Two stars, each of mass M and separated by a distance d, orbit about their center of mass. A planetoid of mass m `(m lt ltM)` moves along the axis of this system perpendicular to the orbital plane.

Let `T_(p)` be the period of simple harmonic motion for the planetoid for small displacements from the center of mass along the z-axis, and let `T_(s)` be the period of motion for the two stars. Find the value `((T_(S))/(T_(P)))^(2)`

Answer

8

Explanation

`R=(d)/(2)`
`(GM^(2))/(d^(2))=M((2pi)/(T_(S)))^(2)((d)/(2))`
`T_(S)=2pisqrt((d^(3))/(2GM))`
`F=(2GMmZ)/((R^(2)+Z^(2))^(3//2))e=(2GMmZ)/(R^(3))=((16GMm)/(d^(3)))Z`
`(d)/(2)=Rgt gt gt Z`
`T_(P)=2pisqrt((m)/(k))=2pisqrt((d^(3))/(16GM))=(pi)/(2)sqrt((d^(3))/(GM))," "(T_(S))/(T_(P))=2sqrt(2)`
`((T_(S))/(T_(P)))^(2)=8`