a solid sphere of mass m and radius r is placed inside a hollow thin spherical shell of mass M and radius R as shown in figure. A particle of mass m is placed on the line joining the two centers as a distance x from the point of contact of the sphere and the shell. Find the magnitude of the resultant gravitational force on this particle due to the sphere and the shell if
a). `rltxlt2r`,
b). `2rltxlt2R`
and c). `xgt2R`.

Answer

A, B, C

Explanation

a. `m_1` is placed at a distance x from O
If rltxlt2r`, Let's consider a thin shell of mass. ltbr. `dm=m/((4/3)pir^3)xx4/3pix^3=(mx^3)/r^3`
Thus `int dm=(mx^3)/r^3`
Then gravitatioN/Al force
`F=(Gmdm)/x^2`
`=(8(mx^3)/r^3)/x^2=(Gmx)/r^3`
`b `2rltxlt2R`, then F is due to only the sphere,
`:. F=(Gmm^1)/((x-r)^2)`
c. If xlt2R then gravitatioN/Al force is due to both sphere and shelll then
due to shell `F=(GMm^2)/((x-R)^2)`
Due to the sphere `=(gmm^1)/((x-r)^2)`
So, resultant force `e=(gmm^1)/((x-r)^2) +(GMm^1)/((x-R)^2)`