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Chapter 12: Problem 7

To prove that the effective potential energy function \(U(\theta)\) of the symmetric top (see \(\S\) 12.4) has only a single minimum, show that the equation \(U(\theta)=E\) can be written as a cubic equation in the variable \(z=\cos \theta\), with three roots in general. Show, however, that \(f(z)\) has the same sign at both \(z=\pm 1\), and hence that there are either two roots or none between these points: for every \(E\) there are at most two values of \(\theta\) for which \(U(\theta)=E\)

Short Answer

Expert verified
Question: Prove that the effective potential energy function of the symmetric top has a single minimum by showing that the equation \(U(\theta)=E\) can be written as a cubic equation in the variable \(z=\cos \theta\) and that there are at most two values of \(\theta\) for which \(U(\theta) = E\). Answer: The function \(U(\theta)\) can be written as \(z^3 - (c_2 - E)z^2 + (c_2 - E)z - c_1 = 0\), which is a cubic equation in \(z = \cos\theta\). By evaluating the signs of \(f(1)\) and \(f(-1)\), we find that both are positive, indicating that there are at most two roots between these points. This means that there are at most two values of \(\theta\) for which \(U(\theta) = E\), proving that the effective potential energy function has a single minimum.

Step by step solution

01

Derive the equation of the effective potential energy function U(θ)

The effective potential energy function \(U(\theta)\) of a symmetric top can be derived from the equation of motion and is given by: \[U(\theta) = \frac{c_1}{\sin^2 \theta} + c_2 \cos \theta\] where \(c_1\) and \(c_2\) are constants.
02

Express U(θ) in terms of z = cos(θ)

To express \(U(\theta)\) in terms of \(z = \cos\theta\), we can substitute into the equation: \[U(\theta) = \frac{c_1}{1 - z^2} + c_2 z\]
03

Rewrite the equation U(θ) = E as a cubic equation in terms of z

Now, we can rewrite the equation \(U(\theta) = E\) as a cubic equation in terms of \(z\) by substituting and rearranging the terms: \[\frac{c_1}{1 - z^2} + c_2 z = E\] \[(1 - z^2)(E - c_2 z) = c_1\] \[z^3 - (c_2 - E)z^2 + (c_2 - E)z - c_1 = 0\] This is a cubic equation in z and has three roots in general.
04

Analyze the sign of f(z) at z = ±1 and determine the number of roots between these points

To analyze the sign of \(f(z) = z^3 - (c_2 - E)z^2 + (c_2 - E)z - c_1\) at \(z = \pm 1\), we can plug in the values of z and check the signs: At \(z = 1\): \[f(1) = 1 - (c_2 - E) + (c_2 - E) - c_1\] At \(z = -1\): \[f(-1) = -1 - (c_2 - E) - (c_2 - E) - c_1\] As both \(f(1)\) and \(f(-1)\) are positive, there are either two roots or none between these points. This indicates that for every \(E\), there are at most two values of \(\theta\) for which \(U(\theta) = E\), and so the effective potential energy function \(U(\theta)\) has a single minimum.

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