Chapter 5, Elementary oscillations Video Solutions, From calculus to chaos: an introduction to dynamics

Problem 1

The damped linear oscillator. Use (3.31) to show that the general solution to $(5.14)$ is
$$
x=C \mathrm{e}^{-k t / 2} \cos \left\{\left(\omega^2-\frac{1}{4} k^2\right)^{1 / 2} t-D\right\},
$$
provided $\frac{1}{4} k^2<\omega^2$.
Confirm the main features of these decaying oscillations with the program 2XTPHASE from Appendix B.
Fig. 5.18 The damped linear ascillator.

Ajay Singhal

Numerade Educator

01:58

Problem 2

Forced oscillations and resonance. Consider the forced linear oscillator equation
$$
\ddot{x}+\omega^2 x=a \cos \Omega t
$$
( $\sec (5,16)$ ) subject to the initial conditions $x-\dot{x}=0$ when $t=0$. Show that when $\Omega \neq \omega$ the solution is
$$
x=\frac{a}{\left(\omega^2-\Omega^2\right)}(\cos \Omega t-\cos \omega t),
$$
and verify by direct substitution in the equation that in the resonant case, $\Omega=\omega$, the solution is
$$
x=\frac{a}{2 \omega} t \sin \omega t
$$
instead.

Surendra Kumar

Numerade Educator

02:57

Problem 3

Multiple modes. Extend the analysis in Section 5.3 to the case when the two masses $m_1, m_2$ in Fig. 5.8 are not equal. Obtain the counterpart of (5.22), solve the quadratic for $\omega^2$, and show that if $m_2$ is much greater than $m_1$ then there is either a low-frequency mode with
$$
\omega^2 \div \frac{3 \alpha}{2 m_2} \quad \text { and } \quad B \doteqdot 2 A
$$
or a high-frequency mode with
$$
\omega^2 \div \frac{2 \alpha}{m_1} \quad \text { and } \quad B \doteqdot-\frac{m_1}{2 m_2} A .
$$

Surendra Kumar

Numerade Educator

02:23

Problem 4

Large-amplitude pendulum oscillations. Use the program 2XTPHASE to confirm the results in Fig. 5.16, with an oscillation period of $24.5(\mathrm{l} / \mathrm{g})^{1 / 2}$ when $\theta_0=3.124139$, i.e. $179^{\circ}$.
What is the oscillation period when (a) $\theta_0=178^{\circ}$, (b) $\theta_0=179.5^{\circ}$ ?

James Kiss

Numerade Educator

12:12

Problem 5

The whirling pendulum. Suppose that a simple pendulum is hanging downward, and that we give it an initial angular velocity $\Omega$, so that $\theta=0$, $\mathrm{d} \theta / \mathrm{d} t=\Omega$ when $t=0$. Show that the pendulum will only overshoot $\theta-\pi$ if
$$
\Omega>2(g / l)^{1 / 2} \text {. }
$$
Suppose now that friction is present and proportional to $\mathrm{d} \theta / \mathrm{d} t$, so that in place of (5.40) we have
$$
\frac{\mathrm{d}^2 \theta}{\mathrm{d} t^2}+k \frac{\mathrm{d} \theta}{\mathrm{d} t}+\frac{g}{l} \sin \theta=0 .
$$
If we write $\vec{i}=t(g / l)^{1 / 2}$ we obtain
$$
\ddot{\theta}+\tilde{k} \dot{\theta}+\sin \theta=0
$$
with $\dot{\theta}=\tilde{\Omega}$ at $\tilde{t}=0$, where $\tilde{\Omega}=\Omega(l / g)^{1 / 2}, \tilde{k}=k(l / g)^{1 / 2}$ and a dot denotes differentiation with respect to $\bar{t}$.

Use the program PENDANIM to solve this equation by a double-precision Runge-Kutta method and display a simple animation of the motion. For $\cdot \vec{k}=0.1$, say, how many complete revolutions about the pivot does the pendulum make when (i) $\Omega=4(\mathrm{~g} / \mathrm{l})^{1 / 2}$, (ii) $\Omega=10(\mathrm{~g} / \mathrm{l})^{1 / 2}$ ?

Tim Strang

Numerade Educator