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)`