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Exercises

  1. A particle of mass $ m$ is attached to a rigid support by means of a spring of spring constant $ k$ . At equilibrium, the spring hangs vertically downward. An identical oscillator is added to this system, the spring of the former being attached to the mass of the latter. Calculate the normal frequencies for one-dimensional vertical oscillations about the equilibrium state, and describe the associated normal modes.

  2. Consider a mass-spring system of the general form shown in Figure 15 in which the two masses are of mass $ m$ , the two outer springs have spring constant $ k$ , and the middle spring has spring constant $ k'$ . Find the normal frequencies and normal modes in terms of $ \omega_0=\sqrt{k/m}$ and $ \alpha=k'/k$ .

  3. Consider a mass-spring system of the general form shown in Figure 15 in which the two masses are of mass $ m$ , the two leftmost springs have spring constant $ k$ , and the rightmost spring is absent. Find the normal frequencies and normal modes in terms of $ \omega_0=\sqrt{k/m}$ .

  4. Consider a mass-spring system of the general form shown in Figure 15 in which the springs all have spring constant $ k$ , and the left and right masses are of mass $ m$ and $ m'$ , respectively. Find the normal frequencies and normal modes in terms of $ \omega_0=\sqrt{k/m}$ and $ \alpha=m'/m$ .

    \begin{figure}
\epsfysize =1.3in
\centerline{\epsffile{Chapter03/fig05.eps}}
\end{figure}

  5. Find the normal frequencies and normal modes of the coupled LC circuit shown in the preceding figure terms of $ \omega_0=1/\sqrt{L\,C}$ and $ \alpha=L'/L$ .

  6. Consider two simple pendula with the same length, $ l$ , but different bob masses, $ m_1$ and $ m_2$ . Suppose that the pendula are connected by a spring of spring constant $ k$ . Let the spring be unextended when the two bobs are in their equilibrium positions. Demonstrate that the equations of motion of the system (for small amplitude oscillations) are

    $\displaystyle m_1\,\ddot{\theta}_1$ $\displaystyle =-m_1\,\frac{g}{l}\,\theta_1+ k\,(\theta_2-\theta_1),$    
    $\displaystyle m_2\,\ddot{\theta}_2$ $\displaystyle =-m_2\,\frac{g}{l}\,\theta_2+ k\,(\theta_1-\theta_2),$    

    where $ \theta_1$ and $ \theta_2$ are the angular displacements of the respective pendula from their equilibrium positions. Show that the normal coordinates are $ \eta_1=(m_1\,\theta_1+m_2\,\theta_2)/(m_1+m_2)$ and $ \eta_2=\theta_1-\theta_2$ . Find the normal frequencies and normal modes. Find a superposition of the two modes such that at $ t=0$ the two pendula are stationary, with $ \theta_1=\theta_0$ , and $ \theta_2=0$ .

  7. A linear triatomic molecule (e.g., carbon dioxide) consists of a central atom of mass $ M$ flanked by two identical atoms of mass $ m$ . The atomic bonds are represented as springs of spring constant $ k$ . Find the molecule's normal frequencies and modes of linear oscillation.

next up previous
Next: Transverse Standing Waves Up: Coupled Oscillations Previous: Three Spring-Coupled Masses
Richard Fitzpatrick 2013-04-08