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 3.1 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 3.1 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 3.1 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 $\alpha=m'/m$.

   Figure 3.7: Figure for Exercise 5.

   

5. Find the normal frequencies and normal modes of the coupled LC circuit shown in Figure 3.7 in 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.
   1. Demonstrate that the equations of motion of the system (for small amplitude oscillations) are

      $$m_1\,\skew{3}\ddot{\theta}_1 =-m_1\,\frac{g}{l}\,\theta_1+ k\,(\theta_2-\theta_1),$$

      $$m_2\,\skew{3}\ddot{\theta}_2 =-m_2\,\frac{g}{l}\,\theta_2+ k\,(\theta_1-\theta_2),$$

      where $\theta_1$ and $\theta_2$ are the angular displacements of the respective pendula from their equilibrium positions.

   2. 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.

   3. 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. Two masses, $m_1$ and $m_2$, slide over a horizontal frictionless surface, and are connected via a spring of force constant $k$. Mass $m_1$ is acted on by a horizontal force $F_0\,\cos(\omega\,t)$. In the absence of the second mass, this force causes the first mass to execute simple harmonic motion of amplitude $F_0/(m_1\,\omega^{\,2})$. Find an appropriate choice of the combination of values $m_2$ and $k$ that reduces the oscillation amplitude of $m_1$ as much as possible. What is the oscillation amplitude of $m_2$ in this case?

8. 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.

9. Consider the mass-spring system discussed in Section 3.2.
   1. Show that, when written in terms of the physical coordinates, the total energy of the system takes the form
      $$E = m\left[\frac{1}{2}\left(\dot{x}_1^{\,2}+\dot{x}_2^{\,2}\right) + \omega_0^{\,2}\left(x_1^{\,2}-x_1\,x_2+x_2^{\,2}\right)\right].$$

   2. Furthermore, show that the total energy takes the form
      $$E= m\left[\left(\dot{\eta}_1^{\,2}+\dot{\eta}_2^{\,2}\right) + \omega_0^{\,2}\left(\eta_1^{\,2}+3\,\eta_2^{\,2}\right)\right]$$
      when expressed in terms of the normal coordinates.

   3. Hence, deduce that

      $$E = m\left({\cal E}_1+{\cal E}_2\right),$$

      $${\cal E}_1 = \dot{\eta}_1^{\,2}+ \omega_0^{\,2}\,\eta_1^{\,2},$$

      $${\cal E}_2 =\dot{\eta}_2^{\,2} + 3\,\omega_0^{\,2}\,\eta_2^{\,2},$$

      $$\frac{d{\cal E}_1}{dt} =0,$$

      $$\frac{d{\cal E}_2}{dt} =0.$$

      Here, ${\cal E}_1$ and ${\cal E}_2$ are the separately conserved energies per unit masses of the first and second normal modes, respectively.