A uniform metal sphere of radius a and mass M is surrounded by a thin uniform spherical shell of equal mass and radius 4a (figure 11−E2). The centre of the shell falls on the surface of the inner sphere. Find the gravitational field at the points P₁ and P₂ shown in the figure

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Solution

At point P₁, the gravitational field due to the sphere and the shell is given by
F $= \frac{G M}{{(3 a + a)}^{2}} + 0 = \frac{G M}{16 a^{2}}$

At point P₂, the gravitational field due to the sphere and the shell is given by
$F = \frac{GM}{{(a + 4 a + a)}^{2}} + \frac{GM}{{(4 a + a)}^{2}} \Rightarrow F = \frac{GM}{36 a^{2}} + \frac{GM}{25 a^{2}} \Rightarrow F = \frac{GM}{a^{2}} (\frac{1}{36} + \frac{1}{25}) \Rightarrow F = \frac{GM}{a^{2}} (\frac{25 + 36}{900}) \Rightarrow F = (\frac{61}{900}) \frac{GM}{a^{2}}$

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Q. A uniform metal sphere of radius

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and mass

(m)

is surrounded by a thin, uniform spherical shell of equal mass and radius

(4 a)

. The centre of the shell falls on the surface of the inner sphere. Find the gravitational field ratio at the points

P_{1} a n d P_{2}

shown in figure:

A uniform metal sphere of radius a and mass M is surrounded by a thin uniform spherical shell of equal mass and radius 4a (figure 11-E2). The centre of th shell on the surface of the inner sphere. Find th egravitational field at the points $P_{1}$ and $P_{2}$ shown in the figure.