A particle placed
1.inside a uniform spherical shell of mass M, but not at the center
2.inside a uniform spherical shell of mass M, at the center
3.outside a uniform spherical shell of mass M, a distance r from the center
4.outside a uniform solid sphere of mass M, a distance 2r from the center
If `F_1,F_2,F_3` and `F_4` are gravitational forces acting on the particle in four cases.

Explanation

In this case, we replace dm by `lambda dx`, where `lambda` is not constant. Therefore,`x_(cm)` is `x_(cm)=1/Mint_(0)^(L)lamdadm=1/Mint_(0)^(L)xlamdadx=alpha/Mint_(0)^(L)x^2dx=(lamdaL^3)/(3M)`We can eliminate a by noting that the total mass of the rod is elated to a through the relationship `M=int dm=int_(0)^(L)lamdadx=int_(0)^(L)alphaxdx=(alphaL^3)/2`Substituting this into the expression for `x(CM)` gives `x_(cm)=(alphaL^3)/((3alphaL^2
)/2)=2/3L=4m`.