Simple Harmonic Motion

Equation 3 is the sinusoidal function of a cosine

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Unformatted text preview: is the sinusoidal function of a cosine function shown in Figure 1. 1 Figure 1 A graph of equation (3) taken from Wikipedia. Figure 2 A handy reference to the quantities in equation (3) for simple harmonic motion. The quantity xm, called the amplitude of the motion, is a positive constant whose value depends on how the motion was started. The subscript m stands for maximum because the amplitude is the magnitude of the maximum displacement of the particle in either direction. The cosine function in equation (3) varies between the limits ±1; so the displacement x(t) varies between the limits ±xm. The time-varying quantity (ω t + φ ) in equation (3) is called the phase of the motion, and the constant φ is called the phase constant (or phase angle). The value of φ depends on the displacement and velocity of the particle at time t=0. To interpret the constant ω, called the angular frequency of the motion, we first note that the displacement x(t) must return to its initial value after one period T of the motion. That is (4) for all t. To simplify this analysis, let us suppose that φ = 0 in equation (3). Then using equations (3) and (4) we can write cos cos (5) It is important to note that the cosine function repeats itself when its argument (the phase) has increased by 2π rad; so equation (5) yields 2 2 1 2 2 (6) (7) (8) (9) The SI unit of angular frequency is radians per second (rad/s). 2.2 The Velocity of SHM 2.2.1 Method 1 There are two different methods for finding the velocity for simple harmonic motion. The first method involves no calculus. To begin, recall that the velocity of an object in uniform circular motion of radius r has a magnitude equal to (10) 2 In addition, the velocity is tangential to the object’s circular path, as indicated in Figure 3. Therefore we see that when P is at the angular position θ, the velocity vector makes an angle θ with the vertical. As a result, the x component of the velocity is –v sin θ. Combining these results we find that the velocity of point P, along the x axis, is — sin sin (11) (12) In what follows, we shall drop the x subscripts, since we are assuming that point P is only moving along the x axis....
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