15.1 Simple harmonic motion  (Page 3/16)

Equations of shm

Consider a block attached to a spring on a frictionless table ( [link] ). 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. [link] shows the motion of the block as it completes one and a half oscillations after release. [link] 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 [link] .