第15章振動

Consider a block attached to a spring on a frictionless table (Figure 15.4). The equilibrium position (the position where the spring is neither stretched nor compressed) is marked as $x=0$. At the equilibrium position, the net force is zero.

Work is done on the block to pull it out to a position of $x=+A,$ and it is then released from rest. The maximum x-position (A) is called the amplitude of the motion. The block begins to oscillate in SHM between $x=+A$ and $x=\text{−}A,$ where A is the amplitude of the motion and T is the period of the oscillation. The period is the time for one oscillation. Figure 15.5 shows the motion of the block as it completes one and a half oscillations after release. Figure 15.6 shows a plot of the position of the block versus time. When the position is plotted versus time, it is clear that the data can be modeled by a cosine function with an amplitude A and a period T. The cosine function $\text{cos}\theta$ repeats every multiple of $2\pi ,$ whereas the motion of the block repeats every period T. However, the function $\text{cos}\left(\frac{2\pi }{T}t\right)$ repeats every integer multiple of the period. The maximum of the cosine function is one, so it is necessary to multiply the cosine function by the amplitude A.

Recall from the chapter on rotation that the angular frequency equals $\omega =\frac{d\theta }{dt}$. In this case, the period is constant, so the angular frequency is defined as $2\pi$ divided by the period, $\omega =\frac{2\pi }{T}$.

The equation for the position as a function of time $x\left(t\right)=A\text{cos}\left(\omega t\right)$ is good for modeling data, where the position of the block at the initial time $t=0.00\text{s}$ is at the amplitude A and the initial velocity is zero. Often when taking experimental data, the position of the mass at the initial time $t=0.00\text{s}$ is not equal to the amplitude and the initial velocity is not zero. Consider 10 seconds of data collected by a student in lab, shown in Figure 15.7.

The data in Figure 15.7 can still be modeled with a periodic function, like a cosine function, but the function is shifted to the right. This shift is known as a phase shift and is usually represented by the Greek letter phi $(\varphi )$. The equation of the position as a function of time for a block on a spring becomes

This is the generalized equation for SHM where t is the time measured in seconds, $\omega$ is the angular frequency with units of inverse seconds, A is the amplitude measured in meters or centimeters, and $\varphi$ is the phase shift measured in radians (Figure 15.8). It should be noted that because sine and cosine functions differ only by a phase shift, this motion could be modeled using either the cosine or sine function.

The velocity of the mass on a spring, oscillating in SHM, can be found by taking the derivative of the position equation:

Because the sine function oscillates between –1 and +1, the maximum velocity is the amplitude times the angular frequency, $v_{\text{max}}=A\omega$. The maximum velocity occurs at the equilibrium position $\left(x=0\right)$ when the mass is moving toward $x=+A$. The maximum velocity in the negative direction is attained at the equilibrium position $\left(x=0\right)$ when the mass is moving toward $x=\text{−}A$ and is equal to $\text{−}{v}_{\text{max}}$.

The acceleration of the mass on the spring can be found by taking the time derivative of the velocity:

The maximum acceleration is $a_{\text{max}}=A{\omega }^2$. The maximum acceleration in the positive direction occurs at the position $\left(x=\text{−}A\right)$, and is equal to $a_{\text{max}}$.