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Consider the mass confined to the surface of a cone
Chapter 13, Problem 13.14(choose chapter or problem)
Consider the mass confined to the surface of a cone described in Example 13.4 (page 533). We saw there that there have to be maximum and minimum heights \(z_{\max }\) and \(z_{\min }\), beyond which the mass cannot stray. When z is a maximum or minimum, it must be that \(\dot{z}=0\). Show that this can happen if and only if the conjugate momentum \(p_{z}=0\), and use the equation \(\mathcal{H}=E\), where H is the Hamiltonian function (13.33), to show that, for a given energy E, this occurs at exactly two values of z. [Hint: Write down the function H for the case that \(p_{z}=0\) and sketch its behavior as a function of z for \(0<z<\infty\). How many times can this function equal any given E?] Use your sketch to describe the motion of the mass.
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QUESTION:
Consider the mass confined to the surface of a cone described in Example 13.4 (page 533). We saw there that there have to be maximum and minimum heights \(z_{\max }\) and \(z_{\min }\), beyond which the mass cannot stray. When z is a maximum or minimum, it must be that \(\dot{z}=0\). Show that this can happen if and only if the conjugate momentum \(p_{z}=0\), and use the equation \(\mathcal{H}=E\), where H is the Hamiltonian function (13.33), to show that, for a given energy E, this occurs at exactly two values of z. [Hint: Write down the function H for the case that \(p_{z}=0\) and sketch its behavior as a function of z for \(0<z<\infty\). How many times can this function equal any given E?] Use your sketch to describe the motion of the mass.
ANSWER:Step 1 of 2
The equation for conjugate momentum is given by,
\(\begin{array}{l} p_{z}=m\left(c^{2}+1\right) \dot{z}\\ \dot{z}=\frac{p_{z}}{m\left(I^{2}+1\right)} \end{array}\)
From the above two equations, the \(p_{z}\) can be zero if \(\dot{z}\) is zero and \(\dot{z}\) is zero when \(p_{z}\) is zero.
Therefore, \(\dot{z}=0\) if and only if \(p_{z}=0\).
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Review this written solution for 104800) viewed: 988 isbn: 9781891389221 | Classical Mechanics - 0 Edition - Chapter 13 - Problem 13.14
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