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4. Show that functions (a) x = A sin wt, (b) x = A sin wt + B cos wt, (c) x = A cos(@t + ¢)

4. Show that functions (a) x = A sin wt, (b) x = A sin wt + B cos wt, (c) x = A cos(@t + ¢)

University Physics Volume 1
University Physics Volume 1
18th Edition
ISBN:9781938168277
Author:William Moebs, Samuel J. Ling, Jeff Sanny
Publisher:William Moebs, Samuel J. Ling, Jeff Sanny
Chapter15: Oscillations
Section: Chapter Questions
Problem 67AP: A 2.00-kg block lies at rest on a frictionless table. A spring, with a spring constant of 100 N/m is...
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4. Show that functions (a) x = A sin wt, (b) (c) x = A cos(@t + 4) х — A sin wt + B cos wt, all satisfy Eq. (1.7), provided w = Vks/M. A, B, and ø are constants.
Transcribed Image Text:4. Show that functions (a) x = A sin wt, (b) (c) x = A cos(@t + 4) х — A sin wt + B cos wt, all satisfy Eq. (1.7), provided w = Vks/M. A, B, and ø are constants.
Review of Oscillations x >0 FIGURE 1-2 Displacement x > 0. The spring is stretched and pulls the mass toward the equilibrium position, x = 0. M x <0 FIGURE 1-3 1- lat Displacement x < 0. The spring is squeezed and pushes the mass toward the equilibrium position. M second law (mass x acceleration = force), we find d²x M = -k,x dt? (1.7) This is the equation of motion for the mass to follow. The position x of the mass is to be found from this differential equation as a function of time t. Remember that the mass was located at x = xo at t = 0, when the oscillation starts, or x(0) = xo (1.8) where x(0) means the value of x at t = 0. We know that the second order derivatives of sinusoidal functions, sin ae and cos ae, are d? de3 sin ae = -a² sin ae sin að = -a² sin a® d? J0z cos a0 = -a cos ae Therefore it is very likely that Eq. (1.7) has a sinusoidal solution. Let the solution for x(t) be x(t) = A cos ot (1.9) where A and w are constants that are to be determined. Since cos 0 = 1, we find x(0) = A (1.10) 1.2 Mass Spring System 5 Comparing Eq. (1.8) with Eq. (1.10), we must have A = xo. This quantity is called the amplitude of the oscillation. To find aw, we calculate the second derivative of x(t) = xo cos ot: dx cos wt = -xow sin ot dt (1.11) d²x dr2 d? cos wt = -Xow- sin wt = -xow cos ot dt2 d dt After substituting Eq. (1.11) into Eq. (1.7), we find -Moxo cos ot = -k, xo cos ot which yields k, M (1.12) This quantity w is called the angular frequency and has the dimensions of radians/second. Since the function cos ot has a period of 27 radians, the temporal period T is given by 27 T = (s) (1.13) In 1 second, the oscillation repeats itself 1/T times (Figure 1-4). This number is defined as the frequency, v (cycles/s = Hertz). It is obvious that V = - (Hz) (1.14) 27
Transcribed Image Text:Review of Oscillations x >0 FIGURE 1-2 Displacement x > 0. The spring is stretched and pulls the mass toward the equilibrium position, x = 0. M x <0 FIGURE 1-3 1- lat Displacement x < 0. The spring is squeezed and pushes the mass toward the equilibrium position. M second law (mass x acceleration = force), we find d²x M = -k,x dt? (1.7) This is the equation of motion for the mass to follow. The position x of the mass is to be found from this differential equation as a function of time t. Remember that the mass was located at x = xo at t = 0, when the oscillation starts, or x(0) = xo (1.8) where x(0) means the value of x at t = 0. We know that the second order derivatives of sinusoidal functions, sin ae and cos ae, are d? de3 sin ae = -a² sin ae sin að = -a² sin a® d? J0z cos a0 = -a cos ae Therefore it is very likely that Eq. (1.7) has a sinusoidal solution. Let the solution for x(t) be x(t) = A cos ot (1.9) where A and w are constants that are to be determined. Since cos 0 = 1, we find x(0) = A (1.10) 1.2 Mass Spring System 5 Comparing Eq. (1.8) with Eq. (1.10), we must have A = xo. This quantity is called the amplitude of the oscillation. To find aw, we calculate the second derivative of x(t) = xo cos ot: dx cos wt = -xow sin ot dt (1.11) d²x dr2 d? cos wt = -Xow- sin wt = -xow cos ot dt2 d dt After substituting Eq. (1.11) into Eq. (1.7), we find -Moxo cos ot = -k, xo cos ot which yields k, M (1.12) This quantity w is called the angular frequency and has the dimensions of radians/second. Since the function cos ot has a period of 27 radians, the temporal period T is given by 27 T = (s) (1.13) In 1 second, the oscillation repeats itself 1/T times (Figure 1-4). This number is defined as the frequency, v (cycles/s = Hertz). It is obvious that V = - (Hz) (1.14) 27
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