A cylindrical log of wood of height h and area of cross-section A floats in water. It is pressed and then released. Show that the log would execute SHM with a time period. T= 2pi sqrt((m)/(A p g)) where, m is mass of the body and p is density of the liquid. .

A cylindrical log of wood of height h and area of cross-section A floa

1.	A cylindrical log of wood of height h and area of cross-section A floats in water. It is pressed and then released. Show that the log would execute SHM with a time period. T= 2pi sqrt((m)/(A p g)) where, m is mass of the body and p is density of the liquid. .
Answer» Solution :Suppose, LOG is pressed DOWNWARD to y, the volume of liquid displaced by block will be yA. The mass of displaced water `M= Vp` `=yA p""[therefore y xx A= " volume V "]` Buoyant force the upward by displaced water = weight of displaced water `therefore F= -Mg""`(Weight and Buoyant force are opposite to each other) `= -y A pg` `therefore = -(A pg )y` `therefore F propto -y` where `Ap g =k` constant HENCE, force acting on log is directly proportional to the displacement and opposite it. So motion of log is SHM Now, period of SHM particle `T= 2PI sqrt((m)/(k))` but `k= A p g` `therefore T= 2pi sqrt((m)/(A p g))` is PROVED.