Question

Verify that the decay parameter \(\beta-\sqrt{\beta^{2}-\omega_{0}^{2}}\) for an overdamped oscillator \(\left(\beta > \omega_{\mathrm{o}}\right) d e\) creases with increasing \(\beta\). Sketch its behavior for \(\omega_{0} < \beta < \infty\).

Short answer

The expression is decreasing for \(\beta > \omega_0\).

Step-by-step solution

Analyze the expression

The expression we need to verify for decreasing behavior is \(\beta - \sqrt{\beta^2 - \omega_0^2}\). To understand if it decreases with increasing \(\beta\), we need to explore how each component changes as \(\beta\) increases.

Differentiate the expression

Let's differentiate the expression \(\beta - \sqrt{\beta^2 - \omega_0^2}\) with respect to \(\beta\):1. Differentiate \(\beta\) to get \(1\).2. Differentiate \(-\sqrt{\beta^2 - \omega_0^2}\) using the chain rule. The derivative of \(\sqrt{u}\) where \(u = \beta^2 - \omega_0^2\) is \(\frac{1}{2\sqrt{u}}\cdot 2\beta = \frac{\beta}{\sqrt{\beta^2 - \omega_0^2}}\).So, the derivative of the whole expression is:\[1 - \frac{\beta}{\sqrt{\beta^2 - \omega_0^2}}\]

Determine the sign of the derivative

We need to check the sign of the derivative \(1 - \frac{\beta}{\sqrt{\beta^2 - \omega_0^2}}\). For a function to be decreasing, this derivative must be negative:\[1 - \frac{\beta}{\sqrt{\beta^2 - \omega_0^2}} < 0\]Rewriting this, we have:\[\beta > \sqrt{\beta^2 - \omega_0^2}\]Since \(\beta > \omega_0\), this inequality holds true, confirming that our original function is decreasing.

Sketch the behavior

Since the derivative is negative, \(\beta - \sqrt{\beta^2 - \omega_0^2}\) is a decreasing function for \(\beta > \omega_0\). As \(\beta\) approaches infinity, the expression approaches \(0\). The sketch would have a steep decrease starting near \(\omega_0\) and gradually leveling off as it approaches zero.

Key concepts

Decay Parameter

Understanding the decay parameter is critical when studying an overdamped oscillator. The decay parameter is expressed as \( \beta - \sqrt{\beta^2 - \omega_0^2} \). This parameter describes the rate at which the oscillations decay over time. For overdamped oscillators, this decay parameter helps us determine how the system returns to its equilibrium position without oscillating.
To examine how the decay parameter behaves, particularly its dependence on \( \beta \), it's key to see that \( \beta > \omega_0 \) in an overdamped system. With increasing \( \beta \), the decay parameter tends to decrease because each increment in \( \beta \) makes the square root term larger, thereby reducing the difference in the expression.
This gives insight into the system's behavior as the resistance in the damping force changes; the stronger the damping \( \beta \), the quicker the oscillations decay and the system stabilizes.

Differentiation

Differentiation is a fundamental concept in calculus, used to determine the rate of change of a function or its slope at any given point. In the context of the decay parameter, differentiating is essential to confirm whether it decreases as \( \beta \) increases. By differentiating \( \beta - \sqrt{\beta^2 - \omega_0^2} \) with respect to \( \beta \), we are performing a calculus operation that helps us understand how the decay parameter changes with changes in \( \beta \).
Here, differentiating \( \beta \) yields 1 immediately since the derivative of a linear term with respect to itself is just the coefficient. The more complex task is the differentiation of \( -\sqrt{\beta^2 - \omega_0^2} \), which involves applying the chain rule. By the end of this differentiation process, we obtain the expression \( 1 - \frac{\beta}{\sqrt{\beta^2 - \omega_0^2}} \). This result is crucial for the next step, which involves analyzing the derivative's sign.

Chain Rule

The chain rule is a rule in calculus used for computing the derivative of the composition of two or more functions. It's particularly useful when dealing with nested functions, like \( \sqrt{\beta^2 - \omega_0^2} \) in our case.
To differentiate \( -\sqrt{\beta^2 - \omega_0^2} \), we identify an outer function, \( \sqrt{u} \), and an inner function, \( u = \beta^2 - \omega_0^2 \). First, we differentiate the outer function with respect to \( u \), yielding \( \frac{1}{2\sqrt{u}} \). Then, we differentiate the inner function, \( \beta^2 - \omega_0^2 \), with respect to \( \beta \), which results in \( 2\beta \).
Applying the chain rule, we multiply these derivatives resulting in \( \frac{\beta}{\sqrt{\beta^2 - \omega_0^2}} \). The chain rule simplifies handling complex expressions by breaking them into simpler parts tackled individually, which in turn makes it clearer to see how each part contributes to the change of the whole.

Decreasing Function Analysis

Analyzing whether a function is decreasing involves studying the sign of its derivative. For the decay parameter \( \beta - \sqrt{\beta^2 - \omega_0^2} \), once differentiated, we end up with \( 1 - \frac{\beta}{\sqrt{\beta^2 - \omega_0^2}} \). To determine if this indicates a decrease, this derivative must be less than zero.
Thus, we need to verify that \( \beta > \sqrt{\beta^2 - \omega_0^2} \) is true. Given the constraint that \( \beta > \omega_0 \), this inequality holds. Therefore, the function is indeed decreasing for values of \( \beta \) greater than \( \omega_0 \).
In practical terms, as the value of \( \beta \) increases, the decay parameter decreases, meaning the rate of decay becomes less over time. This explains the slower approach to stability as damping increases, leading to a smoother transition to equilibrium.