24.1: Introduction to Physical Pendulums
- Page ID
- 25582
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.
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.