1.
Follow the steps from the end of the video above to find an equation of motion for the simple pendulum.
Section 6.8 The Simple Pendulum
Exercises Warm-up Activity
Consider a mass \(m\) that hangs from a string that has one end fixed in space and is free to pivot (i.e., no friction). Assume the string has negligible mass compared to the object’s mass \(m\) such that we can apply the massless string approximation to our analysis.
The string has length \(L \) and the mass is initially pulled out an angle \(\theta_{max}\) and then released. You know from experience that the pendulum will undergo oscillatory motion—what is the angular position as a function of time \(\theta(t)\) and the oscillation frequency \(\omega_p\) for the pendulum.
Using torque analysis, first draw a pictorial representation of the situation and choose a coordinate system. Assume the mass can be treated as a particle and draw an extended free-body diagram, choosing an axis directed radially toward the pivot point and an axis tangential to the arc of the motion. Assume tension keeps the string taut so it does not bend or bow.
Recall the Rotational Law of Motion
\begin{equation*}
\sum \tau = I\alpha
\end{equation*}
The gravitational torque \(\tau = -Lmg\sin\theta(t)\) is the torque about the pivot point, \(I=mL^2 \) is the moment of inertia of a point mass about the pivot point and \(\alpha= \frac{d^2}{dt^2}\theta(t)\) is the angular acceleration of the pendulum mass. Putting this all together and simplifying:
\begin{equation}
mL^2 \frac{d^2}{dt^2}\theta(t) = -Lmg \sin\theta(t) \tag{6.8.1}
\end{equation}
\begin{equation}
\frac{d^2}{dt^2}\theta(t)=-\frac{g}{L} \sin\theta(t)\tag{6.8.2}
\end{equation}
This is the equation of motion for the pendulum in terms of the angular position as a function of time.
Exercises Activities
1. Analyzing the Equation of Motion.
Looking at equation (6.8.2), does this equation of motion satisfy the condition proposed in (6.5.1)? Explain why or why not.
Answer.
The equation of motion for the simple pendulum does not satisfy the condition. The restoring force is not directly proportional to the angular displacement. Instead, it depends on \(\sin\theta(t)\text{.}\)
2. Approximating the Restoring Force.
Sketch quick graphs of \(\sin\theta\) and \(\theta(t)\) on the same set of axes. Can you use the graphs to come up with a reasonable approximation in which the restoring force for the simple pendulum is directly proportional to the angular displacement?
Answer.
The small-angle approximation can be used to approximate the values of the sine and cosine functions (or any combination) if the angle is restricted to a certain domain. This technique is discussed in detail in the next section.
References References
[1]
Figure 6.8.1 created by Rebecka Tumblin.
