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Learning Introductory Physics with Activities

Paul J. Emigh, Rebecka Tumblin, Kathryn Hadley, Danielle Skinner

Section 6.9 Small-Angle Approximation

The small-angle approximation is used to approximate the values of the main trigonometric functions when the angles involved are restricted to a certain domain. Any function can be expanded into an infinite sum of polynomials. This is called a series expansion. Consider a series expansion (Taylor Series) of the sine and cosine functions:
\begin{equation*} \sin(\theta) = \sum^\infty_{n=0} \frac{(-1)^n}{2n+1}\theta^{2n+1} = \theta - \frac{\theta^3}{3!}+\frac{\theta^5}{5!} -\frac{\theta^7}{7!}-.... \end{equation*}
\begin{equation*} \cos(\theta) = \sum^\infty_{n=0} \frac{(-1)^n}{2n}\theta^{2n} = 1 - \frac{\theta^2}{2!}+\frac{\theta^4}{4!} -\frac{\theta^6}{6!}-.... \end{equation*}
Both expressions above are valid only when \(\theta\) is measured in radians. The small-angle approximation considers only the first nontrivial term in the sum.
\begin{equation} \sin(\theta) \approx \theta\tag{6.9.1} \end{equation}
\begin{equation} \cos(\theta) \approx 1 - \frac{\theta^2}{2!}\tag{6.9.2} \end{equation}
In Figure 6.9.1 you can see a graphical representation of the small-angle approximation for the \(\sin(\theta)\) function. You can see the linear function \(\theta\) and the trigonometric function \(\sin(\theta)\) closely match each other when the angle is small.
Figure 6.9.1. Graphical representation of the small-angle approximation for the \(\sin\theta\) function.

Exercises Activities

1. Analyzing the Equation of Motion.

Looking at equation (6.8.2) apply the small-angle approximation to determine an equation of motion. Then determine the oscillation frequency of the simple pendulum.
Answer.
Applying the small-angle approximation you find that \(\sin\theta \approx \theta\) and the equation of motion becomes,
\begin{equation} \frac{d^2}{dt^2}\theta(t)=-\frac{g}{L} \theta(t)\tag{6.9.3} \end{equation}
from this and your model of simple harmonic motion, you can directly see that the oscillation frequency is
\begin{equation} \omega_p = \sqrt{\frac{g}{L}}\text{.}\tag{6.9.4} \end{equation}

2. Sensemaking.

Make sense of the expression you found for the angular frequency of the simple pendulum using units, numbers, and symbols.

3. What is the error introduced?

Determine the error introduced by the small-angle approximation for \(\sin\theta\) and \(\cos\theta \) when \(\theta = 0.2, 0.4, 0.8 \) radian.

References References

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