Question

In one dimension, it is obvious that a force obeying Hooke's law is conservative (since \(F=-k x\) depends only on the position \(x,\) and this is sufficient to guarantee that \(F\) is conservative in one dimension). Consider instead a spring that obeys Hooke's law and has one end fixed at the origin, but whose other end is free to move in all three dimensions. (The spring could be fastened to a point in the ceiling and be supporting a bouncing mass \(m\) at its other end, for instance.) Write down the force \(\mathbf{F}(\mathbf{r})\) exerted by the spring in terms of its length \(r\) and its equilibrium length \(r_{\mathrm{o}} .\) Prove that this force is conservative. [Hints: Is the force central? Assume that the spring does not bend.]

Short answer

The force is conservative with potential energy \( U(r) = \frac{1}{2}k (r - r_o)^2 \).

Step-by-step solution

Understand Hooke's Law in One Dimension

Hooke's Law states that the force exerted by a spring is proportional to the displacement from its equilibrium position. In one dimension, this is given by the equation \( F = -k x \), where \( F \) is the force, \( k \) is the spring constant, and \( x \) is the displacement.

Extend to Three Dimensions

In three dimensions, the position vector \( \mathbf{r} \) represents the displacement of the spring's end from the origin, with its magnitude being the length of the spring \( r = ||\mathbf{r}|| \). The equilibrium length of the spring is \( r_o \). The force exerted by the spring is directed along the line of displacement and can be expressed as \( \mathbf{F}(\mathbf{r}) = -k \left( \frac{r - r_o}{r} \right) \mathbf{r} \).

Express the Force in Vector Form

Since \( \mathbf{F}(\mathbf{r}) = -k (r - r_o) \frac{\mathbf{r}}{r} \), this representation shows that the force is equal to the spring constant \( k \) times the difference in lengths \( (r - r_o) \), scaled by the unit vector \( \frac{\mathbf{r}}{r} \) ensuring the force direction aligns with the displacement.

Determine if the Force is Central

A central force is a force that is directed along the line connecting two points (in this case the origin and the point \( \mathbf{r} \)) and depends only on the distance between them, \( r \). Our force \( \mathbf{F} \) satisfies these conditions, hence it is a central force.

Prove the Force is Conservative Using Potential Energy

For a force to be conservative, it must be derivable from a potential energy function \( U \), such that \( \mathbf{F}(\mathbf{r}) = -abla U \). Integrate \( \mathbf{F}(\mathbf{r}) = -k(r - r_o) \frac{\mathbf{r}}{r} \) with respect to \( \mathbf{r} \) to find \( U(r) = \frac{1}{2}k (r - r_o)^2 \).

Conclusion on Conservativeness

Since \( U(r) \) exists and \( \mathbf{F}(\mathbf{r}) = -abla U \), \( \mathbf{F} \) is indeed a conservative force.

Key concepts

Hooke's Law

Hooke's Law is a principle of physics that states the force needed to extend or compress a spring by some distance is proportional to that distance. In simpler terms, it can be imagined as a spring trying to pull or push back to its original length when stretched or compressed. The mathematical expression for Hooke's Law in one dimension is \( F = -k x \), where:

• \( F \) is the force exerted by the spring, which is measured in Newtons (N).
• \( k \) is the spring constant, indicating the stiffness of the spring, measured in N/m.
• \( x \) is the displacement from the equilibrium position, measured in meters (m).

The negative sign in the formula signifies that the force exerted by the spring is in the opposite direction to the displacement. This opposition ensures that the force is always working to return the spring to its equilibrium state. When expanded into three dimensions, the principle remains the same. It still aims to return the system to equilibrium, while the direction and magnitude can be more complex due to vector quantities.

Central Force

A central force is defined as a force that is directed along the line joining two points. It only depends on the distance between these two points, and not on their specific positions. In the context of a spring that extends in three dimensions, the force behaves as a central force, because it focuses on the relative position and distance between the origin (the fixed point of the spring) and the end of the spring.
This can be shown mathematically by expressing the force vector, \( \mathbf{F}(\mathbf{r}) \), as \( -k (r - r_o) \frac{\mathbf{r}}{r} \). This formula shows:

• The force acts directly along the radial direction from the origin.
• The magnitude of the force is dependent solely on the difference in lengths \( (r - r_o) \).

Thus, the force remains central, indicating it constantly points towards or away from the center depending on whether the spring is compressed or stretched.

Potential Energy

Potential energy associated with a force is a form of energy stored within a system. For forces like those described by Hooke's Law, determining potential energy helps in confirming whether these forces are conservative. A force is considered conservative if it can be derived from a potential energy function, \( U \), such that the force can be obtained by the negative gradient of \( U \).
In our system, the potential energy function is given by: \[ U(r) = \frac{1}{2}k (r - r_o)^2 \]This function represents:

• \( k \), the spring constant, revealing how stiff or soft the spring is.
• \( r \), the current length of the spring.
• \( r_o \), the natural or equilibrium length of the spring.

The squared term \((r - r_o)^2\) highlights that any displacement from equilibrium results in stored potential energy, either through compression or extension of the spring. This potential energy characterizes the system's ability to do work, i.e., returning the spring to its equilibrium position. By integrating the force with respect to \( \mathbf{r} \), we observe that it mirrors the change in potential energy, confirming the force's conservative nature. Such forces, including those governed by Hooke's Law, conserve mechanical energy within a system.