Problem 46 - Algebra and Trigonometry

Question

A guitar string is pulled at point \(P\) a distance of 3 cm above its rest position. It is then released and vibrates in damped harmonic motion with a frequency of 165 cycles per second. After 2 s, it is observed that the amplitude of the vibration at point \(P\) is 0.6 cm. (a) Find the damping constant \(c\) . (b) Find an equation that describes the position of point \(P\) above its rest position as a function of time. Take \(t=0\) to be the instant that the string is released.

Step-by-Step Solution

Answer

\( c \approx 0.8047 \) and the position function is \( y(t) = 3 e^{-0.8047t} \cos(2\pi \times 165 t) \).

Step 1: Identify the Damped Harmonic Motion Formula

The position of point \( P \) as a function of time \( t \) can be described using the damped harmonic motion formula: \( y(t) = A e^{-ct} \cos(2\pi ft) \), where \( A \) is the initial amplitude, \( c \) is the damping constant, and \( f \) is the frequency.

Step 2: Understand Given Values

The initial amplitude \( A \) is 3 cm, the frequency \( f \) is 165 Hz, and after 2 seconds, the amplitude is reduced to 0.6 cm.

Step 3: Relate Amplitudes

Given that the amplitude decays exponentially, we use the relationship \( 0.6 = 3 e^{-2c} \). Solve for \( c \).

Step 4: Solve for Damping Constant \( c \)

Rearrange the equation from Step 3 to solve for \( c \): \( e^{-2c} = \frac{0.6}{3} = 0.2 \). Take the natural logarithm of both sides: \(-2c = \ln(0.2) \). Thus, \( c = -\frac{\ln(0.2)}{2} \).

Step 5: Calculate \( c \)

Calculate \( c \) using \( \ln(0.2) \approx -1.6094 \). Hence, \( c \approx -\frac{-1.6094}{2} \approx 0.8047 \).

Step 6: Write the Function for Position \( y(t) \)

Substitute the values into the damped harmonic motion equation from Step 1: \( y(t) = 3 e^{-0.8047t} \cos(2\pi \times 165 t) \). This describes the position of point \( P \) over time.

Key Concepts

Damping Constant

In the context of damped harmonic motion, the damping constant, denoted by \( c \), plays a crucial role in determining how quickly the amplitude of vibration decreases over time. This reflects the resistance against the motion. For example, in a guitar string, energy is lost due to factors like air resistance and internal friction, leading to damping.

The damping constant is calculated by considering the decay in amplitude over a specific period. As explained in the problem, the amplitude of the guitar string's vibration decreases from 3 cm to 0.6 cm in 2 seconds. The relationship connecting these changes involves exponential decay, boiled down to this equation:

• \( 0.6 = 3 e^{-2c} \)
• Rearrange this to find \( c \): \( e^{-2c} = 0.2 \)
• Take the natural logarithm: \(-2c = \ln(0.2) \)

The final step gives the damping constant \( c \approx 0.8047 \). This value indicates a medium rate of damping, reflecting the moderate slowing down of the vibration's amplitude over time.

Frequency

Frequency, denoted by \( f \), describes how many cycles of vibration occur per second. In simple terms, it's the speed at which the vibration repeats itself. For the guitar string, the frequency is given as 165 Hz. This tells us that the string undergoes 165 complete vibrations each second.

Frequency is a fundamental property of sound and vibration. It directly contributes to the pitch we hear when the string vibrates. In terms of the mathematical model for damped harmonic motion, frequency appears in the cosine term of the formula:

• Formula: \( y(t) = A e^{-ct} \cos(2\pi ft) \)
• The term \( 2\pi f \) corresponds to the angular frequency, \( \omega \).

This factor defines how the string cycles across its positions from peak to peak over time. Because the frequency remains constant (165 Hz), we know that any change in the sound or vibrational pattern is due to damping, not a change in tempo.

Amplitude

Amplitude measures the extent of a vibration or wave from its rest position. In the exercise, the initial amplitude of the guitar string is 3 cm, which means it was pulled 3 cm from its resting position before release.

Amplitude indicates the energy and intensity of the vibration. At the start, the string has high energy (3 cm). However, as the damping takes effect over time, the amplitude diminishes. This decay follows an exponential trend, as seen with the observed amplitude reducing to 0.6 cm after 2 seconds:

• The amplitude progression is captured by: \( A e^{-ct} \)
• Initial amplitude \( A \): 3 cm
• Observed amplitude after time \( t \): 0.6 cm

This shrinking amplitude mirrors the energy loss due to damping. Thus, the string produces softer sounds as the amplitude drops. Expressing this in the formula: \( y(t) = 3 e^{-c t} \cos(2\pi f t) \), highlights how the amplitude gradually lessens while frequency components remain unaltered.