Exercises

1. A block of mass $m$ slides along a horizontal surface which is lubricated with heavy oil such that the block suffers a viscous retarding force of the form
   
   $$F = - c\,v^n,$$
   
   
   where $c>0$ is a constant, and $v$ is the block's instantaneous velocity. If the initial speed is $v_0$ at time $t=0$, find $v$ and the displacement $x$ as functions of time $t$. Also find $v$ as a function of $x$. Show that for $n=1/2$ the block does not travel further than $2\,m\,v_0^{3/2}/(3\,c)$.
2. A particle is projected vertically upward in a constant gravitational field with an initial speed $v_0$. Show that if there is a retarding force proportional to the square of the speed then the speed of the particle when it returns to the initial position is
   
   $$\frac{v_0\,v_t}{\sqrt{v_0^{\,2} + v_t^{\,2}}},$$
   
   
   where $v_t$ is the terminal speed.

3. A particle of mass $m$ moves (in 1D) in a medium under the influence of a retarding force of the form $m\,k\,(v^3+a^2\,v)$, where $v$ is the particle speed, and $k$ and $a$ are positive constants. Show that for any value of the initial speed the particle will never move a distance greater than $\pi/(2\,k\,a)$, and that the particle comes to rest only for $t\rightarrow \infty$.

4. Two light springs have spring constants $k_1$ and $k_2$, respectively, and are used in a vertical orientation to support an object of mass $m$. Show that the angular frequency of oscillation is $[(k_1+k_2)/m]^{1/2}$ if the springs are in parallel, and $[k_1\,k_2/(k_1+k_2)\,m]^{1/2}$ if the springs are in series.

5. A body of uniform cross-sectional area $A$ and mass density $\rho$ floats in a liquid of density $\rho_0$ (where $\rho<\rho_0$), and at equilibrium displaces a volume $V$. Show that the period of small oscillations about the equilibrium position is
   
   $$T = 2\,\pi\,\sqrt{\frac{V}{g\,A}}.$$
   
   

6. Show that the ratio of two successive maxima in the displacement of a damped harmonic oscillator is constant.

7. If the amplitude of a damped harmonic oscillator decreases to $1/e$ of its initial value after $n$ periods show that the ratio of the period of oscillation to the period of the same oscillator with no damping is
   
   $$\left(1+\frac{1}{4\,\pi^2\,n^2}\right)^{1/2}\simeq 1 + \frac{1}{8\,\pi^2\,n^2}.$$
   
   

8. Show that for a lightly damped linear oscillator of natural frequency $\omega_0$ and damping coefficient $\nu$ driven by a sinusoidal forcing function of frequency $\omega$ that the height of the resonance peak (in amplitude versus $\omega$) scales apprimately as $1/\nu$ whereas its width scales approximately as $\nu$.

9. Consider a damped driven oscillator whose equation of motion is
   
   $$\ddot{x} + 2\,\nu\,\dot{x} + \omega_0^{\,2} \,x = F(t).$$
   
   
   Let $x=0$ and $\dot{x} = v_0$ at $t=0$.
   1. Find the solution for $t>0$ when $F = \sin(\omega\,t)$.
   2. Find the solution for $t>0$ when $F= \sin^2(\omega\,t)$.

10. Obtain the response of a damped linear oscillator of natural frequency $\omega_0$ and damping coefficient $\nu$ to a square-wave periodic forcing function of amplitude $F = m\,\omega_0^{\,2}\,X_0$ and frequency $\omega$.