A simple pendulum is a type of oscillatory motion in which a mass, assumed to be a point mass, is suspended from a fixed point and swings under the influence of gravity after being displaced from its vertical equilibrium position. Typically, the pendulum consists of a mass attached to a non-elastic, massless string or rod. The oscillations are considered under the condition of small angular displacements.
Definition and Theoretical Background
Periodic motion refers to a type of motion in which an object returns to its original position at regular time intervals. The simple pendulum exhibits periodic motion. The time taken for the pendulum to complete one full cycle is called the period (<span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9694em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">T</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">p</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span>), and the number of cycles per unit time is called the frequency (f). These quantities are related by the following expression:
- <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.3932em;vertical-align:-0.5481em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8451em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.3488em;margin-left:-0.1389em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathnormal mtight" style="margin-right:0.10764em;">f</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2901em;"><span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.5481em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span>
- <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="mord mathnormal">re</span><span class="mord mathnormal">kan</span><span class="mord mathnormal">s</span><span class="mord mathnormal">bi</span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="mord mathnormal">imi</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">:</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">Her</span><span class="mord mathnormal">t</span><span class="mord mathnormal" style="margin-right:0.04398em;">z</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.04398em;">Hz</span><span class="mclose">)</span></span></span></span>
- <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mord mathnormal" style="margin-right:0.02778em;">er</span><span class="mord mathnormal">i</span><span class="mord mathnormal">yo</span><span class="mord mathnormal">t</span><span class="mord mathnormal">bi</span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="mord mathnormal">imi</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">:</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">s</span><span class="mord mathnormal">ani</span><span class="mord mathnormal">ye</span><span class="mopen">(</span><span class="mord mathnormal">s</span><span class="mclose">)</span></span></span></span>
Equation of Motion
The analysis of simple pendulum motion is based on the parameters: pendulum length 𝐿 , mass 𝑚 , and gravitational acceleration 𝑔.
Assuming small angular displacements, the approximation sin 𝜃 ≈ 𝜃 sinθ≈θ (in radians) can be applied. Under this assumption, the motion is described by a second-order differential equation:
- <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9694em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">T</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">p</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.84em;vertical-align:-0.6594em;"></span><span class="mord">2.</span><span class="mord mathnormal" style="margin-right:0.03588em;">π</span><span class="mord sqrt"><span class="root"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.3127em;"><span style="top:-2.3127em;"><span class="pstrut" style="height:2em;"></span><span class="sizing reset-size6 size1 mtight"><span class="mord mtight"></span></span></span></span></span></span></span><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.1806em;"><span class="svg-align" style="top:-3.8em;"><span class="pstrut" style="height:3.8em;"></span><span class="mord" style="padding-left:1em;"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8723em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">g</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">L</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.4811em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:-3.1406em;"><span class="pstrut" style="height:3.8em;"></span><span class="hide-tail" style="min-width:1.02em;height:1.88em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="1.88em" viewBox="0 0 400000 1944" preserveAspectRatio="xMinYMin slice"><path d="M983 90
l0 -0
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H1013.1s-83.4,268,-264.1,840c-180.7,572,-277,876.3,-289,913c-4.7,4.7,-12.7,7,-24,7
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According to this expression, in small-angle oscillations, the period of the pendulum depends only on the length of the string and the gravitational acceleration, and does not depend on the mass of the object.
Energy Transformation
During the motion of a pendulum, continuous energy transformation occurs. At the lowest point of the swing, the kinetic energy reaches its maximum, while at the highest points, the potential energy is at its maximum.
When frictional effects (such as air resistance and pivot friction) are not neglected, the amplitude of oscillation decreases over time, and the motion becomes damped.
