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Standard X

Physics

Work Done by Force of Gravity

A planet of m...

Question

A planet of mass M has a uniform density in a spherical volume of radius R. Calculate the work done by the external agent to disassemble the planet in eight identical spherical part against gravitational pull amongst its constitute particle.

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Solution

the total self-energy of the 8 smaller spheres is
$\frac{8 x - 3 G {(\frac{M}{8})}^{2}}{5 (\frac{R}{2})}$
$= \frac{- 3 G M^{2}}{8 x} \times \frac{2}{(5 R)}$
$= \frac{- 3 G M^{2}}{4 x} \times \frac{1}{(5 R)}$
$= \frac{- 3 G M^{2}}{(20 R)}$

The work done is the increase in potential enenrgy which is
$\frac{- 3 G M^{2}}{(20 R)} - (\frac{- 3 G M^{2}}{(5 R)})$
$= \frac{3 G M^{2}}{(5 R)} - \frac{- 3 G M^{2}}{(20 R)}$
$= (\frac{3 G M^{2}}{(R)}) (\frac{1}{5} - \frac{1}{20})$
$= (\frac{3 G M^{2}}{(R)}) (\frac{3}{20})$
$= \frac{\frac{3 G M^{2}}{(R)}}{20}$

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A planet of mass M has a uniform density in a spherical volume of radius R. Calculate the work done by the external agent to disassemble the planet in eight identical spherical part against gravitational pull amongst its constitute particle.

the total self-energy of the 8 smaller spheres is8x−3G(M8)25(R2)=−3GM28x×2(5R) =−3GM24x×1(5R) =−3GM2(20R) The work done is the increase in potential enenrgy which is −3GM2(20R)−(−3GM2(5R)) =3GM2(5R)−−3GM2(20R) =(3GM2(R))(15−120) =(3GM2(R))(320) =3GM2(R)20