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24.1: Introduction to Physical Pendulums

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    25582
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    We have already used Newton’s Second Law or Conservation of Energy to analyze systems like the spring-object system that oscillate. We shall now use torque and the rotational equation of motion to study oscillating systems like pendulums and torsional springs.

    Simple Pendulum: Torque Approach

    Recall the simple pendulum from Chapter 23.3.1.The coordinate system and force diagram for the simple pendulum is shown in Figure 24.1.

    clipboard_e28fb2900bddbc6708aaa81855adb3d05.png clipboard_e2c2f64a2ba8c7a3073abccee6921f6b0.png
    Figure 24.1 (a) Coordinate system and (b) torque diagram for simple pendulum

    The torque about the pivot point P is given by

    \[\overrightarrow{\boldsymbol{\tau}}_{P}=\overrightarrow{\mathbf{r}}_{P, m} \times m \overrightarrow{\mathbf{g}}=l \hat{\mathbf{r}} \times m g(\cos \theta \hat{\mathbf{r}}-\sin \theta \hat{\boldsymbol{\theta}})=-\operatorname{lm} g \sin \theta \hat{\mathbf{k}} \nonumber \]

    The z -component of the torque about point P

    \[\left(\tau_{P}\right)_{z}=-m g l \sin \theta \nonumber \]

    When \(\theta>0, \quad\left(\tau_{P}\right)_{z}<0\) and the torque about P is directed in the negative \(\hat{\mathbf{k}}\)-direction (into the plane of Figure 24.1b) when \(\theta<0, \quad\left(\tau_{P}\right)_{z}>0\) and the torque about P is directed in the positive \(\hat{\mathbf{k}}\)-direction (out of the plane of Figure 24.1b). The moment of inertia of a point mass about the pivot point P is \(I_{P}=m l^{2}\). The rotational equation of motion is then

    \[\begin{array}{l}
    \left(\tau_{P}\right)_{z}=I_{P} \alpha_{z} \equiv I_{P} \frac{d^{2} \theta}{d t^{2}} \\
    -m g l \sin \theta=m l^{2} \frac{d^{2} \theta}{d t^{2}}
    \end{array} \nonumber \]

    Thus we have

    \[\frac{d^{2} \theta}{d t^{2}}=-\frac{g}{l} \sin \theta \nonumber \]

    agreeing with Equation 23. 3.14. When the angle of oscillation is small, we may use the small angle approximation

    \[\sin \theta \cong \theta \nonumber \]

    and Equation (24.1.4) reduces to the simple harmonic oscillator equation

    \[\frac{d^{2} \theta}{d t^{2}} \cong-\frac{g}{l} \theta \nonumber \]

    We have already studied the solutions to this equation in Chapter 23.3. A procedure for determining the period when the small angle approximation does not hold is given in Appendix 24A.


    This page titled 24.1: Introduction to Physical Pendulums is shared under a not declared license and was authored, remixed, and/or curated by Peter Dourmashkin (MIT OpenCourseWare) via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.