7.3 Representing and Analyzing SHM

Learning objectives

7.3A—Describe the displacement, velocity, and acceleration of an object exhibiting SHM. TOPIC 7.3 Representing and…Describe the displacement, velocity, and acceleration of an object exhibiting SHM. TOPIC 7.3 Representing and Analyzing SHM• For an object exhibiting SHM, the displacement of that object measured from its equilibrium position can be represented by the equations xA =co s2() ft or xA =si n2() .- i. Minima, maxima, and zeros of displacement, velocity, and acceleration are features of harmonic motion.- ii. Recognizing the positions or times at which the displacement, velocity, and acceleration for SHM have extrema or zeros can help in qualitatively describing the behavior of the motion.• Changing the amplitude of a system exhibiting SHM will not change the period of that system.• Properties of SHM can be determined and analyzed using graphical representations.

Represent displacement sinusoidally

$x=A\cos(2\pi ft)\quad\text{or}\quad x=A\sin(2\pi ft)$

$A$ is maximum displacement from equilibrium and $f$ sets the cycle rate. Sine and cosine describe the same SHM with different choices of where the clock starts.

Match the three key positions

Position	Velocity	Acceleration
$x=+A$	$v=0$	Maximum magnitude toward negative/equilibrium direction
$x=0$	Maximum speed	$a=0$
$x=-A$	$v=0$	Maximum magnitude toward positive/equilibrium direction

Trace one cosine-start cycle

Time for cosine start	Displacement	Motion state
$0$	$+A$	Released from rest; acceleration toward equilibrium
$T/4$	$0$	Moving fastest in the negative direction; acceleration zero
$T/2$	$-A$	Instantaneously at rest; acceleration toward equilibrium
$3T/4$	$0$	Moving fastest in the positive direction; acceleration zero
$T$	$+A$	One complete cycle

Read zeros and extrema

On an $x$ –time graph, velocity is zero at displacement peaks and greatest in magnitude where the curve crosses equilibrium. Acceleration always has the opposite sign to $x$ , so its extrema align with displacement extrema but point oppositely.

Change amplitude without changing period

Increasing amplitude makes the displacement graph taller, but it does not change the period of an ideal SHM system. Do not infer a longer cycle merely because the object travels farther.

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—7.4 Energy of Simple Harmonic OscillatorsRead