$A$ capacitor of capacitance $C_0$ is charged to a potential $V_0$ and is connected with another capacitor of capacitance $C$ as shown. After closing the switch $S$,the common potential across the two capacitors becomes $V$. The capacitance $C$ is given by

The height at which the weight of a body becomes $\frac{1}{9}^{th}$ of its weight on the surface of the Earth (radius of Earth $R$) is:

The sensation of fatigue in the muscles after prolonged strenuous physical work is caused by

The work functions of metals $A$ and $B$ are in the ratio $1 : 2$. If light of frequencies $f$ and $2f$ are incident on the surfaces of $A$ and $B$ respectively,the ratio of the maximum kinetic energies of photoelectrons emitted is (given that $f > f_{0A}$ and $2f > f_{0B}$).

$A$ small ball of mass $m$ is released at a height $R$ above the Earth's surface,as shown in the figure. The ball enters a narrow groove and reaches a maximum depth of $R/2$ inside the Earth before coming to rest momentarily. The groove contains an ideal spring of spring constant $K$ and natural length $R$. Find the value of $K$. ($R$ is the radius of the Earth and $M$ is the mass of the Earth).

What is the coefficient of mutual inductance when the magnetic flux changes by $2 \times 10^{-2} \, Wb$ and the change in current is $0.01 \, A$?