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https://texercises.com/exercise/identical-massive-spheres/
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The following quantities appear in the problem: Masse \(m\) / Kraft \(F\) / Volumen \(V\) / Radius \(r\) / Dichte \(\varrho\) /
The following formulas must be used to solve the exercise: \(\varrho = \dfrac{m}{V} \quad \) \(F = G \dfrac{m_1m_2}{r^2} \quad \) \(V = \dfrac{4}{3}\pi r^3 \quad \)
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Exercise:
Two identical massive spheres each with rO radius touch each other. What density would the material of the homogeneous spheres have to have so that they attract each other with FO due to gravity? One inch is .cm one pound-foce is .N.

Solution:
center tikzpicture shadeball color gray! opacity. - circle ; draw - circle ; draw - arc :: and .; drawdashed arc :: and .; shadeball color gray! opacity. circle ; draw circle ; draw arc :: and .; drawdashed arc :: and .; drawthick red - --- nodemidway above Rr; tikzpicture center Geg r rO  r F FO  F % GesDensityvarrho sikgpcm % The two center pos of the two spheres are separated by SolQtyR*rXm al R r r R. In order to attract each other with the given force the spheres have to have SolQtym*rX*sqrtFX/ncGnkg al m RsqrtfracFG rsqrtfracFG R sqrtfracFG m. mass. Because the volume of a sphere is SolQtyV/*pi*rX^cubicmeter al V frac pi r^ fracpi qtyr^ V their density has to be: SolQtypmX/VXkgpmk al varrho fracmV fracsqrtfracr^ FGfrac pi r^ fracmV p varrho p Ausrufbox The element with the highest density Osmium no. has a density of .ekgpmk standard temperature and pressure. Ausrufbox
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Exercise:
Two identical massive spheres each with rO radius touch each other. What density would the material of the homogeneous spheres have to have so that they attract each other with FO due to gravity? One inch is .cm one pound-foce is .N.

Solution:
center tikzpicture shadeball color gray! opacity. - circle ; draw - circle ; draw - arc :: and .; drawdashed arc :: and .; shadeball color gray! opacity. circle ; draw circle ; draw arc :: and .; drawdashed arc :: and .; drawthick red - --- nodemidway above Rr; tikzpicture center Geg r rO  r F FO  F % GesDensityvarrho sikgpcm % The two center pos of the two spheres are separated by SolQtyR*rXm al R r r R. In order to attract each other with the given force the spheres have to have SolQtym*rX*sqrtFX/ncGnkg al m RsqrtfracFG rsqrtfracFG R sqrtfracFG m. mass. Because the volume of a sphere is SolQtyV/*pi*rX^cubicmeter al V frac pi r^ fracpi qtyr^ V their density has to be: SolQtypmX/VXkgpmk al varrho fracmV fracsqrtfracr^ FGfrac pi r^ fracmV p varrho p Ausrufbox The element with the highest density Osmium no. has a density of .ekgpmk standard temperature and pressure. Ausrufbox
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Attributes & Decorations
Branches
Gravitation
Tags
density, force, gravitation, mechanis, physics, pound-force, sphere
Content image
Difficulty
(3, default)
Points
5 (default)
Language
ENG (English)
Type
Calculative / Quantity
Creator uz
Decoration
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Link