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Chapter 10: Problem 1

\( \mathrm{What}\) is the moment of inertia of (a) a solid disc of mass \(m,\) radius \(R,\) about its axis, (b) a solid sphere of mass \(m,\) radius \(R,\) about its centre?

Short Answer

Expert verified
The moment of inertia of a solid disc of mass \(m,\) radius \(R,\) about its axis is \(0.5 * m * R^2\), and the moment of inertia of a solid sphere of mass \(m,\) radius \(R,\) about its centre is \(0.4 * m * R^2\).

Step by step solution

01

Moment of Inertia of Disc

Using the formula for the moment of inertia of a solid disc, \(I = 0.5 * m * R^2\), where \(m\) is the mass of the disc and \(R\) is its radius, substitute the given values of \(m\) and \(R\) into the formula and calculate the moment of inertia.
02

Moment of Inertia of Sphere

The formula for the moment of inertia of a solid sphere is \(I = 0.4 * m * R^2\). Substitute the given values of \(m\) and \(R\) into the formula and calculate the moment of inertia.

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Most popular questions from this chapter

The \(J+1 \leftarrow J\) rotational transitions of \(^{16} \mathrm{O}^{12} \mathrm{C}^{32} \mathrm{S}\) and \(^{16} \mathrm{O}^{12} \mathrm{C}^{34} \mathrm{S}\) occur at the following frequencies \((v / \mathrm{GHz})\) $$\begin{array}{lllll} \hline J & 1 & 2 & 3 & 4 \\ \hline^{16} \mathrm{O}^{12} \mathrm{C}^{32} \mathrm{S} & 24.32592 & 36.48882 & 48.65164 & 60.81408 \\ ^{16} \mathrm{O}^{12} \mathrm{C}^{34} \mathrm{S} & 23.73223 & & 47.46240 & \\ \hline \end{array}$$ Find (a) the rotational constants, (b) the moments of inertia, and (c) the CS and CO bond lengths. Hint. Begin by finding expressions for the moment of inertia \(I\) through \(I=m_{\Lambda} R_{A}^{2}+m_{B} R_{B}^{2}+m_{C} R_{C}^{2},\) where \(R_{X}\) is the distance of atom \(\mathrm{X}\) from the centre of mass. The easiest procedure is to use the result established in Exercise \(10.6,\) which leads to \(I=\left(m_{\mathrm{A}} m_{\mathrm{C}} / m\right)\left(R_{\mathrm{AB}}+R_{\mathrm{BC}}\right)^{2}+\left(m_{\mathrm{B}} / m\right)\left(m_{\mathrm{A}} R_{\mathrm{AB}}^{2}+m_{\mathrm{C}} R_{\mathrm{BC}}^{2}\right)\) The lengths \(R_{\text {AB }}\) and \(R_{\mathrm{BC}}\) may be found only if two values of \(I\) are known. Assume the bond lengths are the same in isotopomeric molecules.

\( \mathrm{An}\) infrared spectrum of gaseous DCl revealed lines at the following wavenumbers for the lowest four transitions from the \(v=0\) state: 2091,4128,6111,8043 Determine the spectroscopic constants \(\omega\) and \(\omega x_{e^{-}}\).

Identify the conditions for the existence and locations of heads in the \(P\) - and \(R\) -branches of a diatomic molecule.

A diatomic molecule for which \(\tilde{v}=4401.2 \mathrm{cm}^{-1}\) and \(\bar{B}=121.3 \mathrm{cm}^{-1}\) is initially in the state \((\nu=1, J=2)\) In a Raman experiment utilizing \(15873.0 \mathrm{cm}^{-1}\) incident radiation, determine the wavenumber of the scattered radiation for (a) the Q-branch Stokes line, (b) the O-branch Stokes line, (c) the Q-branch anti-Stokes line. How will the wavenumber computed in part (a) change if the effects of anharmonicity are included?

Find expressions for the moments of inertia of an \(\mathrm{AB}_{3}\) molecule that is (a) planar, (b) trigonal pyramidal.
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