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Simple Harmonic Motion Explained

Simple harmonic motion (SHM) describes the periodic motion of an object under the influence of a restoring force proportional to its displacement from equilibrium, such as a mass attached to a spring. The motion is characterized by oscillations between the maximum displacement (amplitude) and the equilibrium position, with the acceleration always directed towards equilibrium. The period and frequency of SHM are inversely related to the square root of the ratio of the spring's stiffness to the object's mass. Damped harmonic motion accounts for energy lost to friction, dividing it into underdamped, overdamped, and critically damped cases based on the degree of damping.

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155 views2 pages

Simple Harmonic Motion Explained

Simple harmonic motion (SHM) describes the periodic motion of an object under the influence of a restoring force proportional to its displacement from equilibrium, such as a mass attached to a spring. The motion is characterized by oscillations between the maximum displacement (amplitude) and the equilibrium position, with the acceleration always directed towards equilibrium. The period and frequency of SHM are inversely related to the square root of the ratio of the spring's stiffness to the object's mass. Damped harmonic motion accounts for energy lost to friction, dividing it into underdamped, overdamped, and critically damped cases based on the degree of damping.

Uploaded by

feras.barbar06
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© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Simple Harmonic Motion

A particle or object under the influence of a linear restoring force described


by Hooke’s Law (𝐹𝑥 = 𝑘𝛥𝑥) undergoes Simple Harmonic Motion (SHM).
Simple Harmonic Motion is a periodic (“back-and-forth” or “up-and-down”)
motion in which the force (and therefore the acceleration) is directly
proportional to the displacement.
Equations of Simple Harmonic Motion
As with all periodic motion, SHM has a property called amplitude.

 This is the maximum displacement from the equilibrium position


 This occurs at the turning point (where the direction changes)
At the turning point, the spring only has elastic potential energy (its velocity is
momentarily zero), therefore:

At the turning point, the displacement is equal to the amplitude (Δx = A), therefore:
(Equation 1)

At the equilibrium point, the situation is different – the ball is at its maximum velocity because the spring
only has kinetic energy (the spring is neither stretched nor compressed). Therefore:
(Equation 2)

Combining Equations 1 & 2 and solving for v (which we can call 𝑣𝑚𝑎𝑥 ) gives:
(Equation 3)

Like all types of periodic motion, SHM has the properties of period and frequency. Can we derive equations
to describe these properties?
Consider a ball on a spring undergoing SHM compared to a ball undergoing
uniform circular motion.
 If the circle is turned on its side, they look almost identical!
Recall that the speed of an object undergoing uniform circular motion is:
therefore

We will make two assumptions:


 the amplitude of the SHM is the same as
the radius of the circular motion.
 the velocity of the circular motion is the
same as the vmax of the SHM.
This gives:
(Equation 4)

Substituting in Equation 3 for vmax gives:

 T is the period of SHM (s)


 m is the mass (kg)
Which simplifies to:
 k is the spring constant (Nm-1)
Example: An 85 kg diver stands on a diving board with k = 8.1 x 103 N·m-1. Calculate the period and
frequency at which the board vibrates. Ignore the mass of the board.

Damped Harmonic Motion


The previous derivations assume friction is zero. In reality, the SHM of a periodic system is the real world is
affected by friction. This is known as damping (when energy leaves a system that is oscillating).
 Friction converts the mechanical energy of the system into thermal energy
There are three types of damped harmonic motion:
Underdamped (Curve 3) experiences low friction and undergoes many oscillations before reaching
equilibrium gradually decreasing in amplitude
 Ex: pendulum, mass on spring

Overdamped (Curve 1) involves quite a lot of friction, such as a piston moving through thick hydraulic fluid.
The mass moves slowly back to equilibrium without passing through zero
 Ex: shock absorber in heavy door

Critically damped (Curve 2) falls to zero as quickly as possible


 Ex: car shock absorbers

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