15.1 Simple harmonic motion  (Page 4/16)

The data in [link] 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 ( [link] ). 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 occurs at the position $\left(x=\text{−}A\right)$ , and the acceleration at the position $\left(x=\text{−}A\right)$ and is equal to $\text{−}{a}_{\text{max}}$ .

Summary of equations of motion for shm

In summary, the oscillatory motion of a block on a spring can be modeled with the following equations of motion:

Here, A is the amplitude of the motion, T is the period, $\varphi$ is the phase shift, and $\omega =\frac{2\pi }{T}=2\pi f$ is the angular frequency of the motion of the block.