Q: A uniform cylinder of length L and mass M having cross sectional area A is suspended with its vertical length, from fixed point by massless spring, such that it is half submerged in a liquid of density d at equilibrium position. When the cylinder is given a small downward push and released, it starts ossilating vertcally with a small amplitude. If the force constant of the spring is K, the frequency of ossilation of the cylinder is - ?
Sol^n : let x be the extension in the string...
So restoring force in string = -kx if there was no liquid...(- sign for opp. direction)
Now the the buoyancy force will push it upwards...
.
mass of liquid initially displaced = (L/2).Ad
So. initial buoyant force = -(L/2).Adg ...(- sign for opp. direction)
.
mass of liquid displaced finally = (L/2).Ad + x.Ad
weight of liquid displaced finally = (L/2 + x)Adg
So final buoyant force = -(L/2 + x)Adg ...(- sign for opp. direction)
.
.
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The body is already under the effect of a buoyant force under initial condition of equilibrium. On further displacing by x, the buoyant force will try to bring it to the prev. position of half emerged.
.
.
So net force acting due to liquid....
= Final buoyant force - initial buoyant force
= -(L/2 + x)Adg + (L/2).Adg
= -x.Adg
= -k'.x ... ( - sign for opp. direction)
where k' = Adg = force constant due to liquid...
.
Now net force pulling the cylinder upward is = - (kx + k'x)
= -(k+k')x
So time pd. = 1/2(pi) (eq. force constant / mass )^1/2
= 1/2(pi) [[K + Adg)/M]^1/2
Moderation Team
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