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Exercise 2.4.1: Justify the identity A cos(wot) + B sin(wot) = C cos(wot - y) and verify the equations for C and y. Hint: Start with cos(a - p) = cos(a) cos(p) + sin(a) sin(ß) and multiply by C. Then what should a and ß be? While it is generally easier to use the first form with A and B to solve for the initial conditions, the second form is much more natural. The constants C and y have nice physical interpretation. Write the solution as

Exercise 2.4.1: Justify the identity A cos(wot) + B sin(wot) = C cos(wot - y) and verify the equations for C and y. Hint: Start with cos(a - p) = cos(a) cos(p) + sin(a) sin(ß) and multiply by C. Then what should a and ß be? While it is generally easier to use the first form with A and B to solve for the initial conditions, the second form is much more natural. The constants C and y have nice physical interpretation. Write the solution as

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Exercise 2.4.1: Justify the identity A cos(wot) + B sin(wot) = C cos(wot - y) and verify the equations for C and y. Hint: Start with cos(a − p) = cos(a) cos(p) + sin(a) sin(p) and multiply by C. Then what should a and ß be? While it is generally easier to use the first form with A and B to solve for the initial conditions, the second form is much more natural. The constants C and y have nice physical interpretation. Write the solution as x(t) = C cos(wot - y). This is a pure-frequency oscillation (a sine wave). The amplitude is C, wo is the (angular) frequency, and y is the so-called phase shift. The phase shift just shifts the graph left or right. We call wo the natural (angular) frequency. This entire setup is called simple harmonic motion. Let us pause to explain the word angular before the word frequency. The units of wo are radians per unit time, not cycles per unit time as is the usual measure of frequency. Because one cycle is 27 radians, the usual frequency is given by . It is simply a matter of where we put the constant 27, and that is a matter of taste. The period of the motion is one over the frequency (in cycles per unit time) and hence it is 2. That is the amount of time it takes to complete one full cycle. wo
Transcribed Image Text:Exercise 2.4.1: Justify the identity A cos(wot) + B sin(wot) = C cos(wot - y) and verify the equations for C and y. Hint: Start with cos(a − p) = cos(a) cos(p) + sin(a) sin(p) and multiply by C. Then what should a and ß be? While it is generally easier to use the first form with A and B to solve for the initial conditions, the second form is much more natural. The constants C and y have nice physical interpretation. Write the solution as x(t) = C cos(wot - y). This is a pure-frequency oscillation (a sine wave). The amplitude is C, wo is the (angular) frequency, and y is the so-called phase shift. The phase shift just shifts the graph left or right. We call wo the natural (angular) frequency. This entire setup is called simple harmonic motion. Let us pause to explain the word angular before the word frequency. The units of wo are radians per unit time, not cycles per unit time as is the usual measure of frequency. Because one cycle is 27 radians, the usual frequency is given by . It is simply a matter of where we put the constant 27, and that is a matter of taste. The period of the motion is one over the frequency (in cycles per unit time) and hence it is 2. That is the amount of time it takes to complete one full cycle. wo
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