A long cylindrical tube of length L, and density ?t , is very slightly weighted

Question

A long cylindrical tube of length L, and density ?t, is very slightly weighted

on one end so that it floats upright in a fluid having density ?f. Assume the

tube is constant density and the slight weighting on one end is very small

(negligible), with the only purpose to stablize the cylinder vertically in the

fluid. It is pushed down a distance x from its equilibrium position and

released. (a) Show that it will execute simple harmonic motion if the

resistive effects of the fluid are neglected, and (b) determine the period of

the oscillation in terms of L, ?t, and ?f

A long cylindrical tube of length L, and density ?t, is very slightly weighted on one end so that it floats upright in a fluid having density ?f. Assume the tube is constant density and the slight weighting on one end is very small (negligible), with the only purpose to stablize the cylinder vertically in the fluid. It is pushed down a distance x from its equilibrium position and eleased. (a) Show that it will execute simple harmonic motion if the resistive effects of the fluid are neglected, and (b) determine the period of the oscillation in terms of L, ?t, and ?f

Explanation / Answer

Suppose the block is pushed x into the water the ehtra force is equal to the weight of the water displaced

F = g*df* pi*r^2 * x

a) Hene force on bock is directly proportional to its displacement. it is a simple harmonic motion

b) k = pi *g* df r^2

T = 2 pi sqrt( m/k) = 2 pi sqrt( pi*r^2*L*dt / pi*g*df*r^2 ) = 2 pi*sqrt ( 2Ldt/g df)

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