202111250910 Solution to 2020-DSE-PHY-1A-10

The diameter of Neptune is about $4$ times that of the Earth and its mass is about $17$ times that of the Earth. Estimate the acceleration due to gravity on Neptune’s surface.

Given: acceleration due to gravity on Earth’s surface $g=9.81\,\mathrm{m\, s^{-2}}$

Background.

From Newton’s law of gravitation, Eq. (B7):

$F=\displaystyle{\frac{Gm_{1}m_{2}}{r^2}}$

_{(pg. 15, List of data, formulae and relationship)}

Solution.

On the Earth, for some object of mass $m$ , we see that:

$\begin{aligned} F_{E} & = \frac{GmM_{E}}{r_{E}} \\ mg_{E} & = \frac{GmM_{E}}{r_{E}} \\ g_{E} & = \frac{GM_{E}}{r_{E}} \\ \end{aligned}$

Similarly, for some object of mass $m$ on Neptune, we have

$\begin{aligned} F_{N} & = \frac{GmM_{N}}{r_{N}} \\ mg_{N} & = \frac{GmM_{N}}{r_{N}} \\ g_{N} & = \frac{GM_{N}}{r_{N}} \\ \end{aligned}$

We compare $g_{N}$ to $g_{E}$ by the assumption:

$\begin{aligned} g_{N} & = \frac{GM_{N}}{r_{N}} \\ & = \frac{G(17M_E)}{(4r_{E})} \\ & = \frac{17}{4}\bigg(\frac{GM_{E}}{r_{E}}\bigg) \\ & = \frac{17}{4}g_{E} \\ \end{aligned}$

And the answer is D.

M	T	W	T	F	S	S
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15	16	17	18	19	20	21
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