LC Oscillator

The LC circuit shown in Figure 1.7(b) is an example of an electrical circuit which is a SHO. It is governed by the equation

$$L \dot{I} + \frac{Q}{C}=0$$ (1.24)

where $L$ refers to the inductance, $C$ capacitance, $I$ current and $Q$ charge. This can be written as

$$\ddot{Q} + \frac{1}{LC} Q=0$$ (1.25)

which allows us to identify

as the angular frequency.

Problems

1. An empty tin can floating vertically in water is disturbed so that it executes vertical oscillations. The can weighs $100 \, {\rm gm}$, and its height and base diameter are $20$ and $10$ ${\rm cm}$ respectively. [a.] Determine the period of the oscillations. [b.] How much water need one pour into the can to make the time period ?
2. A SHO with $\omega_0= 2 \, {\rm s}^{-1}$ has initial displacement and velocity $0.1 \, {\rm m}$ and $2.0 \, {\rm m s^{-1}}$ respectively. [a.] At what distance from the equilibrium position does it come to rest? [b.] What are the rms. displacement and rms. velocity? What is the displacement at $t= \pi/4 \, {\rm s}$?
3. A SHO with $\omega_0= 3 \, {\rm s}^{-1}$ has initial displacement and velocity $0.2 \, {\rm m}$ and $2\, {\rm m s^{-1}}$ respectively. [a.] Expressing this as $\tilde{x}(t)=\tilde{A} e^{i \omega_0 t}$, determine $\tilde{A}=a + i b$ from the initial conditions. [b.] Using $\tilde{A}= A e^{i \phi}$, what are the amplitude $A$ and phase $\phi$ for this oscillator? [c.] What are the initial position and velocity if the phase is increased by ?
4. A particle of mass $m=0.3 \, {\rm kg}$ in the potential $V(x)=2 e^{x^2/L^2} \, {\rm J}$ $(L=0.1 \, {\rm m})$ is found to behave like a SHO for small displacements from equilibrium. Determine the period of this SHO.
5. Calculate the time average for the SHO $x=A \cos \omega t$.