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Standard XII

Physics

Displacement of COM:Application

From asolid s...

Question

From asolid sphere of mass M & radius r,as sphere of diameter r is cut off as shown. What is the intensity at a point x distance away from the centre.

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Solution

REF.Image
Let $μ^{'}$ be the mass of remaining part.
Then,
$E = \frac{G M^{'}}{x^{2}}$
As the mass of sphere was
uniformly distributed :
$f = \frac{M}{\frac{4 π}{3} R^{3}} = \frac{M^{'}}{\frac{4 π}{3} [R^{3} - r^{3}]} \Rightarrow M^{'} = M [\frac{R^{3} - r^{3}}{R^{3}}]$
So, $E = \frac{G M}{x^{3}} [\frac{R^{3} - r^{3}}{R^{3}}]$

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From asolid sphere of mass M & radius r,as sphere of diameter r is cut off as shown. What is the intensity at a point x distance away from the centre.

REF.ImageLet μ′ be the mass of remaining part.Then, E=GM′x2As the mass of sphere wasuniformly distributed :f=M4π3R3=M′4π3[R3−r3]⇒M′=M[R3−r3R3]So, E=GMx3[R3−r3R3]