Problem 45
Question
A tuning fork is struck and oscillates in damped harmonic motion. The amplitude of the motion is measured, and 3 s later it is found that the amplitude has dropped to \(\frac{1}{4}\) of this value. Find the damping constant \(c\) for this tuning fork.
Step-by-Step Solution
Verified Answer
The damping constant is \(c = \frac{2}{3} \ln(2)\).
1Step 1: Understanding Damped Harmonic Motion
In damped harmonic motion, the amplitude of vibration decreases exponentially over time. The displacement, or amplitude, at any given time \(t\) can be described by the equation \(A(t) = A_0 e^{-ct}\), where \(A_0\) is the initial amplitude, \(c\) is the damping constant, and \(t\) is time in seconds.
2Step 2: Set Up the Equation with Given Values
According to the problem, after 3 seconds, the amplitude \(A(t)\) becomes \(\frac{1}{4}A_0\). This means we can set up the equation \(\frac{1}{4}A_0 = A_0 e^{-3c}\).
3Step 3: Cancel Out the Initial Amplitude
Since \(A_0\) appears on both sides of the equation \(\frac{1}{4}A_0 = A_0 e^{-3c}\), we can divide through by \(A_0\) to obtain \(\frac{1}{4} = e^{-3c}\).
4Step 4: Solve for the Exponent
To solve \(\frac{1}{4} = e^{-3c}\), we take the natural logarithm on both sides, giving us \(\ln\left(\frac{1}{4}\right) = -3c\). Then, solve for \(c\) by dividing both sides by \(-3\).
5Step 5: Calculate the Damping Constant
Calculate \(c\) using the expression: \[ c = -\frac{1}{3} \ln\left(\frac{1}{4}\right) \]\(\ln(\frac{1}{4})\) is equal to \(-2\ln(2)\), hence\[ c = \frac{2}{3} \ln(2) \].
Key Concepts
Damping ConstantExponential DecayNatural Logarithm
Damping Constant
The damping constant is a crucial factor in understanding damped harmonic motion. It represents how quickly the amplitude of an oscillating system decreases over time. In simpler terms, it measures the resistance against the oscillation, such as friction or air resistance that slows down the system.
To see how it features in calculations, let's consider a tuning fork. When struck, it vibrates, producing sound. However, over time, the sound fades due to damping. The greater the damping constant, the quicker this fade occurs, resulting in shorter sound times.
In the context of the given exercise, the amplitude decreases to \( \frac{1}{4} \) of its original in 3 seconds. By knowing the equation \( A(t) = A_0 e^{-ct} \, \) we can calculate the damping constant by using logarithm techniques, ultimately showing how resistance levels affect vibration longevity.
To see how it features in calculations, let's consider a tuning fork. When struck, it vibrates, producing sound. However, over time, the sound fades due to damping. The greater the damping constant, the quicker this fade occurs, resulting in shorter sound times.
In the context of the given exercise, the amplitude decreases to \( \frac{1}{4} \) of its original in 3 seconds. By knowing the equation \( A(t) = A_0 e^{-ct} \, \) we can calculate the damping constant by using logarithm techniques, ultimately showing how resistance levels affect vibration longevity.
Exponential Decay
Exponential decay is a mathematical concept where a quantity reduces over time at a rate proportional to its current value. This is commonly seen in natural phenomena, like radioactive decay or the cooling of a hot object. In the case of damped harmonic motion, the amplitude of the motion diminishes exponentially.
In our exercise, the amplitude becomes exponentially smaller as time progresses. Mathematically, this is described as \( A(t) = A_0 e^{-ct} \), where \( A(t) \) is the amplitude at any time \( t \, \ A_0 \) is the initial amplitude, and \( c \) is the damping constant.
Due to exponential decay, after 3 seconds, the tuning fork's amplitude drops to only \( \frac{1}{4} \) of its initial level. This rapid reduction demonstrates how damping creates a quick drop-off in sound volume, connecting the mathematical concept of exponential decay to real-world observations.
In our exercise, the amplitude becomes exponentially smaller as time progresses. Mathematically, this is described as \( A(t) = A_0 e^{-ct} \), where \( A(t) \) is the amplitude at any time \( t \, \ A_0 \) is the initial amplitude, and \( c \) is the damping constant.
Due to exponential decay, after 3 seconds, the tuning fork's amplitude drops to only \( \frac{1}{4} \) of its initial level. This rapid reduction demonstrates how damping creates a quick drop-off in sound volume, connecting the mathematical concept of exponential decay to real-world observations.
Natural Logarithm
The natural logarithm, denoted as \( \ln \, \) is a logarithm to the base of the mathematical constant \( e \, \) approximately equal to 2.718. It is a powerful tool to solve equations involving exponential functions. In damped harmonic motion, natural logarithms are particularly useful in determining damping constants.
In the exercise, to find out the damping constant \( c \,\) we manipulate the equation \( \frac{1}{4} = e^{-3c} \) by taking the natural logarithm of both sides. This step converts the exponential equation into a linear form, making it solvable:
In the exercise, to find out the damping constant \( c \,\) we manipulate the equation \( \frac{1}{4} = e^{-3c} \) by taking the natural logarithm of both sides. This step converts the exponential equation into a linear form, making it solvable:
- Apply logarithms: \( \ln(\frac{1}{4}) = -3c \).
- Rearrange to solve for \( c \): \( c = -\frac{1}{3} \ln(\frac{1}{4}) \).
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