Problem 47 - Algebra and Trigonometry

Question

Tuning Fork 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

Answer

The damping constant \(c\) is approximately 0.4621.

Step 1: Understand the Problem

We need to find the damping constant \(c\) of a tuning fork whose amplitude decreases over time in a damped harmonic motion. We know that after 3 seconds, the amplitude is \(\frac{1}{4}\) of the initial amplitude.

Step 2: Write the Amplitude Decay Equation

The amplitude \(A(t)\) of a damped harmonic oscillator is given 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.

Step 3: Set Up the Equation at t = 3

We know that \(A(3) = \frac{1}{4} A_0\). Substituting into the amplitude decay equation gives: \( \frac{1}{4} A_0 = A_0 e^{-3c}\).

Step 4: Simplify and Solve for c

Cancel \(A_0\) from both sides: \( \frac{1}{4} = e^{-3c} \). Take the natural logarithm on both sides to solve for \(c\): \(\ln\left(\frac{1}{4}\right) = -3c\).

Step 5: Calculate the Damping Constant

Rearrange the equation \(\ln\left(\frac{1}{4}\right) = -3c\) to find \(c\): divide both sides by \(-3\), \( c = -\frac{1}{3} \ln\left(\frac{1}{4}\right) \).

Step 6: Simplify the Logarithmic Expression

Knowing that \(\ln\left(\frac{1}{4}\right) = -\ln(4)\), we can substitute into our expression for \(c\): \(c = \frac{1}{3} \ln(4)\).

Step 7: Final Calculation

Calculate \(\ln(4)\) and then multiply by \(\frac{1}{3}\) to find \(c\). \(\ln(4) \approx 1.3863\), so \(c \approx \frac{1}{3} \times 1.3863 = 0.4621\).

Key Concepts

Understanding the Damping Constant

The damping constant, often denoted by the symbol \(c\), is a crucial parameter in the study of damped harmonic motion. This constant is used to describe how quickly the energy of an oscillating system, such as a tuning fork, dissipates over time due to the resistance or damping forces at play. The idea is that in the real world, no oscillating system remains in perpetual motion because of these damping forces, such as air resistance or internal friction.

When calculating the damping constant, understanding the amplitude's behavior is key. The equation \(A(t) = A_0 e^{-ct}\) encapsulates this relationship, where \(A(t)\) is the amplitude after time \(t\), and \(A_0\) is the initial amplitude. As time progresses, the value of \(e^{-ct}\) decreases, indicating that the amplitude diminishes. This exponential decrease characterizes damped harmonic motion and directs us to the damping constant, \(c\), which quantifies how fast the amplitude diminishes.

• The higher the damping constant, the quicker the amplitude decreases.
• A smaller damping constant indicates slower decay and more sustained oscillations.

Amplitude Decay in Damped Motion

Amplitude decay refers to the reduction in the peak value of the oscillation of a system undergoing damped harmonic motion. Over time, the maximum extent of the oscillation, or amplitude, gradually diminishes as energy is lost to damping forces.

The decay of amplitude in a damped system is often modeled using the equation \(A(t) = A_0 e^{-ct}\), where \(A(t)\) is the amplitude at time \(t\), and \(A_0\) is the initial amplitude. The equation suggests that amplitude decay follows an exponential pattern, which is characterized by the natural base \(e\).

• At the initial moment (\(t = 0\)), the amplitude is \(A_0\).
• As time increases, the exponential term \(e^{-ct}\) decreases, leading to a reduction in amplitude.
• This decay continues until external energy is added to the system or the system comes to rest.

Role of the Natural Logarithm in Solving Damped Motion Problems

The natural logarithm, denoted as \( \ln\), plays an essential role in solving equations related to damped harmonic motion. It helps in decoupling the exponential part of the equation, allowing us to isolate the damping constant \(c\).

In the context of our problem, when the amplitude at a certain time is set using the equation \(A(t) = A_0 e^{-ct}\), we end up with an exponential equation in terms of \(c\). To find \(c\), we take the natural logarithm of both sides. This process transforms the equation into a linear one, making it easier to solve.

• Taking \(\ln\) of an exponential expression like \(e^x\) simplifies it to \(x\).
• In our problem, we used \(\ln\left(\frac{1}{4}\right) = -3c\) to solve for \(c\), demonstrating this method's power.
• The natural logarithm efficiently extracts \(c\) from its exponential enclosure, simplifying the process of finding the damping constant.