Problem 23 - Applied Physics

Question

A pendulum of length \(1.50 \mathrm{~m}\) swings through an arc of \(5.0^{\circ} .\) Find the length of the arc through which the pendulum swings.

Step-by-Step Solution

Answer

The arc length is approximately \(0.13 \text{ m}\).

Step 1: Understanding the Problem

We need to find the length of the arc through which a pendulum swings. We have the length of the pendulum (radius of the circle) as \(1.50\, m\) and the angle of swing as \(5.0^{\circ}\). The arc length \(s\) can be calculated using the formula \(s = r \theta\) where \(r\) is the radius and \(\theta\) is the angle in radians.

Step 2: Convert Degrees to Radians

First, we need to convert the angle from degrees to radians since the formula \(s = r \theta\) requires the angle in radians. We use the conversion factor \(\frac{\pi}{180}\) rad/degree:\[\theta = 5.0^{\circ} \times \frac{\pi}{180^{\circ}} = \frac{5\pi}{180} = \frac{\pi}{36} \, \text{radians}\]

Step 3: Calculate the Arc Length

Now, we can use the arc length formula \(s = r \theta\):\[s = 1.50 \, \text{m} \times \frac{\pi}{36} = \frac{1.50\pi}{36} \, \text{m}\]

Step 4: Simplify the Result

Simplify the expression to find the final arc length:\[s \approx \frac{4.712}{36} \, \text{m} \approx 0.1309 \, \text{m}\]

Key Concepts

Arc Length Calculation

The concept of arc length is essential in understanding circular motion and pendulum mechanics. In a circle, the arc length is the distance measured along the curved line forming a part of the circumference. When calculating the arc length for a pendulum, we consider a section of the circle's circumference that the pendulum's weight would trace as it swings.

The formula for arc length \(s\) is given by:

• \(s = r \theta\)

Here, \(r\) represents the radius of the circle, and \(\theta\) is the angle in radians. For pendulum problems, \(r\) would be the length of the pendulum, as it describes the radius of the circular path traced by the pendulum's bob. Arc length tells us how far the pendulum travels across its path, a fundamental measure in analyzing motion in physics. Understanding how to apply the arc length formula helps in solving many real-world problems involving circular motion.

Radians Conversion

Radians are a unit of angular measure used commonly in many areas of mathematics and physics. Unlike degrees, which divide a circle into 360 portions, radians relate an angle directly to the circle's radius. This makes them very useful for calculations involving circles, such as with the arc length formula.
The conversion between degrees and radians is crucial since many calculus-based formulas require angles to be expressed in radians. The conversion uses the relation:

• 1 radian \( = \frac{180}{\pi}\) degrees

To convert an angle from degrees to radians:

• Multiply the degree measure by \(\frac{\pi}{180}\)

In the current exercise, an angle of \(5.0^{\circ}\) was converted to radians by multiplying by \(\frac{\pi}{180}\), resulting in \(\frac{\pi}{36}\) radians. This conversion enabled the proper application of the formula \(s = r \theta\), highlighting the importance of employing the correct units in trigonometry-related problems.

Trigonometry in Physics

Trigonometry plays a vital role in understanding phenomena in physics, especially in analyzing systems like pendulums, waves, and oscillations. The pendulum serves as a classic example where trigonometry is essential to determine distances and angles during its swinging motion. In physics, the relationship between different sides and angles in right-angled triangles helps describe various aspects of motion and force. While dealing with pendulums, trigonometry supports the calculation of quantities such as:

• Displacement
• Velocity
• Acceleration

For the pendulum, the length of the pendulum (or radius of swing) and the angle it swings through are used to find the arc length, a straightforward application of trigonometry. Besides calculating distances or lengths, trigonometry also helps in resolving vector components, essential in describing forces and directions of motion in physics.