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Simple Harmonic Motion

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10.2Simple Harmonic Motion


In these revision notes for Simple Harmonic Motion, we cover the following key points:

  • Definition of simple harmonic motion
  • Differences between periodic motion and simple harmonic motion
  • Types of simple harmonic motion
  • Conditions for the existence of simple harmonic motion
  • The equation of simple harmonic motion
  • How to calculate the velocity and acceleration in a simple harmonic motion
  • How does simple harmonic motion occur in oscillating springs?

Simple Harmonic Motion Revision Notes

By definition, "Simple harmonic motion (in short SHM) is a repetitive movement back and forth through an equilibrium (or central) position, so that the maximum displacement on one side of this position is equal to the maximum displacement on the other side."

In other words, in simple harmonic motion the object moves back and forth along a line. These back and forth movements are known as "oscillations". Simple harmonic motion is part of a wider category of motion known as "periodic motion", which includes other types of motions that repeat themselves such as circular, vibrational and other similar motions. However, simple harmonic motion and periodic motion are not the same thing. There are two main differences between them.

  1. In periodic motion, the object's displacement may or may not be in the direction of the restoring force. On the other hand, in simple harmonic motion, the object's displacement is always in the opposite direction of the restoring force.
  2. In periodic motion, the movement may or may not be oscillatory, while in simple harmonic motion the movement is only oscillatory.
    1. Based on the trajectory (path) of the oscillating object, there are two types of simple harmonic motion.

  3. If oscillations occur according a straight line, such as when we pull a spring and then we release it, we obtain a linear simple harmonic motion, For example, an oscillating spring makes linear simple harmonic motion.
  4. If a hanged object swings around a central line (usually a vertical line dictated by the existence of gravity), we have an angular simple harmonic motion

The function that describes better mathematically a SHM, is the sine (or cosine) function. This is because of two main reasons:

  1. Both sine and cosine functions have an equilibrium position and two extremities: one maximum and one minimum, just like the SHM. When used to describe SHM, the maximum and the minimum of the sine function represent the turning points, i.e. the distance from the origin to these points gives the amplitude A, which is the maximum displacement from the origin. In many cases, the amplitude is also denoted as x0 or xmax.
  2. Like in rotational motion (which has many similarities to SHM as they both are periodic), we use the concept of period T to describe the time needed to do one complete oscillation. As a result, we can also make use of related concepts such as frequency f, where f = 1/T, angular velocity (which here is known as angular frequency) ω = 2π / T = 2π × f, and so on.

In mathematics, the simplest sine function in which the time t is the independent variable and the position x is the dependent one, has the form

x(t) = x0 × sin⁡ω × t

If the graph does not start at the origin but it is shifted from it, we insert the angular displacement φ in the above equation, so it becomes

x(t) = x0 × sin⁡(ω × t + φ)

The initial angular displacement φ here is known as the "phase shift".

From calculus it is known that the first derivative of position with respect to time (dx/dt) represents the velocity v and the second derivative of position with respect to time d2x/dt2 (or the first derivative of velocity in respect to time dv/dt) gives the acceleration a.

Applying the above derivation rules for the SHM motion, we obtain for the velocity v in respect to the time, v(t):

v(t) = dx/dt
= x0 × ω × cos⁡(ω × t + φ)

and for the acceleration a in respect to the time, a(t):

a(t) = d2 x/dt2
= dv/dt
= -x0 × ω2 × sin⁡(ω × t + φ)
= -ω2 × x(t)

Therefore, we obtained a very important formula of SHM that relates acceleration a, angular frequency ω and displacement x:

a = -ω2 × x

Since the restoring force F is in the opposite direction to the acceleration a, we obtain from Newton's Second Law of Motion for the acting forces in springs:

F = m × a = -k × x

where m is the mas of hanged object and k is the spring constant. Thus,

-m × ω2 × x = -k × x
m × ω2 = k

Hence,

ω2 = k/m

Or

ω = √k/m

Then we can substitute this value of angular frequency in the sine equation of SHM if an object hanged on an elastic spring causes oscillations.

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Enjoy the "Simple Harmonic Motion" revision notes? People who liked the "Simple Harmonic Motion" revision notes found the following resources useful:

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  2. Oscillations Physics tutorial: Simple Harmonic Motion. Read the Simple Harmonic Motion physics tutorial and build your physics knowledge of Oscillations
  3. Oscillations Practice Questions: Simple Harmonic Motion. Test and improve your knowledge of Simple Harmonic Motion with example questins and answers
  4. Check your calculations for Oscillations questions with our excellent Oscillations calculators which contain full equations and calculations clearly displayed line by line. See the Oscillations Calculators by iCalculator™ below.
  5. Continuing learning oscillations - read our next physics tutorial: Pendulums. Energy in Simple Harmonic Motion

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