A spring that obeys Hooke's Law is an example of simple harmonic motion. If you displace the spring a maximum amount x = A, the amplitude, release it from rest (vo = 0), photograph and plot the position as function of time, you find, as shown in Fig. 3a below that x(t) = A cos (2pt/T), where T, the period is the time for one complete oscillation. To save time I shall write: x(t) = A cos (2pt/T), as x(t) = A cos wt then, v(t) = dx/dt = - wA sin wt, as shown in Fig. 3b below:
Note: Maximum value of v = wA because maximum value of sine =1. a(t) = dv/dt = - w2 (A cos wt), as shown in Fig. 3c below:
Note: Maximum value of a = w2A because maximum value of cosine =1. a(t) = -[w2] x(t) (Equation 2)
A. Write x(t) for this graph. First find A, T, f, and w. From Fig. 3a, you see that the maximum value of x or the amplitude A is 0.20 m. The graph repeats itself after one period = T = 2 s, f = 1/T = 0.5 s-1 and w = 2pf = p s-1. In general, x(t) = A cos wt. For this case x(t) = 0.20 m cos p s-1 t B. Find a general expression for the velocity, apply it to this case, and check with Fig. 3b to see if it is correct. What is the maximum value of the velocity for Fig. 3b? Find x when v = -0.1 p m/s. Since x(t) = 0.20 m cos p s-1 t, dx/dt = v(t) = -(0.20p m/s) sin p s-1 t. In Fig. 3b we see that v as a function of t is a negative sine curve with a maximum value of 0.2(3.14) m/s. v = -0.1p m/s = -(0.20p m/s) sin p s-1t. Or 1/2 = sin p s-1t. The sine of an angle is 1/2 when the angle is 300 or p/6 radians. So p/6 = p s-1t or t = 1/6 s. x(1/3 s) = 0.20 m cos p/6= 0.173 cm. From Fig. 3a and 3b, you can see that these are the correct values. C. Find a general expression for the acceleration, apply it to this case, and check with Fig. 3c to see if it is correct. What is the maximum value of the acceleration for Fig. 3b? Since v(t) = -(0.20p m/s) sin p s-1 t, dv/dt = a(t) = -(0.20p2 m/s) cos p s-1 t. In Fig. 3c we see that v as a function of t is a negative cosine curve with a maximum value of 0.2(3.14)2 m/s2 approximately equal to 2 m/s2. D. Sample Problems in 104 Problem Set for Simple Harmonic Motion: 1-6, 10, and 12-16.
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