A small ball of mass $m=21 \mathrm{g}$ suspended by an insulating thread at a height $h=12 \mathrm{cm}$ from a large horizontal conducting plane performs small oscillations. After a charge $q$ had been imparted to the ball, the oscillation period changed $\mathrm{\eta }=2.0$ times. Find $q.$Take  $g=9.8 m/s^2$ and $\frac{1}{4{\mathrm{\pi \epsilon }}_{\mathrm{o}}}=9\times {10}^9 N m^2 C^{-2}$.

A loop is formed by two parallel conductors connected by a solenoid, with inductance $L$, and a conducting rod of mass $m$, which can freely (without friction) slide over the conductors. The conductors are located in a horizontal plane in a uniform vertical magnetic field with induction $B.$ The distance between the conductors is equal to $l.$ At the moment $t=0$, the rod is imparted an initial velocity $v_0$, directed to the right. Find the law of its motion $x\left(t\right)$, if the electric resistance of the loop is negligible.

In an oscillating circuit shown in the figure, the coil inductance is equal to $L=2.5 \mathrm{mH}$, and the capacitor have capacitances $C_1=2.0 \mathrm{\mu F}$ and $C_2=3.0 \mathrm{\mu F}$. The capacitors were charged to a voltage $V=180 \mathrm{V}$, and then the switch $Sw$ was closed. Find, 
$\left(a\right)$ the natural oscillation frequency, 
$\left(b\right)$ the peak value of the current flowing through the coil.

An electric circuit shown in has a negligibly small active resistance. The left-hand capacitor was charged to a voltage $V_0$ and then at the moment $t=0$ the switch $S_{\mathrm{w}}$ was closed. Find the time dependence of the voltages in left and right capacitors.