Question

1. A cylinder of density $\rho_c$, length $l$, and cross-section area $A$ floats in between the interface of two liquids of densities $\rho_1$ and $\rho_2$ ($\rho_2 > \rho_1$) 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 $h = \frac{(\rho_c - \rho_1)}{(\rho_2 - \rho_1)} l$. 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 $y$? 2. By what percent does the frequency of a piano string change if the force that the peg exerts on it decreases by 10%?

          1. A cylinder of density $\rho_c$, length $l$, and cross-section area $A$ floats in between the interface of two liquids of densities $\rho_1$ and $\rho_2$ ($\rho_2 > \rho_1$) 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 $h = \frac{(\rho_c - \rho_1)}{(\rho_2 - \rho_1)} l$.
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 $y$?
2. By what percent does the frequency of a piano string change if the force that the peg exerts on it decreases by 10%?

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1. A cylinder of density , length l, and cross-section area A floats in between the interface of two liquids of densities ρ1 and ρ2 (ρ2 > ρ1) 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 = (( - ρ1))/((ρ2 - ρ1)) 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 s̈pring constant$̈k?
d. What is the period of oscillationTafter the cylinder is released from positiony?
2. By what percent does the frequency of a piano string change if the force that the peg exerts on it decreases by 10%?

Added by Karen S.

University Physics with Modern Physics

Hugh D. Young 14th Edition

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Solved by Expert Averell Hause

06/13/2020

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The cylinder is in equilibrium, so the buoyant force equals the weight of the cylinder. The buoyant force is the sum of the forces due to the displaced liquid 1 and liquid 2. Show more…

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1. A cylinder of density Pe, 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. P1 Pc P2 a. Derive an expression for h to show that h = ( (p;-p1) b. Suppose the cylinder is a small distance y above its equilibrium position. Find an expression for (Fnet ),, 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 Tafter the cylinder is released from position y? 2.By what percent does the frequency of a piano string change if the force that the peg exerts on it decreases by 10%?

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00:01 Part a, we know the gravitational force on the cylinder would be equaling then the density not alg.

00:11 The weight of the displaced fluid, which would essentially be the buoyancy force, would be equalling the density of the fluid, multiplied by ahg.

00:22 And so in static equilibrium, the gravitational force of the cylinder is equalling the buoyancy force for the liquid.

00:28 And so the density of the fluid, ahg, equalling the density of the cylinder, alg, of course, the cross -sectional areas and the acceleration due to gravity cancel out.

00:44 And so h is simply equal to l multiplied by the density of the cylinder divided by the density of the fluid.

00:54 And so for part b, the volume of the displaced, we can say volume of the displaced liquid would be equaling then a multiplied by h minus y.

01:09 Applying newton's second law in the y direction, we could say that the sum of forces in the y direction would be equalling to the negative gravitational force plus the buoyancy force.

01:21 And so this would be, of course, the sum of forces in the y direction, equaling negative row not, alg, plus the density of the fluid.

01:35 A multiplied by h minus y, g.

01:41 And so using that the density of the fluid, ahg equals row not alg, of course, we can find that then the net force in the y direction would be equaling to the negative density of the fluid, a, g, y.

02:01 So, and essentially for part c, the result from part b, f is equaling negative k y where essentially k would be equaling to the density of the fluid times a cross -sectional area times the acceleration due to gravity and of course this is simply hook's law now if hooks law determines the cylinder's motion the angular frequency for would be just in simple harmonic motion and so we can say that then for part d, the angular frequency for an object following simple harmonic motion would be the square root of the spring constant divided by the mass...

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