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A cylindrical log of wood of height h an...

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.
.

Text Solution

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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.
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