Question

The force on a mass \(m\) at position \(x\) on the \(x\) axis is \(F=-F_{0} \sinh \alpha x,\) where \(F_{0}\) and \(\alpha\) are constants. Find the potential energy \(U(x),\) and give an approximation for \(U(x)\) suitable for small oscillations. What is the angular frequency of such oscillations?

Short answer

Potential energy: \( U(x) = \frac{F_0}{\alpha} \cosh(\alpha x) \). For small oscillations: \( U(x) \approx U(0) + \frac{F_0 \alpha x^2}{2} \). Angular frequency: \( \omega = \sqrt{\frac{F_0 \alpha}{m}} \).

Step-by-step solution

Understand the relationship between force and potential energy

The potential energy function, \( U(x) \), is related to the force \( F(x) \) by the negative derivative with respect to position. Mathematically, this is expressed as: \[ F(x) = -\frac{dU}{dx} \]Given that \( F = -F_0 \sinh(\alpha x) \), we substitute to set up an equation for \( U(x) \): \[ -F_0 \sinh(\alpha x) = -\frac{dU}{dx} \]

Integrate to find the potential energy function

To find \( U(x) \), integrate the expression for \( F(x) \):\[ \frac{dU}{dx} = F_0 \sinh(\alpha x) \]Integrating both sides with respect to \( x \), we get:\[ U(x) = \int F_0 \sinh(\alpha x) \, dx = \frac{F_0}{\alpha} \cosh(\alpha x) + C \]Where \( C \) is the integration constant. For simplicity, let's assume \( C = 0 \), as the reference potential can be set to any constant.

Approximate potential energy for small oscillations

For small oscillations around \( x = 0 \), the hyperbolic cosine can be approximated by its Taylor expansion:\[ \cosh(x) \approx 1 + \frac{x^2}{2} \]Thus, \( \cosh(\alpha x) \approx 1 + \frac{(\alpha x)^2}{2} \).So, the potential energy function for small \( x \) is:\[ U(x) \approx \frac{F_0}{\alpha} \left( 1 + \frac{\alpha^2 x^2}{2} \right) \approx U(0) + \frac{F_0 \alpha x^2}{2} \]

Find the angular frequency of oscillations

The potential can be written in the form of a harmonic potential \( \frac{1}{2} kx^2 \), where \( k = F_0 \alpha \).The angular frequency \( \omega \) for a harmonic oscillator is given by:\[ \omega = \sqrt{\frac{k}{m}} \]Substitute \( k = F_0 \alpha \) to get:\[ \omega = \sqrt{\frac{F_0 \alpha}{m}} \]

Key concepts

Force and Potential Energy

Force and potential energy are fundamentally related in classical mechanics. Here, the force on an object at position \(x\) is given by \( F = -F_0 \sinh(\alpha x) \). The negative sign indicates that the force is a restoring force, which animates motion back to equilibrium.

The relationship between force and potential energy \(U(x)\) can be understood through the equation \( F = -\frac{dU}{dx} \). This implies that the force is the derivative of the potential energy concerning displacement, but with a negative sign. This relationship allows us to determine the potential energy from a known force.

• The force \(F\) is a measure of how much the potential energy changes with position.
• If \(F\) is negative, the potential energy increases with increasing \(x\), signaling a stable, bound state.

By integrating \( F = -F_0 \sinh(\alpha x) \), we find the potential energy function, \( U(x) \), to be \( U(x) = \frac{F_0}{\alpha} \cosh(\alpha x) + C \). The constant \(C\) is often set to zero, simplifying further calculations.

Small Oscillations

In many physics problems, especially those dealing with small oscillations, the system's behavior can be significantly simplified. For small displacements from equilibrium, complex forces and potentials can often be approximated as linear or quadratic.

In this problem, the potential energy for small oscillations near \(x=0\) requires an approximation of \(\cosh(\alpha x)\) using the Taylor series expansion: \(\cosh(x) \approx 1 + \frac{x^2}{2}\) for very small \(x\). Applying this approximation, \(\cosh(\alpha x) \approx 1 + \frac{(\alpha x)^2}{2}\), simplifies the potential energy to:

• \( U(x) \approx U(0) + \frac{F_0 \alpha x^2}{2} \)

This is similar to a quadratic potential, typical for simple harmonic oscillators.
The assumption of small oscillations is significant because it leads to simpler dynamics described by linear differential equations, making predictions and calculations more manageable.

Angular Frequency

Angular frequency \(\omega\) is a key concept when analyzing oscillatory motion. It describes how many radians an oscillator completes per unit time, similar to how frequency describes oscillations per second.

In the context of harmonic oscillations, we derive \(\omega\) using the effective spring constant \(k\) from the potential energy approximation. For our problem, this implies a potential energy of form \(\frac{1}{2}kx^2\). Matching this to \(U(x) \approx \frac{F_0 \alpha x^2}{2}\) suggests \(k = F_0 \alpha\).

• Angular frequency \(\omega\) for a harmonic oscillator is given by \(\omega = \sqrt{\frac{k}{m}}\).
• Substituting \(k\) gives \(\omega = \sqrt{\frac{F_0 \alpha}{m}}\).

This frequency is crucial for understanding how quickly and under what conditions the system returns to equilibrium during small oscillations. Understanding \(\omega\) allows us to predict the time behavior and rhythms of various physical systems in mechanics.