Derive tidal force upon star (approximation: divide star in 2)

[tumconn]

Homework Statement

Spherical,homogeneous star with radius R orbiting black hole at distance $r_p >>R$ .Derive the tidal force acting upon the star by dividing the star into two equal parts and making the necessary approximations.

Homework Equations

The tidal force equation of $a \propto \frac{ R }{ {r_p}^3}$ is not a given.

The Attempt at a Solution

I calculated the force acting on the nearest and farthest hemisphere from the black hole.
Nearest: $dF_1= \frac{G M_o dm}{{(r_p-R+r)}^2}$ where $M_o$ is the mass of the black hole and $dm= \rho \pi r^2 dr$ the mass of a disk with thickness dr . By integrating from 0 to R I got $F_1= \frac{3GM_o M(2r_p-R)}{4r_p R^2}$.
For the farthest hemisphere $dF_2=\frac{G M_o dm}{{(r_p+R-r)}^2} \Longrightarrow F_2= \frac{3GM_o M(2r_p+R)}{4r_p R^2}$
By substracting these,I get the tidal force $\Delta F= F_1-F_2=\frac{3}{2} \frac{GM_oM}{R r_p}$ which is definitely not in agreement with the actual physics of the problem,since it should be proportional to R.
Can someone point me to where I went wrong?

 

[haruspex]

dm=ρπr²dr the mass of a disk with thickness dr .

You seem to have defined r here as the distance from the point of the star nearest the black hole to the disk. The disk's radius will not be r.