Must get answer provided below 12. A block with a mass m = 4.0kg is suspended from a nonlinear spring withf₁ = F, sin() where F = 50N and x = 15mm .Write a single MatLab program which:a) Plots the spring force f, for (-15mm

Answered: 12. A block with a mass m = 4.0kg is…

Elements Of Electromagnetics

7th Edition

ISBN:9780190698614

Author:Sadiku, Matthew N. O.

Publisher:Sadiku, Matthew N. O.

ChapterMA: Math Assessment

Section: Chapter Questions

Problem 1.1MA

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12. A block with a mass m = 4.0kg is suspended from a nonlinear spring with
f₁ = F, sin() where F = 50N and x = 15mm .
Write a single MatLab program which:
a) Plots the spring force f, for (-15mm<x<15mm ) on the same figure.
b) Determine the equilibrium position, x, of the mass.
c) What is the unstretched spring constant (at x=0) ?
d) What is the equivalent spring constant for the stretched spring of b).
e) Plot the equivalent spring constant for any equilibrium position.
g
Answer:
b) x = 13.54mm
c) k, = 3.33 N/mm@x=0
d) k, = 2.07 N/mm@x=13.54mm
Unstretched position
X

Expert Solution

Step 1: Formula

Let's first derive the relation for the spring constant from the given data.

Work done by the spring is given by,

$dW equals straight f subscript straight s dx straight f subscript straight s space is space given space in space the space problem comma dW equals open square brackets straight F subscript 0 cross times sin space open parentheses straight x over straight x subscript straight m close parentheses close square brackets dx Integrating space this space above space equation comma straight W equals integral subscript straight x subscript 1 end subscript superscript straight x subscript 2 end superscript integral open square brackets straight F subscript 0 cross times sin space open parentheses straight x over straight x subscript straight m close parentheses close square brackets dx straight W equals straight F subscript 0 cross times straight x subscript straight m cross times open square brackets negative cos space open parentheses straight x over straight x subscript straight m close parentheses close square brackets subscript straight x subscript 1 end subscript superscript straight x subscript 2 end superscript straight W equals straight F subscript 0 cross times straight x subscript straight m cross times open square brackets negative cos space open parentheses straight x subscript 2 over straight x subscript straight m close parentheses plus cos space open parentheses straight x subscript 1 over straight x subscript straight m close parentheses close square brackets Work space done space can space also space defined space using space the space spring space constant space as space below comma straight W equals 1 half cross times straight k cross times open parentheses straight x subscript 2 squared minus straight x subscript 1 squared close parentheses Equating space these space two space equations space of space work space done comma 1 half cross times straight k cross times open parentheses straight x subscript 2 squared minus straight x subscript 1 squared close parentheses equals straight F subscript 0 cross times straight x subscript straight m cross times open square brackets negative cos space open parentheses straight x subscript 2 over straight x subscript straight m close parentheses plus cos space open parentheses straight x subscript 1 over straight x subscript straight m close parentheses close square brackets bold k bold equals fraction numerator bold 2 bold cross times bold F subscript bold 0 bold cross times bold x subscript bold m over denominator stretchy left parenthesis x subscript 2 squared minus x subscript 1 squared stretchy right parenthesis end fraction bold cross times stretchy left square bracket negative cos space open parentheses x subscript 2 over x subscript m close parentheses plus cos space open parentheses x subscript 1 over x subscript m close parentheses stretchy right square bracket bold This bold space bold is bold space bold the bold space bold relation bold space bold of bold space bold spring bold space bold constant bold space bold with bold space bold the bold space bold length bold.$

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