Question

Two particles whose reduced mass is \(\mu\) interact via a potential energy \(U=\frac{1}{2} k r^{2},\) where \(r\) is the distance between them. (a) Make a sketch showing \(U(r),\) the centrifugal potential energy \(U_{\mathrm{cf}}(r)\) and the effective potential energy \(U_{\text {eff }}(r) .\) (Treat the angular momentum \(\ell\) as a known, fixed constant.) (b) Find the "equilibrium" separation \(r_{\mathrm{o}},\) the distance at which the two particles can circle each other with constant \(r .\left[\text { Hint: This requires that } d U_{\text {eff }} / d r \text { be zero. }\right]\left(\text { c) By making a Taylor expansion of } U_{\text {eff }}(r)\right.\) about the equilibrium point \(r_{\mathrm{o}}\) and neglecting all terms in \(\left(r-r_{\mathrm{o}}\right)^{3}\) and higher, find the frequency of small oscillations about the circular orbit if the particles are disturbed a little from the separation \(r_{\mathrm{o}}.\)

Short answer

The equilibrium separation is \( r_{o} = \left(\frac{\ell^2}{\mu k}\right)^{1/4} \), and to find the frequency, use \( \omega = \sqrt{3\frac{\ell^2}{\mu r_o^4} + k} \/ \mu \).

Step-by-step solution

Understand the Problem

We have two charged particles interacting via a potential energy model given by \[ U(r) = \frac{1}{2} k r^{2} \] where \( k \) is a constant and \( r \) is the distance between the particles. Additional aspects to consider are the centrifugal potential energy \( U_{cf}(r) \) and the effective potential energy \( U_{eff}(r) \). We need to sketch these potentials, find the equilibrium separation \( r_o \), and determine the frequency of small oscillations around this equilibrium point.

Identify the Expression for Centrifugal Potential

The centrifugal potential energy is a function of the angular momentum \( \ell \) of the system and is given by \[ U_{cf}(r) = \frac{\ell^2}{2 \mu r^2} \]. This term arises due to the rotation of the system, and it modifies the total potential energy experienced by the system.

Combine Potentials to Find Effective Potential

The effective potential energy combines both the interaction potential and the centrifugal potential: \[ U_{eff}(r) = \frac{\ell^2}{2 \mu r^2} + \frac{1}{2} k r^{2}. \] This function allows us to understand the effective forces acting on the system.

Sketch the Potentials

Draw a graph of the three potential energies as functions of \( r \):1. \( U(r) = \frac{1}{2} k r^2 \) is a parabola starting at the origin.2. \( U_{cf}(r) = \frac{\ell^2}{2 \mu r^2} \) tends to infinity as \( r \rightarrow 0 \) and approaches zero as \( r \rightarrow \infty \).3. \( U_{eff}(r) \) is a combination showing a dip due to the competing effects of attraction (spring) and repulsion (centrifugal).

Find the Equilibrium Separation

At equilibrium, the derivative \( \frac{dU_{eff}}{dr} \) must be zero:\[ \frac{dU_{eff}}{dr} = -\frac{\ell^2}{\mu r^3} + kr = 0. \]Solving for \( r \) gives the equilibrium separation:\[ r_{o} = \left(\frac{\ell^2}{\mu k}\right)^{1/4}. \]

Taylor Expansion around Equilibrium

Perform a Taylor expansion of \( U_{eff}(r) \) about \( r_{o} \). The expansion is:\[ U_{eff}(r) \approx U_{eff}(r_o) + \frac{1}{2} U_{eff}''(r_o) (r - r_o)^2. \]Neglecting higher-order terms, we approximate the potential near \( r_o \) as harmonic.

Calculate Frequency of Small Oscillations

The frequency of oscillations \( \omega \) can be related to the second derivative of \( U_{eff} \):\[ \omega = \sqrt{\frac{U_{eff}''(r_{o})}{\mu}}. \]Calculate \( U_{eff}''(r) = \frac{3\ell^2}{\mu r^4} + k \) and evaluate at \( r = r_{o} \) to find \( \omega \).

Key concepts

Effective Potential Energy

In classical mechanics, effective potential energy is the total interaction energy experienced by a system when considering both direct interactions and rotational dynamics. For two particles orbiting each other, the effective potential combines the force from their mutual potential energy with the force linked to their rotational motion due to angular momentum.
The effective potential energy, which we denote as \( U_{eff}(r) \), is composed of two main components: the interaction potential energy and the centrifugal potential energy.
Mathematically, it is expressed as:

• The interaction term or potential energy: \( U(r) = \frac{1}{2} kr^2 \), representing a harmonic interaction.
• The rotational contribution or centrifugal term: \( U_{cf}(r) = \frac{\ell^2}{2 \mu r^2} \), which accounts for the system's angular momentum \( \ell \).
• Therefore, the effective potential energy is given by: \[ U_{eff}(r) = \frac{\ell^2}{2 \mu r^2} + \frac{1}{2} k r^{2}. \]

This function helps us visualize the forces in the system, particularly highlighting points of stable motion.

Centrifugal Potential Energy

Centrifugal potential energy arises from the rotating frame of a system where two bodies circle each other. It is not an actual force acting outward, but from a rotating reference frame, it appears as if there is a force pushing objects away from the center of rotation.
This potential depends on the angular momentum \( \ell \) and the distance \( r \) between the rotating bodies, expressed by:\[ U_{cf}(r) = \frac{\ell^2}{2 \mu r^2}. \]

• As \( r \to 0 \), the centrifugal potential tends to infinity, indicating a strong resistance to close proximity due to rotational effects.
• As \( r \to \infty \), \( U_{cf}(r) \to 0 \), showing that the centrifugal impact diminishes with increasing separation.

Understanding this term is crucial for combining it with other potential energies to assess the full dynamics of a system.

Equilibrium Separation

The concept of equilibrium separation refers to the stable point where two particles revolve around each other without any change in the distance between them. This equilibrium is established when the effective potential energy's derivative with respect to radius \( r \) is zero.
Applying this to the effective potential:\[ \frac{dU_{eff}}{dr} = -\frac{\ell^2}{\mu r^3} + kr = 0. \]

• From this equation, solving for \( r \) gives the equilibrium separation \( r_{o} \):
• \[ r_{o} = \left(\frac{\ell^2}{\mu k}\right)^{1/4}. \]
• At \( r = r_{o} \), competing forces balance, allowing stable circular motion.

At equilibrium, particles are not pulled closer or pushed farther apart purely from their combined potential and kinetic energies.

Small Oscillations

Small oscillations in a system describe tiny deviations from an equilibrium configuration. If a particle in a system experiences a slight perturbation, it will oscillate about the equilibrium position. The motion in this context is harmonic and is typically analyzed using a Taylor expansion.
By capturing this small oscillation behavior, we understand the system's stability and response to perturbations.
For our system's effective potential energy:

• We perform a Taylor expansion around the equilibrium point \( r_{o} \).
• Express \( U_{eff}(r) \) as \( U_{eff}(r_o) + \frac{1}{2} U_{eff}''(r_o) (r - r_o)^2 \).
• This approximation reveals an effective harmonic potential, simplifying calculations of oscillation frequencies.

The resulting motion provides insight into a system's resilience against small disturbances and potential energy changes.

Taylor Expansion

Taylor expansion is a mathematical technique used to approximate a function around a specific point by converting it into a series.
This is especially useful in dealing with small deviations and understanding system behavior around equilibrium points.
For the problem at hand, the effective potential energy function is expanded around the equilibrium separation \( r_{o} \):

• The expansion is written as \( U_{eff}(r) \approx U_{eff}(r_o) + \frac{1}{2} U_{eff}''(r_o) (r - r_o)^2 \).
• By ignoring higher-order terms, this provides an approximation of \( U_{eff}(r) \) similar to the equation of a harmonic oscillator.
• This makes calculating oscillation frequencies straightforward using the simplified quadratic form.

Taylor expansion thereby allows us to convert complex interactions into simpler, predictable forms, making them valuable for practical analysis.