Question

The energy in an oscillating LC circuit containing a 1.57 H inductor is 5.76 μJ. The maximum charge on the capacitor is 201 μC. For a mechanical system with the same period, find the
(a) mass
(b) spring constant
(c) maximum displacement
(d) maximum speed.

Answer 1

Answer:

(a)1.57 kg

(b) 281.17 N/m

(c) 201 micrometer

(d) $2.69\times 10^{-3}m/sec$

Explanation:

We have given that value of inductor L = 1.57 Henry

Inductive energy $E=5.76\mu j=5.76\times 10^{-6}J$

Maximum charge $Q=201\mu C=201\times 10^{-6}C$

(A) In electrical mechanical system mass corresponds to inductance

So mass will be m = 1.57 kg

(B) We have given energy $E=\frac{Q^2}{2C}$

$C=\frac{Q^2}{2E}=\frac{(201\times 10^{-6})^2}{2\times 5.7\times 10^{-6}}=3543.94\times 10^{-6}$

In electrical mechanical system spring constant is equivalent to $\frac{1}{C}$

So spring constant $k=\frac{1}{C}=\frac{1}{3543.94\times 10^{-6}}=282.17N/m$

(c) Displacement is equivalent to maximum charge

So displacement will be $x=201\mu m$

(d) Maximum speed is correspond to maximum current

As maximum current $i_m=\frac{Q}{\sqrt{LC}}=\frac{201\times 10^{-6}}{\sqrt{1.57\times 3543.94\times 10^{-6}}}=2.69\times 10^{-3}A=2.69\times 10^{-3}m/sec$