Simple Harmonic Motion: A Special Periodic Motion

The oscillations of a system in which the net force can be described by Hooke’s law are of special importance, because they are very common. They are also the simplest oscillatory systems. Simple Harmonic Motion (SHM) is the name given to oscillatory motion for a system where the net force can be described by Hooke’s law, and such a system is called a simple harmonic oscillator. If the net force can be described by Hooke’s law and there is no damping (by friction or other non-conservative forces), then a simple harmonic oscillator will oscillate with equal displacement on either side of the equilibrium position, as shown for an object on a spring in Figure 16.9. 

The maximum displacement from equilibrium is called the amplitude $X$. The units for amplitude and displacement are the same, but depend on the type of oscillation. For the object on the spring, the units of amplitude and displacement are meters; whereas for sound oscillations, they have units of pressure (and other types of oscillations have yet other units). Because amplitude is the maximum displacement, it is related to the energy in the oscillation.

Figure 16.9 An object attached to a spring sliding on a frictionless surface is an uncomplicated simple harmonic oscillator. When displaced from equilibrium, the object performs simple harmonic motion that has an amplitude $𝑋$ and a period $𝑇$. The object’s maximum speed occurs as it passes through equilibrium. The stiffer the spring is, the smaller the period $𝑇$. The greater the mass of the object is, the greater the period $𝑇$.

What is so significant about simple harmonic motion? One special thing is that the period $𝑇$ and frequency $𝑓$ of a simple harmonic oscillator are independent of amplitude. The string of a guitar, for example, will oscillate with the same frequency whether plucked gently or hard. Because the period is constant, a simple harmonic oscillator can be used as a clock.

Two important factors do affect the period of a simple harmonic oscillator. The period is related to how stiff the system is. A very stiff object has a large force constant $𝑘$, which causes the system to have a smaller period. For example, you can adjust a diving board’s stiffness – the stiffer it is, the faster it vibrates, and the shorter its period. Period also depends on the mass of the oscillating system. The more massive the system is, the longer the period. For example, a heavy person on a diving board bounces up and down more slowly than a light one.

In fact, the mass $𝑚$ and the force constant $𝑘$ are the only factors that affect the period and frequency of simple harmonic motion.

The Link between Simple Harmonic Motion and Waves

If a time-exposure photograph of the bouncing car were taken as it drove by, the headlight would make a wavelike streak, as shown in Figure 16.10. Similarly, Figure 16.11 shows an object bouncing on a spring as it leaves a wavelike "trace of its position on a moving strip of paper. Both waves are sine functions. All simple harmonic motion is intimately related to sine and cosine waves.

Figure 16.10 The bouncing car makes a wavelike motion. If the restoring force in the suspension system can be described only by Hooke’s law, then the wave is a sine function. (The wave is the trace produced by the headlight as the car moves to the right.)

Figure 16.11 The vertical position of an object bouncing on a spring is recorded on a strip of moving paper, leaving a sine wave.

The displacement as a function of time t in any simple harmonic motion—that is, one in which the net restoring force can be described by Hooke’s law, is given by

$x(t)=X\: cos\frac{2\pi t}{T}$ [equation 16.20]

where $𝑋$ is amplitude. At 𝑡=0size 12{t=0} {}, the initial position is 𝑥0=𝑋size 12{x rSub { size 8{0} } =X} {}, and the displacement oscillates back and forth with a period 𝑇. (When 𝑡=𝑇, we get 𝑥=𝑋size 12{x=X} {} again because cos2π=1.). Furthermore, from this expression for $𝑥$, the velocity 𝑣size 12{v} {} as a function of time is given by:

$v(t)=-v_{max}\: sin\: (\frac{2\pi t}{T})$ [equation 16.21]

where $v_{max}=\frac{2\pi X}{T}=X\sqrt{\frac{k}{m}}$. The object has zero velocity at maximum displacement – for example, , and at that time $x=X$. The minus sign in the first equation for $v(t)$gives the correct direction for the velocity. Just after the start of the motion, for instance, the velocity is negative because the system is moving back toward the equilibrium point. Finally, we can get an expression for acceleration using Newton’s second law. [Then we have $x(t)$, $v(t)$, $t$, and $A(t)$, the quantities needed for kinematics and a description of simple harmonic motion.] According to Newton’s second law, the acceleration is $a=\frac{F}{m}=\frac{kx}{m}$. So, $a(t)$ is also a cosine function:

$a(t)=-\frac{kX}{m}\: cos\: \frac{2\pi t}{T}$ [equation 16.22]

Hence, $a(t)$ is directly proportional to and in the opposite direction to $x(t)$.

Figure 16.12 shows the simple harmonic motion of an object on a spring and presents graphs of $x(t)$, $v(t)$, and $a(t)$ versus time.

Figure 16.12 Graphs of $x(t)$, $v(t)$, and $a(t)$ versus $t$ for the motion of an object on a spring. The net force on the object can be described by Hooke’s law, and so the object undergoes simple harmonic motion. Note that the initial position has the vertical displacement at its maximum value $𝑋$; $v$ is initially zero and then negative as the object moves down; and the initial acceleration is negative, back toward the equilibrium position and becomes zero at that point.

The most important point here is that these equations are mathematically straightforward and are valid for all simple harmonic motion. They are very useful in visualizing waves associated with simple harmonic motion, including visualizing how waves add with one another.

Source: Rice University, https://openstax.org/books/college-physics/pages/16-3-simple-harmonic-motion-a-special-periodic-motion
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