\\ T=2.11\sec \end{array}$

Time period for Mercury:

  $\begin{array}{l}T=\frac{2\pi }{\sqrt{g_{Mercury}}}\\ g_{Mercury}=3.59\\ T=\frac{2\pi }{\sqrt{3.59}}\\ T=\frac{2\pi }{1.894}\\ T=3.31\sec \end{array}$

Time period for Earth:

  $\begin{array}{l}T=\frac{2\pi }{\sqrt{g_{Earth}}}\\ g_{Earth}=9.80\\ T=\frac{2\pi }{\sqrt{9.80}}\\ T=\frac{2\pi }{3.130}\\ T=2.00\sec \end{array}$

Time period for Mars:

  $\begin{array}{l}T=\frac{2\pi }{\sqrt{g_{Mars}}}\\ g_{Mars}=3.77\\ T=\frac{2\pi }{\sqrt{3.77}}\\ T=\frac{2\pi }{1.941}\\ T=3.23\sec \end{array}$

Time period for Jupiter:

  $\begin{array}{l}T=\frac{2\pi }{\sqrt{g_{Jupiter}}}\\ g_{Jupiter}=25.95\\ T=\frac{2\pi }{\sqrt{25.95}}\\ T=\frac{2\pi }{5.094}\\ T=1.23\sec \end{array}$

Time period for Saturn:

  $\begin{array}{l}T=\frac{2\pi }{\sqrt{g_{Saturn}}}\\ g_{Saturn}=11.08\\ T=\frac{2\pi }{\sqrt{11.08}}\\ T=\frac{2\pi }{3.435}\\ T=1.82\sec \end{array}$

Time period for Uranus:

  $\begin{array}{l}T=\frac{2\pi }{\sqrt{g_{Uranus}}}\\ g_{Uranus}=10.67\\ T=\frac{2\pi }{\sqrt{10.67}}\\ T=\frac{2\pi }{3.266}\\ T=1.92\sec \end{array}$

Time period for Neptune:

  $\begin{array}{l}T=\frac{2\pi }{\sqrt{g_{Neptune}}}\\ g_{Neptune}=14.07\\ T=\frac{2\pi }{\sqrt{14.07}}\\ T=\frac{2\pi }{3.750}\\ T=1.67\sec \end{array}$