# osc - Physics 53 Oscillations You've got to be very careful...

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Unformatted text preview: Physics 53 Oscillations You've got to be very careful if you don't know where you're going, because you might not get there. — Yogi Berra Overview Many natural phenomena exhibit motion in which particles retrace the same trajectory repeatedly. This motion is called periodic motion, or oscillation . The simplest number characteristic of this motion is the time it takes the particle to make one complete trip through the trajectory, returning to the initial point with the initial velocity. This time is the period , usually denoted by T . Alternatively, one can use the number of complete trips made per unit time, called the frequency , denoted by f . Clearly f = 1/ T . If r ( t ) is the position of the particle, and v ( t ) its velocity, then for oscillation with period T we must have both r ( t + T ) = r ( t ) and v ( t + T ) = v ( t ) . That is, the particle must return, after time T , to its original position and velocity. It is important that both the position and velocity return to the original values. In the swing of a pendulum, the bob passes through the bottom point twice in each oscillation, once with the original velocity and once with the same speed but opposite direction. An important type of periodic motion is harmonic motion in which the time dependence is sinusoidal: r ( t ) = r cos ω t , where r is a constant (vector) and ω = 2 π f = 2 π / T . The constant r , giving the maximum value of r , is the amplitude ; the constant ω is the angular frequency . The choice of cosine rather than sine is arbitrary. Our choice makes r (0) = r . The special case where ω is independent of the amplitude is called simple harmonic motion (SHM). We begin with that case, in one dimension for simplicity. PHY 53 1 Oscillations Simple harmonic motion We restrict the motion to the x-axis, so the most general SHM is described by: Simple harmonic motion (1-D) x ( t ) = A cos( ω t + φ ) Here A is the amplitude, ω is the angular frequency, and φ is called the “initial phase”. The term “phase” in harmonic motion means the argument of the cosine, in this case the quantity ( ω t + φ ) . The constant φ is the value of the phase at t = 0. The velocity and acceleration are found as usual by taking time derivatives: v ( t ) = dx dt = − ω A sin( ω t + φ ) a ( t ) = dv dt = − ω 2 A cos( ω t + φ ) We see that a = − ω 2 x . What sort of force gives rise to this motion? From the 2 nd Law the total force must have the general form F = − m ω 2 x ....
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osc - Physics 53 Oscillations You've got to be very careful...

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