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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

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

Step by step solution

01

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.
02

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 \( \mathbf{F}(\mathbf{r}) = -k \left( \frac{r - r_o}{r} \right) \mathbf{r} \).
03

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.
04

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.
05

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 \( U(r) = \frac{1}{2}k (r - r_o)^2 \).
06

Conclusion on Conservativeness

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

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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.

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Most popular questions from this chapter

A charge \(q\) in a uniform electric field \(\mathbf{E}_{0}\) experiences a constant force \(\mathbf{F}=q \mathbf{E}_{0}\). (a) Show that this force is conservative and verify that the potential energy of the charge at position \(\mathbf{r}\) is \(U(\mathbf{r})=-q \mathbf{E}_{0} \cdot \mathbf{r}\). (b) By doing the necessary derivatives, check that \(\mathbf{F}=-\nabla U\).

(a) The force exerted by a one-dimensional spring, fixed at one end, is \(F=-k x,\) where \(x\) is the displacement of the other end from its equilibrium position. Assuming that this force is conservative (which it is) show that the corresponding potential energy is \(U=\frac{1}{2} k x^{2},\) if we choose \(U\) to be zero at the equilibrium position. (b) Suppose that this spring is hung vertically from the ceiling with a mass \(m\) suspended from the other end and constrained to move in the vertical direction only. Find the extension \(x_{\mathrm{o}}\) of the new equilibrium position with the suspended mass. Show that the total potential energy (spring plus gravity) has the same form \(\frac{1}{2} k y^{2}\) if we use the coordinate \(y\) equal to the displacement measured from the new equilibrium position at \(x=x_{\mathrm{o}}\) (and redefine our reference point so that \(U=0\) at \(y=0\) ).

Calculate the gradient \(\nabla f\) of the following functions, \(f(x, y, z):(\mathbf{a}) f=\ln (r),\) (b) \(f=r^{n}\), (c) \(f=g(r),\) where \(r=\sqrt{x^{2}+y^{2}+z^{2}}\) and \(g(r)\) is some unspecified function of \(r .\) [Hint: Use the chain rule.]

Prove that if \(f(\mathbf{r})\) and \(g(\mathbf{r})\) are any two scalar functions of \(\mathbf{r},\) then \(\nabla(f g)=f \nabla g+g \nabla f\)

The proof that the condition \(\nabla \times \mathbf{F}=0\) guarantees the path independence of the work \(\int_{1}^{2} \mathbf{F} \cdot d \mathbf{r}\) done by \(\mathbf{F}\) is unfortunately too lengthy to be included here. However, the following three exercises capture the main points: \(^{16}\) (a) Show that the path independence of \(\int_{1}^{2} \mathbf{F} \cdot d \mathbf{r}\) is equivalent to the statement that the integral \(\oint_{\mathrm{T}} \mathbf{F} \cdot d \mathbf{r}\) around any closed path \(\Gamma\) is zero. (By tradition, the symbol \(\oint\) is used for integrals around a closed path \(-\) a path that starts and stops at the same point.) [Hint: For any two points 1 and 2 and any two paths from 1 to 2 , consider the work done by \(\mathbf{F}\) going from 1 to 2 along the first path and then back to 1 along the second in the reverse direction. \((\) b) Stokes's theorem asserts that \(\oint_{\mathrm{T}} \mathbf{F} \cdot d \mathbf{r}=\int(\nabla \times \mathbf{F}) \cdot \hat{\mathbf{n}} d A,\) where the integral on the right is a surface integral over a surface for which the path \(\Gamma\) is the boundary, and \(\hat{\mathbf{n}}\) and \(d A\) are a unit normal to the surface and an element of area. Show that Stokes's theorem implies that if \(\nabla \times \mathbf{F}=0\) everywhere, then \(\oint_{\mathrm{T}} \mathbf{F} \cdot d \mathbf{r}=0 .\) (c) While the general proof of Stokes's theorem is beyond our scope here, the following special case is quite easy to prove (and is an important step toward the general proof): Let \(\Gamma\) denote a rectangular closed path lying in a plane perpendicular to the \(z\) direction and bounded by the lines \(x=B, x=B+b, y=C\) and \(y=C+c .\) For this simple path (traced counterclockwise as seen from above), prove Strokes's theorem that \(\oint_{\Gamma} \mathbf{F} \cdot d \mathbf{r}=\int(\mathbf{\nabla} \times \mathbf{F}) \cdot \hat{\mathbf{n}} d A\) where \(\hat{\mathbf{n}}=\hat{\mathbf{z}}\) and the integral on the right runs over the flat, rectangular area inside \(\Gamma\). [Hint: The integral on the left contains four terms, two of which are integrals over \(x\) and two over \(y\). If you pair them in this way, you can combine each pair into a single integral with an integrand of the form \(F_{x}(x, C+c, z)-F_{x}(x, C, z)\) (or a similar term with the roles of \(x\) and \(y\) exchanged). You can rewrite this integrand as an integral over \(y\) of \(\partial F_{x}(x, y, z) / \partial y\) (and similarly with the other term), and you're home.]

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