A thin uniform annular disc (see figure) of mass M has outer radius 4R and inner radius 3R. The work required to take a unit mass from point P on its axis to infinity is?

We know that the work required to take a unit mass from P to infinity = −Vp, where −Vp is the gravitational potential at P due to the disc. To find Vp, we divide the disc into small elements, each of thickness dr. Consider one such element at a distance r from the center of the disc as shown. Mass of the element dm = M(2πrdr)/π(4R)² − π(3R)² = 2Mr dr/7R² Thus, Vp = −∫4R3R Gdm/(√r² + 16R²) = −2MG/7R² ∫4R3R rdr/(√r² + 16R²)1/2 Putting r² + 16R² = x², we get 2rdr = 2xdx or rdr = xdx When r = 3R, x = √9R² + 16R² = 5R When r = 4R, x = √16R² + 16R² = 4√2R Vp = −2MG/7R² ∫4√2R5R dx = −2MG/7R² (4√2 − 5)R or −Vp = 2GM/7R(4√2 − 5)