When are Maximum Velocity and Acceleration acheived in Simple Harmonic Motion?

Question

Im trying to get my head around SMH out of curiosity because it seems simple yet I'm not getting the concept behind some ideas.

For a SMH equation :

$$ x=a \sin(\omega t+\phi) $$

• Under what conditions will each of the two ( velocity and acceleration ) be at maximum ?

we know that :

$$ x'=-\omega \cos(\omega t + \phi) $$ $$ x''=-\omega^2 \sin(\omega t + \phi) $$

Also :

• how would it be different for $x = a \cos(\omega t - \phi)$ ?

Answer 1

The easiest way to determine maximum and minumums of a function is to set the derivative equal to zero. Thus, in this case, setting the equation for acceleration equal to zero and solving for the variables of interest will give you what you want. Thus, in this case we have the equation for position:

$$ x = a \sin(\omega t + \phi) $$

One way to see the maximum and minimum is to plot the function. I find that playing around with the constants (in this case $a, \omega,$ and $\phi$) helps you see what effect they have.

You can see that $a$ is an amplitude that determines how large the swings are from the origin. The termn $\omega$ tells you something about how quickly the position moves back and forth, thus it is known as the "angular frequency." Finally, $\phi$ tells you where you are when $t=0$ (or any other time for that matter) and is called the "phase."

For the position function, it is obvious that the value is maximized when the value of the sine function equals $1$ and, from what you learned in trig class, this happens when $\omega t + \phi = \pm \pi/2$ depending on the sign of $a$. Since $\omega$ and $\phi$ are constants set by the conditions and $t$ is the variable, the position is maximum when $t = (\pm \pi/2 - \phi)/\omega$, again depending on the sign of $a$.

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ps- sorry for the curt answer the first time, but Wikipedia has a good article on this topic and there are tons of resources out there for it.

Answer 2

$$x'=-\omega \cos(\omega t + \phi)$$

maximum value of cosine is 1 implies max value of $x'$ is $|w|$.

Same for $x''$.