Problem 36 - Algebra and Trigonometry

Question

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 equation 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

\( \theta(t) = 10^{\circ} \cos(\pi t + \frac{\pi}{2}) \)

Step 1: Understand Simple Harmonic Motion

The motion of the pendulum is periodic and can be described using the equation for simple harmonic motion: \( \theta(t) = \theta_{max} \cos(\omega t + \phi) \), where \( \theta_{max} \) is the maximum angle, \( \omega \) is the angular frequency, and \( \phi \) is the phase constant.

Step 2: Define the period and calculate the angular frequency

The period \( T \) of the pendulum is given as 2 seconds. The angular frequency \( \omega \) is related to the period by the equation \( \omega = \frac{2\pi}{T} \). Substituting the given period, \( \omega = \frac{2\pi}{2} = \pi \) radians per second.

Step 3: Determine the phase constant

The problem states that at \( t = 0 \), the pendulum is vertical, meaning the angle \( \theta \) is zero. In the equation \( \theta(t) = \theta_{max} \cos(\omega t + \phi) \), setting \( t = 0 \) gives \( \theta(0) = \theta_{max} \cos(\phi) = 0 \). Since \( \theta_{max} eq 0 \), the only solution is \( \cos(\phi) = 0 \), which occurs when \( \phi = \frac{\pi}{2} \).

Step 4: Write the final equation

Substitute the values into the general harmonic equation. With \( \theta_{max} = 10^{\circ} \), \( \omega = \pi \), and \( \phi = \frac{\pi}{2} \), the equation becomes \( \theta(t) = 10^{\circ} \cos(\pi t + \frac{\pi}{2}) \).

Key Concepts

Pendulum Motion

A pendulum is a weight suspended from a pivot that swings back and forth under the influence of gravity. In this scenario, the pendulum in a grandfather clock is being considered. The pendulum is key to keeping time accurate, as it moves in a consistent, periodic motion. This motion is typically known as simple harmonic motion, especially when displaced by small angles. The pendulum exhibits periodic motion, meaning it repeats its movement at regular intervals. For the grandfather clock pendulum, the complete swing (or period) occurs every 2 seconds. This is essential because the motion can be described using mathematical equations that incorporate physical principles and allow us to predict the pendulum's position at any point in time.

Angular Frequency

Angular frequency, denoted by \( \omega \), is a crucial factor in describing the pendulum's motion. It represents how many oscillations occur in a given period in radians per second. More technically, angular frequency is calculated as:\[ \omega = \frac{2\pi}{T} \] where \( T \) is the period. For the grandfather clock pendulum, the period \( T \) is 2 seconds. By substituting this value into the formula, we find the angular frequency:\[ \omega = \frac{2\pi}{2} = \pi \] This means the pendulum completes each oscillation with an angular frequency of \( \pi \) radians per second. Understanding \( \omega \) helps us model the pendulum's motion using trigonometric functions efficiently.

Phase Constant

The phase constant, \( \phi \), is an integral part of the harmonic motion equation. It determines the initial position of the pendulum at \( t=0 \). In our equation for pendulum motion, the phase constant adjusts the cosine function to match the pendulum's starting position. Given that the pendulum is vertical when \( t=0 \), the initial angle \( \theta \) is zero. Hence, in \( \theta(t) = \theta_{max} \cos(\omega t + \phi) \), we set \( \theta(0) = 0 \), leading to the equation:\[ \theta_{max} \cos(\phi) = 0 \] The only solution when \( \theta_{max} eq 0 \) is \( \cos(\phi) = 0 \). This condition is satisfied when \( \phi = \frac{\pi}{2} \), indicating that the pendulum starts from the point where its cosine value is zero.

Trigonometric Functions

Trigonometric functions, such as sine and cosine, are mathematical tools used to model periodic phenomena. In the context of the pendulum motion, the cosine function is invaluable to describe its angle over time. The general harmonic motion equation used for the pendulum is:\[ \theta(t) = \theta_{max} \cos(\omega t + \phi) \] Here, the cosine function accounts for the oscillating nature of the pendulum.

• \( \theta_{max} \) represents the maximum displacement from the rest position, in this case, \( 10^{\circ} \).
• \( \omega t + \phi \) embodies the phase and frequency of oscillation, defining the periodic pattern of the pendulum’s motion.

Through trigonometric functions, the motion of the pendulum can be mathematically described, illustrating behaviors that are predictable and constant over time.