1. A cylinder of density Pc, length l, and cross-section area A floats in between the interface of two liquids of densities P1 and P2 (P2P1) with its axis perpendicular to the surface. Length of the cylinder is submerged in liquid 2 (bottom layer) when the cylinder floats at rest.a. Derive an expression for to show that =(PcP1/P2

Question

1. A cylinder of density Pc, length l, and cross-section area A floats in between the interface of two liquids of densities P1 and P2 (P2>P1) with its axis perpendicular to the surface. Length ℎ of the cylinder is submerged in liquid 2 (bottom layer) when the cylinder floats at rest.

a. Derive an expression for ℎ to show that ℎ=(Pc−P1/P2−P1)L.

b. Suppose the cylinder is a small distance Y above its equilibrium position. Find an expression for (Fnet)Y, the Y-component of the net force. Use your expression from part a to cancel some of the terms.

c. You should recognize your result from part b as a version of Hooke's law. What is the "spring constant" k?

d. What is the period of oscillation T after the cylinder is released from position y?

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* 1. A cylinder of density

ρ_{c}

, length

l

, and cross-section area

A

floats in between the interface of two liquids of densities

ρ_{1}

and

ρ_{2} (ρ_{2} > ρ_{1})

with its axis perpendicular to the surface. Length

h

of the cylinder is submerged in liquid 2 (bottom layer) when the cylinder floats at rest. a. Derive an expression for

h

to show that

h = (\frac{ρ _{c} - ρ _{1}}{ρ _{2} - ρ _{1}}) l

. b. Suppose the cylinder is a small distance

y

above its equilibrium position. Find an expression for

(F_{net})_{y}

, the

y

-component of the net force. Use your expression from part a to cancel some of the terms. c. You should recognize your result from part b as a version of Hooke's law. What is the "spring constant"

k

? d. What is the period of oscillation

T

after the cylinder is released from position

y

?

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