Mathematics For Machine Technology
8th Edition
ISBN: 9781337798310
Author: Peterson, John.
Publisher: Cengage Learning,
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Chapter 54, Problem 2A
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6. [10] Evaluate
(2
(2+2y) ds where C is the upper half-circle centered at the origin connecting
the point (2,0) to the point (-2,0).
7. [10] Show that the vector field
F(x, y, z) =
is conservative and integrate it along the curve
П
C(t) =
si
sin t,
nt, cost,t),
te
[0,1]
8. [10] Use Stokes' Theorem to compute the integral
curl F.dS, where F(x, y, z) = xzi+yxj+xy k
I cur
and S is the part of the sphere x² + y²+z² = 9 that lies inside the cylinder x2 + y²
above the xy-plane.
9. [10] Use Green's theorem to evaluate
So
√1+x3 dx+2xy dy
where C is the triangle with vertices (0,0), (1, 0) and (1, 3).
10. [10] Evaluate the surface integral
(x²z + y²z) dS
where S is the hemisphere x² + y²+2² = 4, z> 0.
= 1 and
☐
Part 1:
A linear electrical load draws 1₁ A at a 0.72 lagging power factor. See the table to find ½ for your
student ID. When a capacitor is connected, the line current dropped to 122 A and the power factor
improved to 0.98 lagging. Supply frequency is 50 Hz.
a. Let the current drawn from the source before and after introduction of the capacitor be 1₁ and I₂
respectively. Take the source voltage as the reference and express 11 and 12 as vector
quantities in polar form.
b. Obtain the capacitor current, Ic = 12 − I₁, graphically as well as using complex number
manipulation. Compare the results.
c. Express the waveforms of the source current before (į (t)) and after (i2(t)) introduction of the
capacitor in the form Im sin(2лft + 0). Hand sketch them on the same graph. Clearly label your
plots.
d. Analytically solve i̟2(t) − i₁ (t) using the theories of trigonometry to obtain the capacitor current
in the form, ic(t) = Icm sin(2πft + 0c). Compare the result with the result in Part b.
Go to page 82 for the geometry problem. Use the formula for the area of a triangle to compute
the area given the base and height.
Link:
[https://drive.google.com/file/d/1RQ2OZK-LSxp RyejKEMg 1t2q15dbpVLCS/view?
usp=sharing]
Provide a step-by-step solution.
Chapter 54 Solutions
Mathematics For Machine Technology
Ch. 54 - Determine the size of 1.Ch. 54 - If ABCD,BCDE , and 1=2725 , what are the sizes of...Ch. 54 - What is the supplement of a 1051344 angle?Ch. 54 - Determine the center diameter of a pinion gear...Ch. 54 - Solve 578=234x.Ch. 54 - Determine the value of 406.442-1/3. Round the...Ch. 54 - Determine which of the following pairs of...Ch. 54 - Solve the following exercises: In ABC and...Ch. 54 - Solve the following exercises: In figure,...Ch. 54 - Solve the following exercises: In ABC and...
Ch. 54 - Solve the following exercises: Use the figure to...Ch. 54 - Solve the following exercises: In HPM,PMJK. a....Ch. 54 - Solve the following exercises: Refer to the figure...Ch. 54 - Solve the following exercises: Refer to the figure...Ch. 54 - Solve the following exercises: In this figure,...Ch. 54 - Solve the following exercises: a. Find x. b. Find...Ch. 54 - Solve the following exercises: a. Find x. b. Find...Ch. 54 - All dimensions are in inches. a. Find 1. b. Find ...Ch. 54 - All dimensions are in millimeters. a. Find x. b....Ch. 54 - All dimensions are in inches. a. Find 1. b. Find...Ch. 54 - Refer to this figure. Using the given values, find...Ch. 54 - Using the figure and these given values, find the...Ch. 54 - Using the figure and these given values, find the...Ch. 54 - Three holes are drilled in the plate shown. All...Ch. 54 - All dimensions are in inches. Round the answers to...Ch. 54 - All dimensions are in inches. Round the answers to...Ch. 54 - Solve the following exercises: A template is...Ch. 54 - Refer to polygon ABCD. a. If 2 = 87.0, find 1. b....Ch. 54 - Use the angle values given. a. If 1 = 114, find 2....Ch. 54 - Use the angle values given to find 2. a. If 1 =...
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- Refer to page 79 of the shared document for the algebra problem. Use basic algebraic rules to simplify the given expression. Link: [https://drive.google.com/file/d/1RQ2OZK-LSxp RyejKEMg1t2q15dbpVLCS/view? usp=sharing] Provide all steps clearly.arrow_forwardPlease calculate the shaded areaarrow_forward3. Solve the Heat Equation with Initial and Boundary Conditions Turn to page 71 for the heat equation problem. Solve the partial differential equation using Fourier series or another suitable method, given the initial and boundary conditions. Link: [https://drive.google.com/file/d/1RQ2OZK-LSxpRyejKEMg1t2q15dbpVLCS/view? usp=sharing] Provide all derivations and intermediate steps.arrow_forward
- 2. Classify the Stability of Fixed Points in a Dynamical System The dynamical system problem is located on page 60 of the file. Identify the fixed points and classify their stability using linearization and eigenvalues. Link: [https://drive.google.com/file/d/1RQ2OZk-LSxpRyejKEMg1t2q15dbpVLCS/view? usp=sharing] Provide a detailed explanation of your analysis.arrow_forwardEvaluate the Z-Transform of the Sequence The Z-transform problem is provided on page 70. Compute the Z-transform of the given sequence and determine the region of convergence. Link: [https://drive.google.com/file/d/1RQ2OZK-LSxpRyejKEMg1t2q15dbpVLCS/view? usp=sharing] Show all steps and provide detailed reasoning.arrow_forward7. Apply Green's Theorem to Evaluate the Line Integral Check page 55 for the Green's theorem problem. Use Green's theorem to convert a line integral into a double integral and compute the result. Link: [https://drive.google.com/file/d/1RQ2OZK-LSxp RyejKEMg1t2q15dbpVLCS/view? usp=sharing] Provide a detailed explanation and calculation.arrow_forward
- 8. Perform Singular Value Decomposition (SVD) on a Matrix The SVD problem can be found on page 66. Decompose the given matrix into its singular values, left singular vectors, and right singular vectors. Link: [https://drive.google.com/file/d/1RQ2OZK-LSxpRyejKEMg1t2ql5dbpVLCS/view? usp=sharing] Show the complete process and verify the decomposition.arrow_forward6. Diagonalize the Matrix and Verify the Result Check page 64 of the document for the matrix diagonalization problem. Find the eigenvalues and eigenvectors to diagonalize the matrix, then verify the result. Link: [https://drive.google.com/file/d/1RQ2OZK-LSxpRyejKEMg1t2q15dbpVLCS/view? usp=sharing] Present all calculations systematically.arrow_forward9. Analyze the Fourier Transform of a Discontinuous Function Refer to page 67 for the Fourier transform problem. Compute the Fourier transform of the given piecewise function and discuss its properties. Link: [https://drive.google.com/file/d/1RQ2OZK-LSxpRyejKEMg1t2q15dbpVLCS/view? usp=sharing] Present all steps in detail.arrow_forward
- 4. Determine the Taylor Series Expansion About a Point Refer to page 62 for the Taylor series problem. Find the Taylor series expansion of the function about the specified point and include the radius of convergence. Link: [https://drive.google.com/file/d/1RQ2OZK-LSxpRyejKEMg1t2q15dbpVLCS/view? usp=sharing] Explain the methodology and verify the series.arrow_forward10. Prove the Orthogonality of Eigenfunctions Go to page 68 of the document for the eigenfunction problem. Prove the orthogonality of the eigenfunctions of a given Sturm-Liouville problem. Link: [https://drive.google.com/file/d/1RQ2OZK-LSxpRyejKEMg1t2q15dbpVLCS/view? usp=sharing] Provide a rigorous proof, including necessary integrals and conditions.arrow_forward1. Perform a Change of Variables for the Given Integral Refer to page 59 in the shared document for the integral problem. Apply a specified change of variables to simplify the integral and evaluate it. Link: [https://drive.google.com/file/d/1RQ2OZk-LSxpRyejKEMg1t2q15dbpVLCS/view? usp=sharing] Clearly outline each step in the transformation and solution.arrow_forward
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