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Chapter 1: Problem 83

Show that any linear combination of \(\sin \omega t\) and \(\cos \omega t\) such that \(x(t)=A_{1} \cos \omega t+A_{2}\) \(\sin \omega t\left(A_{1}, A_{2}=\right.\) constants ) represents a simple harmonic motion.

Short Answer

Expert verified
To show that any linear combination of \(\sin \omega t\) and \(\cos \omega t\) represents a simple harmonic motion (SHM), we need to prove that \(x(t)\) satisfies the SHM equation. We take the given function \(x(t) = A_1 \cos \omega t + A_2 \sin \omega t\), find the second derivative \(\frac{d^2x}{dt^2}\), and check if it's equal to \(-\omega^2 x\). After differentiating twice, we get \(\frac{d^2x}{dt^2} = -\omega^2 (A_1 \cos \omega t + A_2 \sin \omega t)\), which is equivalent to \(-\omega^2 x\). Thus, any linear combination of \(\sin \omega t\) and \(\cos \omega t\) represents a simple harmonic motion.

Step by step solution

01

Compute the first derivative of x(t) with respect to t

Differentiate \(x(t)\) with respect to \(t\): \[\frac{dx}{dt} = \frac{d}{dt}( A_1 \cos \omega t + A_2 \sin \omega t)\] Using the chain rule, we get: \[\frac{dx}{dt} = -\omega A_1 \sin \omega t + \omega A_2 \cos \omega t\]
02

Compute the second derivative of x(t) with respect to t

Differentiate the first derivative \(\frac{dx}{dt}\) with respect to \(t\): \[\frac{d^2x}{dt^2} = \frac{d}{dt}(-\omega A_1 \sin \omega t + \omega A_2 \cos \omega t)\] Using the chain rule again, we get: \[\frac{d^2x}{dt^2} = -\omega^2 (A_1 \cos \omega t + A_2 \sin \omega t)\]
03

Check if the second derivative equals -蠅虏x

Compare the second derivative equation with \(-\omega^2 x\): \[-\omega^2 (A_1 \cos \omega t + A_2 \sin \omega t) = -\omega^2 (A_1 \cos \omega t + A_2 \sin \omega t)\] Both equations are equivalent, which means that \(x(t)\) indeed satisfies the equation for SHM. Thus, any linear combination of \(\sin \omega t\) and \(\cos \omega t\) represents a simple harmonic motion.

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Key Concepts

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

Linear Combination of Functions
A linear combination of functions involves adding or subtracting functions that are each multiplied by a constant. In the context of simple harmonic motion (SHM), a linear combination of \(\sin \omega t\) and \(\cos \omega t\) means we express a function \(x(t)\) as:
  • \(x(t) = A_1 \cos \omega t + A_2 \sin \omega t\)
  • Where \(A_1\) and \(A_2\) are constants
This expression combines both sine and cosine functions, utilizing their periodic nature to perfectly describe oscillatory motion, such as that observed in SHM. Changing \(A_1\) and \(A_2\) alters the amplitude and phase of the motion. This versatility makes linear combinations powerful tools in modeling phenomena that include circular or oscillatory components.
The beauty of a linear combination is that it allows one to tailor the mathematical function to fit a variety of waveforms merely by changing the constants. This form serves as a universal template for describing harmonic motion.
Trigonometric Functions
Trigonometric functions, like sine and cosine, are fundamental to describing oscillatory motion. They naturally appear in the mathematics of waves and vibrations.
Here鈥檚 why they are so handy:
  • Periodic Nature: Both \(\sin \) and \(\cos \) functions repeat their values in a regular interval, precisely \(2\pi\) radians or \(360\degree\).
  • Wave Representation: Since SHM resembles wave patterns, sinusoidal functions are perfect for illustrating the smooth, repetitive swings of an oscillating object.
  • Phase Difference: The \(\sin \) and \(\cos \) functions are \(\pi/2\) or \(90\degree\) out of phase with each other, allowing for adjustment in the phase of oscillation.
When combined linearly, these functions can change not only the amplitude of the oscillation but also its starting position due to phase shifting. The coefficients \(A_1\) and \(A_2\) control the specific properties of the motion. Therefore, understanding these trigonometric properties helps in predicting and manipulating oscillatory behaviors found in various physical systems.
Differential Equations
Differential equations are mathematical equations that relate a function to its derivatives, explaining how a system changes over time. In SHM, differential equations are key to understanding motion characteristics.
  • Second Derivative: The second derivative of position \(\frac{d^2x}{dt^2}\) indicates how acceleration relates to position.
  • Characteristic Equation: For SHM, a typical differential equation is \(\frac{d^2x}{dt^2} = -\omega^2 x\).
  • Solving SHM Equations: If a system satisfies this equation, it confirms that the motion is simple harmonic. The standard solution involves trigonometric functions, typically sine and cosine, due to their properties.
In the solution provided, by demonstrating that the second derivative is equal to \(-\omega^2 x\), we prove that the original function \(x(t) = A_1 \cos \omega t + A_2 \sin \omega t\) indeed models SHM. Thus, understanding differential equations enables us to derive significant insights into dynamic systems modeled mathematically.

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Most popular questions from this chapter

The shell-and-tube exchanger shown in Fig. \(1.125(\) a) can be modeled as shown in Fig. \(1.125(\mathrm{~b})\) for a simplified vibration analysis. Find the cross-sectional area of the tubes so that the total stiffness of the heat exchanger exceeds a value of \(200 \times 10^{6} \mathrm{~N} / \mathrm{m}\) in the axial direction and \(20 \times 10^{6} \mathrm{~N}-\mathrm{m} / \mathrm{rad}\) in the tangential direction. Assume that the tubes have the same length and cross section and are spaced uniformly.

In Problem 1.71 , using the given data as reference, find the variation of the damping constant \(c\) when a. \(r\) is varied from \(0.5 \mathrm{~cm}\) to \(1.0 \mathrm{~cm}\). b. \(h\) is varied from \(0.05 \mathrm{~cm}\) to \(0.10 \mathrm{~cm}\). c. \(a\) is varied from \(2 \mathrm{~cm}\) to \(4 \mathrm{~cm}\).

Subtract the complex number \((1+2 i)\) from \((3-4 i)\) and express the result in the form \(A e^{i \theta}\).

Find the torsional damping constant of a journal bearing for the following data: Viscosity of the lubricant \((\mu): 0.35 \mathrm{~Pa}-\mathrm{s}\), Diameter of the journal or shaft \((2 R): 0.05 \mathrm{~m}\), Length of the bearing \((l): 0.075 \mathrm{~m}\), Bearing clearance \((d): 0.005 \mathrm{~m}\). If the journal rotates at a speed \((N)\) of \(3000 \mathrm{rpm},\) determine the damping torque developed.

The foundation of a reciprocating engine is subjected to harmonic motions in \(x\) and \(y\) directions: $$ \begin{array}{l} x(t)=X \cos \omega t \\ y(t)=Y \cos (\omega t+\phi) \end{array} $$ where \(X\) and \(Y\) are the amplitudes, \(\omega\) is the angular velocity, and \(\phi\) is the phase difference. a. Verify that the resultant of the two motions satisfies the equation of the ellipse given by (see Fig. 1.111 ): $$ \frac{x^{2}}{X^{2}}+\frac{y^{2}}{Y^{2}}-2 \frac{x y}{X Y} \cos \phi=\sin ^{2} \phi $$ b. Discuss the nature of the resultant motion given by Eq. (E.1) for the special cases of \(\phi=0, \phi=\frac{\pi}{2},\) and \(\phi=\pi\)
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