Problem 2 - Algebra and Trigonometry

Question

For an object in damped harmonic motion with initial amplitude \(k\) , period \(2 \pi / \omega,\) and damping constant \(c\) , find an equation that models the displacement \(y\) at time \(t\) if (a) \(y=0\) at time \(t=0 : y=\)________ (b) \(y=a\) at time \(t=0 : y=\) _______

Step-by-Step Solution

Answer

(a) \(y(t) = k e^{-ct/2} \sin(\omega t)\); (b) \(y(t) = k e^{-ct/2} \left( \cos(\omega t) \frac{a}{k} - \sin(\omega t) \sqrt{1 - \left(\frac{a}{k}\right)^2} \right) \)."

Step 1: Understanding Damped Harmonic Motion

Damped harmonic motion can be described with an equation of the form \( y(t) = Ae^{-ct/2} \cos(\omega t + \phi) \), where \( A \) is the initial amplitude, \( c \) is the damping constant, and \( \omega \) is the angular frequency related to the period by \( \omega = \frac{2\pi}{T} \). The term \( \phi \) represents the phase angle.

Step 2: Case (a): Initial Displacement is Zero

For case (a), where the initial displacement \( y=0 \) at \( t=0 \), we need to determine the conditions that lead to this result. Solving \( y(0) = A e^{0} \cos(\phi) = 0 \) gives us \( \cos(\phi) = 0 \). Thus, \( \phi = \frac{\pi}{2} + n\pi \) for any integer \( n \). A suitable choice is \( \phi = \frac{\pi}{2} \). We assume \( A=k \), so the equation becomes: \[ y(t) = k e^{-ct/2} \sin(\omega t) \].

Step 3: Case (b): Initial Displacement is a

For case (b), where the initial displacement \( y=a \) at \( t=0 \), solve \( y(0) = A e^{0} \cos(\phi) = a \). This leads to \( A \cos(\phi) = a \). Assume \( A = k \) and so \( \cos(\phi) = \frac{a}{k} \). The equation is then given as \[ y(t) = k e^{-ct/2} \left( \cos(\omega t) \frac{a}{k} - \sin(\omega t) \sqrt{1 - \left(\frac{a}{k}\right)^2} \right) \].

Key Concepts

Displacement Equation in Damped Harmonic Motion

In the context of damped harmonic motion, the displacement equation is a mathematical expression that describes how the position of a moving object changes over time. This type of motion combines oscillation with a gradual reduction in amplitude due to a damping effect. The general form of the displacement equation is written as:
\[ y(t) = A e^{-ct/2} \cos(\omega t + \phi) \]
Here, \( y(t) \) is the displacement at time \( t \), \( A \) is the initial amplitude, \( c \) is the damping constant, \( \omega \) is the angular frequency, and \( \phi \) is the phase angle.

• The term \( e^{-ct/2} \) represents the damping effect, leading to a decreasing amplitude over time.
• \( \cos(\omega t + \phi) \) captures the cyclical nature of the motion.

The formula is versatile and can be adjusted by substituting specific values for different initial conditions, such as cases where the initial displacement is known.

Understanding Harmonic Motion

Harmonic motion refers to a type of periodic oscillation where an object moves back and forth around a central position. It's commonly observed in systems where the force acting on the object is proportional to the displacement from its equilibrium position, such as springs or pendulums.
In the context of damped harmonic motion:

• The periodic motion gradually decreases over time due to external forces like friction or air resistance.
• This results in an oscillation that becomes less severe, eventually coming to rest.

The key to harmonic motion's predictability lies in its cyclical nature, characterized by a consistent period and frequency. The inclusion of damping modifies this to account for real-world scenarios where energy is lost over time.

Role of Damping Constant

The damping constant, denoted \( c \), is a critical parameter in the displacement equation. It quantifies the rate at which energy is lost in the system due to damping forces. This could be due to friction, air resistance, or another dissipating force.

• A larger \( c \) value implies a stronger damping effect, which means the oscillations die out more quickly.
• A smaller \( c \) results in slower energy loss, allowing the motion to persist longer.

Understanding the damping constant is essential for predicting how quickly an object in damped harmonic motion will come to rest. It significantly impacts both the amplitude and the lifecycle of the oscillations in the system.

Significance of Initial Amplitude

The initial amplitude, represented by \( A \), is the maximum extent of displacement from the rest position when time \( t = 0 \). It provides a baseline for how far the object initially moves during its oscillation.

• If \( A = k \), then the motion begins with this amplitude, which subsequently decreases due to damping.
• The larger the initial amplitude, the more energy the system initially possesses.

Initial amplitude is significant because it helps determine the initial energy of the system and influences how the system behaves over time.
In practice, solving specific scenarios requires integrating the initial amplitude with other variables like the damping constant to model the real-life motion accurately.