Problem 85 - University Physics with Modern Physics

Question

A Hooke's law force \(-k x\) and a constant conservative force \(F\) in the \(+x\) -direction act on an atomic ion. (a) Show that a possible potential-energy function for this combination of forces is \(U(x)=\frac{1}{2} k x^{2}-F x-F^{2} / 2 k\) . Is this the only possible function? Explain. (b) Find the stable equilibrium position. (c) Graph \(U(x)\) (in units of \(F^{2} / k )\) versus \(x\) (in units of \(F / k )\) for values of \(x\) between \(-5 F / k\) and 5\(F / k\) . (d) Are there any unstable equilibrium positions? (e) If the total energy is \(E=F^{2} | k,\) what are the maximum and minimum valnes of \(x\) that the ion reaches in its motion? If the ion has mass \(m,\) find its maximum speed if the total energy is \(E=F^{2} / k .\) For what value of \(x\) is the speed maximum?

Step-by-Step Solution

Answer

(a) The function is one possible option; constant shift possible. (b) Stable position: \( x = F/k \). (c) Graph: Parabola from \(-5\) to \(5\). (d) No unstable position. (e) Max: \(4F/k\), Min: \(-2F/k\), Max speed: \(\sqrt{4F^2/km}\) at \(F/k\).

Step 1: Determine the Potential Energy Function

We know the force is the negative gradient of the potential energy, given by \( F(x) = - \frac{dU}{dx} \).Combining both forces, we have \(-kx + F = - \frac{dU}{dx} \). Integrating this equation with respect to \( x \), we find the potential energy function by separating the terms:\[U(x) = \int (-kx + F) \, dx = -\frac{k}{2}x^2 + Fx + C,\]where \( C \) is the integration constant. We can set \( C \) such that \( C = -\frac{F^2}{2k} \) to match the given expression, hence:\[U(x) = \frac{1}{2}kx^2 - Fx - \frac{F^2}{2k}.\]This is a possible function, but other functions differing by a constant are also valid since potentials are defined up to an arbitrary constant.

Step 2: Find the Stable Equilibrium Position

To find the stable equilibrium position, we need the force to be zero:\(-kx + F = 0.\)Solving for \(x\), we get \[ x = \frac{F}{k}. \]This is the stable equilibrium position because the potential energy function is a parabolic function opening upwards.

Step 3: Graph the Potential Energy Function

We want to graph \(U(x)\) in units of \(F^2/k \) versus \(x\) in units of \(F/k\) between \( -5F/k \) and \( 5F/k \). Let \( u = \frac{U(x)}{F^2/k} \) and \( x' = \frac{x}{F/k} \), so\[ U(x) = \frac{1}{2}\left(\frac{kF^2}{k^2} \cdot (x')^2\right) - \left( F \cdot \frac{F}{k} \cdot x' \right) - \frac{F^2}{2k},\]which simplifies to \[u(x') = \frac{1}{2}(x')^2 - x' - \frac{1}{2}.\]Now plot \( u(x') \) for \(-5 \leq x' \leq 5\).

Step 4: Identify Unstable Equilibrium Positions

An unstable equilibrium occurs where the second derivative of \(U(x)\) is positive. Calculate \[ \frac{d^2U}{dx^2} = k. \]Since \(k > 0\), no position in this parabolic function corresponds to an unstable position because the curvature is never downward.

Step 5: Determine Maximum and Minimum x-values

For total energy \( E = F^2/k \), the potential energy at the turning points satisfies\[ \frac{1}{2}kx^2 - Fx - \frac{F^2}{2k} = \frac{F^2}{k}.\]Simplifying yields:\[ \frac{1}{2}kx^2 - Fx - \frac{3F^2}{2k} = 0.\]This quadratic equation in \(x\) gives maximum and minimum \(x\) values upon solving:\[ x = \frac{F \pm \sqrt{9F^2}}{k} = \frac{F \pm 3F}{k}, \]resulting in \( x = \frac{4F}{k} \) or \( x = -\frac{2F}{k}. \)

Step 6: Find Maximum Speed and Its Location

Maximum speed occurs where potential energy is minimum, i.e., kinetic energy is maximum.Using conservation of energy:\[ \frac{1}{2}mv_{max}^2 = E - U(x_{eq}),\]where \(x_{eq} = F/k\) is the equilibrium position and\(U(x_{eq}) = -F^2/(2k)\).Thus,\[ E - U(x_{eq}) = \frac{2F^2}{k},\]\[ \frac{1}{2}mv_{max}^2 = \frac{2F^2}{k},\]hence\[ v_{max} = \sqrt{\frac{4F^2}{km}}.\]The speed is maximum at \( x = F/k \), where potential energy is minimum.

Key Concepts

Potential Energy Function

Potential Energy is an essential concept when it comes to understanding forces and motions. The potential energy function describes how the stored energy in a system changes as the system's configuration changes. For a Hooke's law spring and a constant force combination, the potential energy function can be found by integrating the overall force. The force equation is given by the sum of the forces, \(-kx + F = - \frac{dU}{dx} \), where \(k\) is the spring constant, and \(F\) is the constant force.

Upon integrating this force with respect to position \(x\), we obtain the potential energy function:\[U(x) = \frac{1}{2}kx^2 - Fx - \frac{F^2}{2k}.\]This function provides details on how potential energy changes for different positions \(x\). One important aspect to note is that potential energy functions are defined up to an arbitrary constant, meaning this isn't the only potential energy function possible, but rather a common form for clarity in calculations. Integration constants are adjusted to align the function with energy boundary conditions or specific scenarios.

Equilibrium Position

In physics, the equilibrium position is where the sum of forces on an object is zero, indicating a stable or balanced state. For systems governed by Hooke's Law and additional forces, equilibrium is where the system naturally settles without external disturbances.

To find the stable equilibrium position for the given potential energy function, we set the derivative (force) to zero: \(-kx + F = 0\). Solving this equation yields \[x = \frac{F}{k}.\]This is the point where the system reaches stability because any small movement away from this point will be counteracted by the spring force attempting to restore balance. In this scenario, the stable equilibrium position corresponds to a minimum in the potential energy curve, which resembles a parabola opening upwards.

Force Analysis

Understanding the combined effects of forces helps to predict the behavior of objects within a system. With Hooke's Law, the force on an object is proportional to its displacement and acts in the opposite direction. This is represented as \(-kx\) for a spring force. Additionally, a constant conservative force \(F\) in the positive direction adds to the overall force.

Analyzing these forces helps derive potential energy functions and solve equilibrium equations. When combining both forces without external influences, they create a predictable pattern within the system governed by their sum, \( -kx + F \). This approach is fundamental for predicting the dynamics and stability in physics problems, demonstrating how diverse forces interact through direct addition to exert influence on a system's motion or static condition.

Kinetic Energy

Kinetic Energy is the energy of motion. When an object moves, it possesses kinetic energy proportional to its mass and the square of its velocity. The expression for kinetic energy is \(\frac{1}{2} mv^2\), where \(m\) is mass and \(v\) is the velocity.

In the context of Hooke’s Law forces and potential energy, kinetic energy becomes maximum when potential energy is at its minimum due to energy conservation. Given total energy \(E\) is constant, any drop in potential energy results in increased kinetic energy:\[\frac{1}{2}mv_{max}^2 = E - U(x_{eq}).\]By finding the maximum speed at \(x = \frac{F}{k}\), where potential energy reaches its lowest, the motion dynamics can be better understood. This allows for prediction of speeds and motion trajectories under specific initial energy conditions.

Conservation of Energy

The principle of Conservation of Energy states that the total energy in an isolated system remains constant, though it can change forms. This principle is crucial when analyzing systems like those described by Hooke's Law, where potential energy and kinetic energy exchange roles during motion.

In a system influenced by conservative forces, potential and kinetic energy sum up to a constant total energy \(E\). Determining the range and behavior of an object within an energy-conserving system entails ensuring that the potential energy function aligns with total energy:\[\frac{1}{2}kx^2 - Fx - \frac{F^2}{2k} = E.\]Through these relationships, we can find conditions for maximum and minimum positions (turning points) and determine motion characteristics like speed and direction at different points within the system. Energy is neither created nor destroyed; it is merely transferred between energy states for informative predictions and analyses.