Problem 38 - Precalculus Mathematics for Calculus

Question

Clock Pendulum The pendulum in a grandfather clock makes one complete swing every 2 s. The maximum angle that the pendulum makes with respect to its rest position is \(10^{\circ} .\) We know from physical principles that the angle \(\theta\) between the pendulum and its rest position changes in simple harmonic fashion. Find an cquation that describes the size of the angle \(\theta\) as a function of time. (Take \(t=0\) to be a time when the pendulum is vertical.)

Step-by-Step Solution

Answer

The angle is given by \( \theta(t) = 10^{\circ} \cos(\pi t) \).

Step 1: Identify the Simple Harmonic Motion

The pendulum follows simple harmonic motion (SHM), given by \[ \theta(t) = \theta_0 \cos(\omega t + \phi) \] where \( \theta_0 \) is the maximum angle, \( \omega \) is the angular frequency, and \( \phi \) is the phase angle.

Step 2: Determine the Maximum Angle

The problem states that the maximum angle \( \theta_0 \) is \( 10^{\circ} \). This is the amplitude of the SHM.

Step 3: Calculate the Angular Frequency

For simple harmonic motion, the angular frequency \( \omega \) is related to the period \( T \) by \[ \omega = \frac{2\pi}{T} \] Given that \( T = 2 \) seconds, we have \[ \omega = \frac{2\pi}{2} = \pi \] \text{ radians per second.}

Step 4: Determine the Phase Angle

Since \( t=0 \) corresponds to the pendulum in the vertical position, which is the maximum angle position in the cosine function, the phase angle \( \phi \) is \( 0 \).

Step 5: Assemble the Equation

Plug the values obtained into the SHM equation: \( \theta(t) = 10^{\circ} \cos(\pi t) \) This describes the angle as a function of time where \( \theta \) is measured in degrees.

Key Concepts

Pendulum Motion

A pendulum is a classic physics example that demonstrates simple harmonic motion (SHM). It involves a weight suspended from a pivot so it can swing freely. When displaced from its equilibrium, or rest position, the restoring force that brings it back is gravity. This simple device exhibits oscillatory motion, where the pendulum swings back and forth in a periodic manner.

Key aspects of pendulum motion include:

• Periodic movement: The pendulum swings through its arc in a consistent time period.
• Rest position: The vertical point where the pendulum naturally comes to rest.
• Maximum displacement: The highest point in the swing, which the pendulum reaches on either side of the rest position.

Understanding pendulum motion helps illustrate the general principles of SHM, where objects move back and forth through an equilibrium point. Unlike chaotic motion, this type of movement is predictable and can be described using mathematical models.

Angular Frequency

Angular frequency is a fundamental concept in describing the motion of a pendulum undergoing SHM. It tells us how quickly the pendulum oscillates through its cycles.

In simple harmonic motion, angular frequency \( \omega \) links directly to the motion's period \( T = 2 \text{ seconds} \), using the formula:\[ \omega = \frac{2\pi}{T} \]For our pendulum, we calculate:\[ \omega = \frac{2\pi}{2} = \pi \text{ radians per second} \]Angular frequency is crucial because it helps define how swiftly the pendulum swings from side to side. It represents the oscillation speed, independent of time, and indicates how many radians the pendulum moves per second. By understanding angular frequency, one can predict how the motion behaves, making it a key part of harmonic motion analysis.

Cosine Function

The cosine function plays a central role in describing the pendulum's motion mathematically. It helps model the oscillation's smooth and continuous nature. The SHM for a pendulum involves a cosine function because the motion repeats every cycle.

The general equation for simple harmonic motion is:\[ \theta(t) = \theta_0 \cos(\omega t + \phi) \]In our pendulum problem, this simplifies to:\[ \theta(t) = 10^{\circ} \cos(\pi t) \]This equation includes:

• \( \theta_0 \): Maximum angle (amplitude) - \( 10^{\circ} \)
• \( \omega \): Angular frequency - \( \pi \) radians per second
• \( \phi \): Phase angle - \( 0 \)

The cosine function perfectly describes the pendulum’s path because it naturally represents oscillations and phases, capturing their cyclical characteristics. As time progresses, the cosine value transitions between -1 and 1, reflecting the pendulum’s swinging extent.

Maximum Angle

The maximum angle, often termed the amplitude in the context of SHM, signifies the farthest angle the pendulum reaches away from its equilibrium position. In this scenario, the pendulum's amplitude is identified as \( 10^{\circ} \).

The maximum angle is crucial because it determines the extent of the pendulum's swing and influences energy dynamics in the motion. High amplitudes would lead to a broader swinging arc, while smaller amplitudes mean a narrower swing. Measuring it from the equilibrium (center) position provides a clear boundary to the pendulum's movement.

Understanding the maximum angle helps students see how it forms a fundamental part of the mathematical description for SHM. By knowing this value, you can calculate other parameters such as velocity and acceleration throughout the pendulum's path, offering a comprehensive understanding of its dynamics.