**LC Oscillations O uantitatively**

Here we want to Show explicitly that Eq. 33-4 for the angular frequency of LC oscillations is correct. At the same time, we want to examine even more closely the analogy between LC oscillations and block-spring oscillations. We start by extending somewhat our earlier treatment of the mechanical block-spring oscillator

**The Block-Spring Oscillator**

We analyzed block-spring oscillations in Chapter 16 in terms of energy transfers and did not-at that early stage-derive the fundamental differential equation that governs those oscillations. We do so now.We can write, for the total energy V of a block-spring oscillator at any instant, where U; and U, are, respectively, the kinetic energy of the moving block and the

potential energy of the stretched or compressed spring. If there is no friction – which we assume-the total energy V remains constant with time, even though I” and x vary. In more formal language, . This leads the fundamental differential equation that governs the friction less block-spring oscillations. The general solution to -that is, the function x(t) that describes the block-spring oscillations-is in which X is the amplitude of the mechanical oscillations (represented by Xm inw is the angular frequency of the oscillations, and cp is a phase constant

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