A spherical cavity of radius R/2 is removed from a uniform solid sphere of mass M and radius R, as shown. The cavity is internally tangent to the original surface. Taking V = 0 at infinity, find the gravitational potential at the centre of the cavity.

Question

G1 Physics · Accepted Answer

Step 1: Treat the remaining body as the original full sphere plus a 'negative-mass' sphere representing the removed cavity. The cavity centre is at r=R/2 from the original centre. Step 2: Potential inside the full sphere at r=R/2 is potential of the full sphere=-GM(3R²-r²)/(2R³)=-11GM/(8R). The removed sphere has radius R/2 and mass M/8; its potential at its own centre is -3G(M/8)/[2(R/2)]=-3GM/(8R). Step 3: Rₑmoving it means subtracting this negative potential: V=-11GM/(8R)+3GM/(8R)=-GM/R. Fin…