/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 19 The following equation can be us... [FREE SOLUTION] | 91影视

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Chapter 28: Problem 19

The following equation can be used to model the deflection of a sailboat mast subject to a wind force: \\[ \frac{d^{2} y}{d z^{2}}=\frac{f}{2 E I}(L-z)^{2} \\] where \(f=\) wind force, \(E=\) modulus of elasticity, \(L=\) mast length and \(I=\) moment of inertia. Calculate the deflection if \(y=0\) and \(d y / d z=0\) at \(z=0 .\) Use parameter values of \(f=60, L=30, E=\) \(1.25 \times 10^{8},\) and \(I=0.05\) for your computation.

Short Answer

Expert verified
The equation for the deflection of the sailboat mast, given the parameters and initial conditions, is \(y(z) = \frac{1}{6\times 10^{5}}(-(30-z)^{3}+900z+27000)\).

Step by step solution

01

Integrate the equation once

First, we need to integrate the given equation with respect to \(z\): \( \int \frac{d^{2} y}{d z^{2}} dz = \int \frac{f}{2 E I}(L-z)^{2} dz \) This will result in: \(\frac{d y}{d z} = \frac{f}{2 E I} \int(L - z)^{2} dz + C_{1}\) where \(C_{1}\) is the constant of integration.
02

Integrate the equation a second time

Now, integrate once more with respect to \(z\): \( \int \frac{d y}{d z} dz = \frac{f}{2 E I} \int \left[ (L-z)^{2} + C_{1}\right] dz \) This gives: \( y(z) = \frac{f}{2 E I}\left( - \frac{1}{3}(L-z)^{3} \right) + C_{1}z + C_{2} \) where \(C_{2}\) is another constant of integration.
03

Apply the initial conditions

We are given that \(y = 0\) and \(dy/dz = 0\) at \(z = 0\). We can use these initial conditions to solve for \(C_{1}\) and \(C_{2}\): Case 1 - \(y(0) = 0\): \( 0 = \frac{f}{2 E I}(-\frac{1}{3}L^{3}) + C_{1}(0) + C_{2} \) So, we have: \(C_{2} = \frac{fL^3}{6EI}\) Case 2 - \(\frac{dy}{dz}(0) = 0\): \(0 = \frac{f}{2 E I}(-2L^{2}) + C_{1}\) So, we have: \(C_{1} = \frac{fL^2}{2EI}\)
04

Write the final equation for deflection

Now that we have constants \(C_{1}\) and \(C_{2}\), we can write the final equation for the deflection as: \(y(z) = \frac{f}{2 E I} \left(-\frac{1}{3}(L-z)^{3}\right) + \frac{fL^2}{2EI}z + \frac{fL^3}{6EI}\)
05

Substitute the given parameter values

Finally, substitute the given parameter values into the equation: \(y(z) = \frac{60}{2(1.25 \times 10^{8})(0.05)} \left(-\frac{1}{3}(30-z)^{3}\right) + \frac{60(30)^{2}}{2(1.25 \times 10^{8})(0.05)}z + \frac{60(30)^{3}}{6(1.25 \times 10^{8})(0.05)}\) Simplifying the equation: \(y(z) = \frac{1}{6\times 10^{5}}(-(30-z)^{3}+900z+27000)\) This is the equation that describes the deflection of the sailboat mast for the given parameters and initial conditions.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Differential Equations
Differential equations are mathematical tools we use to describe changes and dynamics of various systems. In our mast deflection problem, the differential equation reflects how the deflection of the mast, denoted by the function y(z), changes as you move along the length of the mast, z.

The equation \(\frac{d^{2} y}{d z^{2}}=\frac{f}{2 E I}(L-z)^{2}\) is a second-order differential equation because it involves the second derivative of y with respect to z. This second derivative expresses the curvature of the mast's shape, and by integrating it, as shown in the step by step solution, we find a function that describes the mast's deflection at any point along its length.

Understanding and solving differential equations is crucial as they appear in many fields of science and engineering. They model everything from the physics of waves and heat to population dynamics in biology and the behavior of financial markets.
Modulus of Elasticity
The modulus of elasticity, symbolized by E in our equation, is a measure of a material's stiffness. It's how we quantify the inherent 'springiness' or rigidity of the material 鈥� high values of E indicate a stiff material, while low values suggest a more flexible one.

In the context of our mast deflection problem, the modulus of elasticity is a key part of the formula as it directly affects the mast's ability to resist bending under the applied wind force, f. Higher modulus of elasticity would mean the mast deflects less under the same force.

\(E = \frac{\text{Stress}}{\text{Strain}} \) is the typical formula for modulus of elasticity. Stress is the force causing deformation, distributed over an area, while strain is the measure of deformation itself. This concept is a foundational element of materials science and structural engineering, essential for designing safe and efficient structures.
Moment of Inertia
The moment of inertia, represented by I in our deflection equation, is a measure of an object's resistance to changes in its rotation. In the context of beams and masts, it provides insight into how the geometry of the cross-section resists bending.

The moment of inertia is defined by the distribution of mass and its distance from a pivot point or axis. For a beam or mast, it's calculated with respect to an axis along its length and depends on the geometric shape of the cross-section and the material distribution. The greater the moment of inertia, the less it will deflect under a given load.

In our mast deflection problem, a higher value of I would lessen the curvature induced by the wind force, reflecting a sturdier structure. In practical design, by manipulating the cross-sectional geometry, we can optimize structures for strength and materials usage.

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Most popular questions from this chapter

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