Question

A thin uniform annular disc of mass m has radius 4R and inner radius 3R the work done to move a unit mass from p on its axis to infinity is​

Answer 1

Answer:

The work required to take unit mass from P to infinity = – VP. where VP is gravitational potential at P due to disc Dividing disc into small elements each of thickness dr and dm. This element is at distance r from centre of disc. hence dm = [(M × 2πr ∙ dr) / {π(4R)2 – π(3R)2}] -------- where M is mass of disc. ∴ dm = [(2Mr ∙ dr) / (7R2)] VP = – 4R∫3R[(G ∙ dm) / {√(r2 + 16R2)}] = – 4R∫3R[(G) / {√(r2 + 16R2)}] × [(2Mr dr) / (7R2)] = [(– 2MG) / (7R2)] 4R∫3R[(r ∙ dr) / {√(r2 + 16R2)1/2}] put r2 + 16R2 = x2 hence 2r ∙ dr = 2x ∙ dx i.e. r dr = x ∙ dx when r = 3R, x = √(9R2 + 16R2) = 5R r = 4R, x = √(16R2 + 16R2) = 4√(2)R ∴ VP = [(– 2GM) / ()] (G√2)R∫5R dx VP = [(– 2GM) / (7R2)](4√2 – 5)R ∴ – VP = [(2GM) / 7R](4√2 – 5)