A cylindrical piece of cork of density rho, base area A and height h, floats in a liquid of density rho_(1). The cork is depressed slightly and then released. Show that the cork oscillates up and down simple harmonically, with a period T=2pisqrt((hrho)/(rho_(1)g)), where 'rho' is the density of the cork.

A cylindrical piece of cork of density rho, base area A and height h,

1.	A cylindrical piece of cork of density rho, base area A and height h, floats in a liquid of density rho_(1). The cork is depressed slightly and then released. Show that the cork oscillates up and down simple harmonically, with a period T=2pisqrt((hrho)/(rho_(1)g)), where 'rho' is the density of the cork.
Answer» Solution : RESTORING force `=-` WEIGHT of liquid DISPLACED `=-(rho_(1)Ay)g` Applied force `F=ma` `"i.e."F=(rhoAh)a` `"Hence,"rhoAha=-rhoAyg` `"or"(y)/(a)=\|(-rhoh)/(rho_(1)g)\|` We know that period of oscillation, `T=2pisqrt(("displacement")/("accelertion"))""therefore T=2pisqrt((RHO)/(rho_(1)g))` If `.rho.` of the FLOATING body is equal to the density of liquid then, `T=2pisqrt((h)/(g)).`