Question
3 The extension, y,of a material with an applied force, F, is given by y=e^Ftimes 1times 10^(-3) a) Calculate the work done if the force increases from 100N to 500N using: i) An analytical integration technique ii) A numerical integration technique [Note: the work done is given by the area under the curve] b) Compare the two answers c) Using a computer spreadsheet increase the number of values used for your numerical m d) Analyse any affect the size of numerical step has on the result. 4 For the function v=12sin4Theta , calculate the: a) Mean b) Root mean square (RMS) Over a range of 0leqslant Theta leqslant (pi )/(4) radians. [Note the trigonometric identity cos2Theta =1-2sin^2Theta ]
Answer
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Mabel
Professional · Tutor for 6 years
Answer
3.a)i)
3.a)i) applies when we integrate from e^1=2.71 (100 N) to e^5=148.41 (500 N). You could state the numerical equal to 5th or within less precision dictated by another given constraint.3.a)ii) depending the method chosen to approximate the numerical integral, analytical versus numerical arouse precision issues, you in the end equal those two methods reaching equal numerical outputs yet provided sufficient precision of descriptor 3.b) Analytic and numerical method will yield similar results unless developers struggle around the problem of precision particularly found on modern day hardware built on 32/64/128 binary basis3.c&d) By ramping up precision or spreading into smaller slices your initial problem space, aim for even out the repercussions form problems occurring from inherent SGAs4.a) Mean is
4.b) RMS =
Explanation
This problem falls under the intersection Physics and Mathematics. There are two separate tasks given here: 1. Dealing with the extension of a material (Hooke's Law) where we define work as the area under a force-extension graph resulting in the necessity for integration, 2. Providing statistics (mean, RMS) for a sinusoidal function. 3a)i) To find the total work done, we integrate the force over the extension which here would mean we integrate F=e^(1000*y). This case uses analytical integration for work done and results in: Integral from e^1 to e^5 of F*dy.3a)ii) For computation of numerical integral, someone would typically use a method such as a Riemann integral, where one uses several small rectangular areas to approximate the integral. For simplicity and given the equation F=e^(1000*y), applying such method would conclude that numerical method would deliver same results.3b) The comparison would probably yield very close values given that you have used sufficiently small steps. But do keep in mind that a numerical method always produces approximation referring to the underlying precision of numerical representation in binary systems (robots, AB etc).3c)&3d) Using spreadsheets for numerical calculations makes sense by applying fractions (continuously refined slices used for finite approximations - slices become more and smaller); influencing variables such as step size would lead to better approximations.4a)&4b) Mean and RMS values can be calculated for the given sin waveform function, always keep in mind and follow rules for computations: since sin(x)^2 results in a value between 0 and 1, use the given identities which enable you to do so.