A wooden cylinder floats in water with length h immersed into it. If it is pushed a little inside water and then released, show that it will perform a simple harmonic motion. Calculate the time period of this motion.

A wooden cylinder floats in water with length h immersed into it. If i

1.	A wooden cylinder floats in water with length h immersed into it. If it is pushed a little inside water and then released, show that it will perform a simple harmonic motion. Calculate the time period of this motion.
Answer» Solution :Let the cross-sectional area of the cylinder be `alpha`. ACCORDING to Archimedes. principle, weight of the cylinder = weight of the displaced water at equilibrium = `halpharhog," "[rho="DENSITY of water"]` `therefore` Mass of the cylinder, `m=halpharho` If the cylinder is pushed through a distance x inside the water, then an extra buoyant force ACTS on the cylinder in the upward direction. It tries to bring the cylinder BACK to its equilibrium POSITION. So the restoring force, F = extra buoyant force = weight of the extra water displaced = `-xalpharhog` `therefore` Acceleration of the cylinder, `a=F/m=(-xalpharhog)/(hrhog)=(-g)/h*x=-omega^(2)x" "["where "omega=sqrt(g/h)]` As the motion of the cylinder obeys the equation `a=-omega^(2)x`, it is simple harmonic. Time period of the motion, `T=(2pi)/omega=2pisqrt(h/g)`.