Chapter 5

A massless spring has unstretched length $l_{\mathrm{o}}$ and force constant $k$. One end is now attached to the ceiling and a mass $m$ is hung from the other. The equilibrium length of the spring is now $l_1$. (a) Write down the condition that determines $l_1$. Suppose now the spring is stretched a further distance $x$ beyond its new equilibrium length. Show that the net force (spring plus gravity) on the mass is $F=-kx$. That is, the net force obeys Hooke's law, when $x$ is the distance from the equilibrium position $-$ a very useful result, which lets us treat a mass on a vertical spring just as if it were horizontal. (b) Prove the same result by showing that the net potential energy (spring plus gravity) has the form $U(x)=$ const $+\frac{1}{2}k{x}^2$

The potential energy of two atoms in a molecule can sometimes be approximated by the Morse function, $$U(r)=A[{(e^{(R-r)/S}-1)}^2-1]$$ where $r$ is the distance between the two atoms and $A,R,$ and $S$ are positive constants with $S\ll R.$ Sketch this function for $0<r<\mathrm{\infty }$. Find the equilibrium separation $r_{\mathrm{o}}$, at which $U(r)$ is minimum. Now write $r=r_{\mathrm{o}}+x$ so that $x$ is the displacement from equilibrium, and show that, for small displacements, $U$ has the approximate form $U=$ const $+\frac{1}{2}k{x}^2.$ That is, Hooke's law applies. What is the force constant $k?$

Write down the potential energy $U(\varphi )$ of a simple pendulum (mass $m,$ length $l$ ) in terms of the angle $\varphi$ between the pendulum and the vertical. (Choose the zero of $U$ at the bottom.) Show that, for small angles, $U$ has the Hooke's law form $U(\varphi )=\frac{1}{2}k{\varphi }^2,$ in terms of the coordinate $\varphi .$ What is $k?$

An unusual pendulum is made by fixing a string to a horizontal cylinder of radius $R$, wrapping the string several times around the cylinder, and then tying a mass $m$ to the loose end. In equilibrium the mass hangs a distance $l_{\mathrm{o}}$ vertically below the edge of the cylinder. Find the potential energy if the pendulum has swung to an angle $\varphi$ from the vertical. Show that for small angles, it can be written in the Hooke's law form $U=\frac{1}{2}k{\varphi }^2.$ Comment on the value of $k$.

(a) If a mass $m=0.2\mathrm{kg}$ is tied to one end of a spring whose force constant $k=80\mathrm{N}/\mathrm{m}$ and whose other end is held fixed, what are the angular frequency $\omega$, the frequency $f$, and the period $\tau$ of its oscillations? (b) If the initial position and velocity are $x_{\mathrm{o}}=0$ and $v_{\mathrm{o}}=40\mathrm{m}/\mathrm{s},$ what are the constants $A$ and $\delta$ in the expression $x(t)=A\cos (\omega t-\delta )?$

The maximum displacement of a mass oscillating about its equilibrium position is $0.2\mathrm{m},$ and its maximum speed is $1.2\mathrm{m}/\mathrm{s}$. What is the period $\tau$ of its oscillations?

The force on a mass $m$ at position $x$ on the $x$ axis is $F=-F_0\sinh \alpha x,$ where $F_0$ and $\alpha$ are constants. Find the potential energy $U(x),$ and give an approximation for $U(x)$ suitable for small oscillations. What is the angular frequency of such oscillations?

Consider a simple harmonic oscillator with period $\tau$. Let $⟨f⟩$ denote the average value of any variable $f(t),$ averaged over one complete cycle: $$⟨f⟩=\frac{1}{\tau }{\int }_0^{\tau }f(t)dt$$ Prove that $⟨T⟩=⟨U⟩=\frac{1}{2}E$ where $E$ is the total energy of the oscillator. [Hint: Start by proving the more general, and extremely useful, results that $⟨{\sin }^2(\omega t-\delta )⟩=⟨{\cos }^2(\omega t-\delta )⟩=\frac{1}{2}.$ Explain why these two results are almost obvious, then prove them by using trig identities to rewrite ${\sin }^2\theta$ and ${\cos }^2\theta \text{ in terms of }\cos (2\theta ).]$

The potential energy of a one-dimensional mass $m$ at a distance $r$ from the origin is $$U(r)=U_0(\frac{r}{R}+{\lambda }^2\frac{R}{r})$$ for $0<r<\mathrm{\infty },$ with $U_{\mathrm{o}},R,$ and $\lambda$ all positive constants. Find the equilibrium position $r_{\mathrm{o}}.$ Let $x$ be the distance from equilibrium and show that, for small $x$, the PE has the form $U=$ const $+\frac{1}{2}k{x}^2$. What is the angular frequency of small oscillations?

Verify that the decay parameter $\beta -\sqrt{{\beta }^2-{\omega }_0^2}$ for an overdamped oscillator $(\beta >{\omega }_{\mathrm{o}})de$ creases with increasing $\beta$. Sketch its behavior for ${\omega }_0<\beta <\mathrm{\infty }$.