The end of a 40 in pendulum describes an arc of 5 in. Through what angle does the pendulum swing

To find the angle \(\theta\) through which the pendulum swings, we can use the formula that relates the arc length \(s\) to the radius \(r\) and the angle \(\theta\) in radians:

\[
s = r\theta
\]

Here:
- \(s = 5\) inches (the arc length)
- \(r = 40\) inches (the length of the pendulum)

We can rearrange the formula to solve for \(\theta\):

\[
\theta = \frac{s}{r} = \frac{5}{40} = \frac{1}{8} \text{ radians}
\]

To convert radians to degrees, use the conversion factor \( \frac{180^\circ}{\pi} \):

\[
\theta = \frac{1}{8} \times \frac{180^\circ}{\pi} \approx \frac{180^\circ}{8\pi} \approx 7.16^\circ
\]

So, the pendulum swings through an angle of approximately \(7.16^\circ\).

Would you like more details or have any questions about this solution? Here are some related questions you might find interesting:

1. How would the angle change if the pendulum length were doubled?
2. What would be the arc length if the angle were \(10^\circ\)?
3. How would you find the angle if the arc length was given in centimeters?
4. What is the relationship between the pendulum length and the arc length for small angles?
5. How do you convert angles between degrees and radians?

**Tip:** When dealing with pendulum problems, small-angle approximations can simplify calculations if the angle is sufficiently small.

Learn how to calculate the angle through which a pendulum swings using the arc length of 5 inches and pendulum length of 40 inches. This problem involves applying the arc length formula and converting radians to degrees. Suitable for students in Grades 9-12.

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