Problem 50 - Precalculus: Functions and Graphs

Question

Pendulum's swing A pendulum in a grandfather clock is 4 feet long and swings back and forth along a 6 -inch arc. Approximate the angle (in degrees) through which the pendulum passes during one swing.

Step-by-Step Solution

Answer

The pendulum swings approximately 7.16 degrees.

Step 1: Understanding the Setup

The problem involves a pendulum with a string length of 4 feet, which swings along a 6-inch arc. Our goal is to determine the angle in degrees through which this pendulum swings.

Step 2: Convert Units for Consistency

Since the pendulum length is given in feet and the arc length in inches, we first convert the arc length to feet for consistency. We use the fact that 1 foot = 12 inches, so the arc length is \( \frac{6}{12} = 0.5 \) feet.

Step 3: Recall and Rearrange the Formula

The arc length formula for a circle is \( s = r \theta \), where \( s \) is the arc length, \( r \) is the radius, and \( \theta \) is the angle in radians. Rearrange this to find \( \theta \): \( \theta = \frac{s}{r} \).

Step 4: Calculate the Angle in Radians

Substitute the known values into the formula: \( \theta = \frac{0.5}{4} = 0.125 \) radians. This computes the angle in radians through which the pendulum swings.

Step 5: Convert from Radians to Degrees

To convert radians to degrees, use the conversion factor \( 1 \, \text{radian} = \frac{180}{\pi} \, \text{degrees} \). Thus, \( \theta = 0.125 \times \frac{180}{\pi} \approx 7.16 \) degrees.

Key Concepts

Arc Length Formula

To analyze a pendulum's swing, we use the arc length formula. This formula relates the arc length of a circle segment to the radius and angle of that segment. The formula is as follows:

• \( s = r \theta \)

Here, \(s\) represents the arc length, \(r\) is the radius (or the pendulum length in this case), and \(\theta\) is the angle in radians. This formula is invaluable in pendulum problems to find the angle through which a pendulum swings. By rearranging it, we get \( \theta = \frac{s}{r} \). This allows us to solve for the angle when the arc length and radius are known. Proper application of this formula is crucial in analyzing the geometry of pendulum motion.

Radians to Degrees Conversion

Radians and degrees are two ways to measure angles. When dealing with a pendulum's swing, angles are often initially calculated in radians. However, they are more commonly expressed in degrees for easier interpretation. The conversion process is simple:

• Multiply the angle in radians by \( \frac{180}{\pi} \) to convert it to degrees.

For example, when an angle is calculated to be 0.125 radians, we convert it to degrees as follows:

• 0.125 radians \(\times \frac{180}{\pi} \approx 7.16 \) degrees.

This conversion helps to present results in a format that is more intuitive to understand and communicate.

Unit Conversion

When solving problems involving a pendulum's swing, unit consistency is crucial. In our problem, the pendulum's length is given in feet, while the arc length is initially in inches. To proceed with calculations, these units need to be consistent:

• 1 foot = 12 inches, so to convert inches to feet, divide by 12.

Therefore, a 6-inch arc becomes \( \frac{6}{12} = 0.5 \) feet. This change allows us to use the arc length formula effectively because all variables are in the same unit system. Such conversions are vital when applying mathematical formulas since mixing units can lead to incorrect results.

Circle Geometry

Understanding circle geometry is essential when analyzing pendulum swings. A pendulum's motion can be visualized as a segment of a circle. The length of the pendulum acts as the radius of this circle, and the path the pendulum follows is the arc.

• The center of the circle is where the pendulum is attached.
• The arc describes the path the pendulum follows.

Understanding this geometric relationship lets us apply the arc length formula, given the circle's radius (the pendulum length) and the arc's measurement (the distance swung). With this knowledge, solving for the swinging angle becomes straightforward and intuitive.