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Chapter 3: Problem 10

(a) Confirm that the radius of gyration of a solid uniform sphere of radius \(R\) is \(r=\left(\frac{2}{5}\right)^{1 / 2} R .\) (b) What is the radius of gyration of a solid uniform cylinder of radius \(R\) and length \(l\) about an axis perpendicular to its principal axis? Hint. For part (a), the moment of incrtia of a sphere is \(^{2} / s M R^{2}\)

Short Answer

Expert verified
The radius of gyration of a solid uniform sphere is as given in the question, \(\left(\frac{2}{5}\right)^{1 / 2} R\). The radius of gyration of a solid uniform cylinder of radius \(R\) and length \(L\) about an axis perpendicular to its principal axis is \(\sqrt{\frac{\frac{1}{4}R^{2} + \frac{1}{12}L^{2}}{M}}.\)

Step by step solution

01

Confirming the Radius of Gyration for a Solid Sphere

The radius of gyration \(r\) for an object of moment of inertia \(I\) and mass \(M\) is given by \(r=\sqrt{\frac{I}{M}}\). The moment of inertia \(I\) for a solid sphere about its diameter is \(I=\frac{2}{5}MR^{2}\), which is given. Substituting this value into the formula: \(r=\sqrt{\frac{2}{5}R^{2}}=\left(\frac{2}{5}\right)^{1 / 2} R\). This confirms the given statement.
02

Finding the Moment of Inertia for a Solid Cylinder

The moment of inertia \(I\) for a solid cylinder of radius \(R\) and length \(L\) about an axis perpendicular to its principal axis is \(I = \frac{1}{4}M R^{2} + \frac{1}{12}M L^{2}\).
03

Finding the Radius of Gyration for a Solid Cylinder

Using the derived moment of inertia from Step 2, the radius of gyration \(r\) for the cylinder is calculated using the same formula as in Step 1: \(r =\sqrt{\frac{I}{M}}=\sqrt{\frac{\frac{1}{4}R^{2} + \frac{1}{12}L^{2}}{M}}\).

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Most popular questions from this chapter

(a) Construct a wavepacket \(\Psi=\mathrm{N} \sum_{m_{\mathrm{r}}=0}^{\infty}\left(1 / m_{l} !\right) \mathrm{e}^{\mathrm{in} \varphi}\) and normalize it to unity, Sketch the form of \(|\Psi|^{2}\) for \(0 \leq \varphi \leq 2 \pi\) (b) Calculate \(\langle\varphi\rangle,\langle\sin \varphi\rangle,\) and \(\left\langle l_{z}\right\rangle\) (c) Why is \(\left\langle l_{z}\right\rangle \leq \hbar ?\) Hint. Draw on a variety of pieces of information, including \(\sum_{n=0}^{\infty} x^{n} / n !=\mathrm{e}^{x}\) and the following integrals: $$\int_{0}^{2 \pi} \mathrm{e}^{\operatorname{scos} \phi} \mathrm{d} \varphi=2 \pi I_{0}(z) \int_{0}^{2 \pi} \cos \varphi \mathrm{e}^{\sec \varphi} \mathrm{d} \varphi=2 \pi I_{1}(z)$$ with \(I_{0}(2)=2.280 \ldots, I_{1}(2)=1.591 \ldots ;\) the \(I(z)\) are modificd Bessel functions.

Calculate the angle that the angular momentum vector makes with the \(z\) -axis when the system is described by the wavefunction \(\psi_{l m_{i}}\). Show that the minimum angle approaches zero as \(l\) approaches infinity. Calculate the allowed angles when \(l\) is \(1,2,\) and 3

Evaluate the probability that a particle of mass \(m\) in the ground state of a circular square well of radius \(a\) will be found within the circular area of radius \(1 / 2 a\)

Calculate (a) the mean radius, (b) the mean square radius, and (c) the most probable radius of the 1 s- \(, 2\) sand 3 s-orbitals of a hydrogenic atom of atomic number \(Z\) Hint. For the most probable radius look for the principal maximum of the radial distribution function. You will find the following integral useful: $$\int_{0}^{\infty} x^{n} \mathrm{c}^{-\alpha x} \mathrm{d} x=\frac{n !}{a^{n+1}}$$
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