Problem 50 - Algebra and Trigonometry with Analytic Geometry

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 through approximately 7.16 degrees.

Step 1: Convert arc length to feet

The pendulum arc length is given in inches. First, convert the 6-inch arc length to feet by dividing by 12 (since there are 12 inches in a foot): \( \frac{6}{12} = 0.5 \) feet.

Step 2: Calculate the radius

The length of the pendulum, which is the radius of the circle along which it swings, is given as 4 feet.

Step 3: Use the formula for arc length

The formula for the arc length \( s \) of a circle is \( s = r \theta \), where \( r \) is the radius and \( \theta \) is the angle in radians. We need to solve for \( \theta \): \( 0.5 = 4 \theta \).

Step 4: Solve for theta (angle in radians)

Rearrange the equation to solve for \( \theta \): \( \theta = \frac{0.5}{4} = 0.125 \) radians.

Step 5: Convert radians to degrees

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

Key Concepts

Pendulum dynamics

Pendulums are fascinating devices that beautifully illustrate the principles of motion in physics. Specifically, a pendulum is a weight suspended from a pivot so that it can swing freely. When displaced to an initial angle and released, the pendulum will swing back and forth due to the force of gravity, moving in a regular periodic fashion. This movement is a great example of simple harmonic motion. For a basic pendulum like the one in a grandfather clock, the length of the pendulum significantly influences its period, which is the time it takes to complete one full back and forth swing.
The period of a simple pendulum is determined by the formula: \[T = 2\pi \sqrt{\frac{L}{g}} \]where \( T \) is the period, \( L \) is the length of the pendulum, and \( g \) is the acceleration due to gravity (about 9.81 m/s² on Earth). The length of the pendulum, therefore, plays a crucial role in the dynamics of the pendulum's swing, controlling its speed and rhythm. Understanding this physical setup allows us to delve deeper into calculating movements such as the arc length traversed by the pendulum during its swing.

Arc length calculation

The arc length of a pendulum's swing is the distance along the curved path that the pendulum's bob travels. This arc length is part of a circle where the length of the pendulum acts as the radius. Knowing how to calculate this arc length allows us to determine how wide the pendulum swings.
The formula used to determine arc length (\( s \)) is a straightforward application of the relationship between the radius of a circle (\( r \)) and the angle (\( \theta \)) in radians:

• Arc Length: \( s = r \theta \)

In our example, with a pendulum length of 4 feet and an arc length of 0.5 feet, we can use this formula to determine that the angle the pendulum swings through (\( \theta \)) is 0.125 radians, by rearranging and solving the equation. Understanding the arc length allows us to turn the measurements of a pendulum's movement into a problem of angles and circles, setting us up for more advanced calculations like converting these angles into degrees.

Radian to degree conversion

Converting between radians and degrees is a common task in trigonometry, as angles can be measured in both units. A radian measures the radius' proportion along a circle's circumference and is the standard unit of angular measure used in many areas of mathematics.
The conversion formula from radians to degrees is derived from the fact that the circumference of a circle is \( 2\pi \) radians or 360 degrees. Therefore, 1 radian equals the fraction \( \frac{180}{\pi} \) degrees.
To convert an angle from radians to degrees, you multiply the angle by the conversion factor:

• Degrees: \( \text{angle in degrees} = \text{angle in radians} \times \frac{180}{\pi} \)

In the context of our pendulum problem, the angle of 0.125 radians is converted to approximately 7.16 degrees. This conversion is crucial for understanding and working with angles in various mathematical contexts, especially when translating real-world phenomena into calculations or predictions.