Saturn has a radius of about 9.0 earth radii, and a mass 95 times the Earth’s mass. Estimate the gravitational field on the surface of Saturn compared to that on the Earth. Show your work.
+26367 Saturn has a radius of about 9.0 earth radii, and a mass 95 times the Earth’s mass.
Estimate the gravitational field on the surface of Saturn compared to that on the Earth.
Newton's law of universal gravitation : \(F = G \dfrac{Mm}{r^2}\)
Newton's laws of motion : \(F=ma\)
\(\begin{array}{|rcll|} \hline F=ma&=& G \dfrac{Mm}{r^2} \\\\ \mathbf{a} &=& \mathbf{\dfrac{GM}{r^2}} \\ \hline \end{array}\)
\(\begin{array}{|rcll|} \hline a_{\text{Earth}} &=& \mathbf{\dfrac{GM_{\text{Earth}}}{R_{\text{Earth}}^2}} \\ a_{\text{Saturn}} &=& \mathbf{\dfrac{GM_{\text{Saturn}}}{R_{\text{Saturn}}^2}} \\\\ \dfrac{a_{\text{Saturn}}} {a_{\text{Earth}}} &=& \dfrac{GM_{\text{Saturn}}}{R_{\text{Saturn}}^2}\above 1pt \dfrac{GM_{\text{Earth}}}{R_{\text{Earth}}^2} \\\\ \dfrac{a_{\text{Saturn}}} {a_{\text{Earth}}} &=& \dfrac{ M_{\text{Saturn}}R_{\text{Earth}}^2}{R_{\text{Saturn}}^2M_{\text{Earth}}} \quad | \quad R_{\text{Saturn}} = 9R_{\text{Earth}},\ M_{\text{Saturn}}= 95M_{\text{Earth}} \\\\ \dfrac{a_{\text{Saturn}}} {a_{\text{Earth}}} &=& \dfrac{ 95M_{\text{Earth}}R_{\text{Earth}}^2}{\left(9R_{\text{Earth}}\right)^2M_{\text{Earth}}} \\\\ \dfrac{a_{\text{Saturn}}} {a_{\text{Earth}}} &=& \dfrac{ 95R_{\text{Earth}}^2}{81R_{\text{Earth}}^2 } \\\\ \dfrac{a_{\text{Saturn}}} {a_{\text{Earth}}} &=& 1.17283950617 \\ a_{\text{Saturn}} &=& 1.17283950617 *a_{\text{Earth}} \quad | \quad a_{\text{Earth}}= 9.8\ \dfrac{m}{s^2} \\ \mathbf{ a_{\text{Saturn}} } &\approx& \mathbf{ 11.5\ \dfrac{m}{s^2} } \\ \hline \end{array}\)
