Resonance Questions

In forced oscillation of a particle, the amplitude is maximum for a frequency ${\omega }_1$ of the force, while the energy is maximum for a frequency ${\omega }_2$ of the force, then:

In a LCR circuit, capacitance is changed from $C$ to $2C$. For the resonant frequency to remain unchanged, the inductance should be changed from $L$ to:

A metal wire of linear mass density $9.8\text{g/m}$ is stretched with a tension of $10\text{kg-wt}$ between two rigid supports $1\text{metre}$ apart. The wire passes at its middle point between the poles of a magnet and it vibrates in resonance when carrying an alternating current of frequency $n$. The frequency $n$ of the alternating source is:

In a series $\text{LCR}$ circuit $R=200\Omega$ and the voltage and the frequency of the main supply is $220\text{V}$ and $50\text{Hz}$ respectively. On taking out the capacitance from the circuit the current lags behind the voltage by $30^{\circ }$. On taking out the inductor from the circuit the current leads the voltage by $30^{\circ }$. The power dissipated in the $\text{LCR}$ circuit is:

A cylindrical tube, open at both ends, has a fundamental frequency, $f$, in air. The tube is dipped vertically in water so that half of it is in water. The fundamental frequency of the air-column is now: