As shown in the figure,a metal rod makes contact and completes the circuit. The circuit is perpendicular to the magnetic field with $B = 0.15 \, T$. If the resistance is $3 \, \Omega$,the force needed to move the rod as indicated with a constant speed of $2 \, m/s$ is:

If $y = \tan^{-1}(\sec x - \tan x)$,then $\frac{dy}{dx} = $

If the electric flux entering and leaving a closed surface are $\phi_1$ and $\phi_2$ respectively,then the electric charge inside the surface will be:

Two rigid boxes containing different ideal gases are placed on a table. Box $A$ contains one mole of nitrogen at temperature $T_0$,while box $B$ contains one mole of helium at temperature $(7/3) T_0$. The boxes are then put into thermal contact with each other,and heat flows between them until the gases reach a common final temperature (ignore the heat capacity of the boxes). The final temperature of the gases $T_f$ in terms of $T_0$ is:

$A$ child is standing at one end of a long trolley moving with a speed $v$ on a smooth horizontal floor. If the child starts running towards the other end of the trolley with a speed $u$,the centre of mass of the system (trolley + child) will move with a speed:

During the titration of $0.1 \, M$ $CH_3COOH$ and $0.1 \, M$ $KOH$ solution,what will be the $pH$ of the solution after $40\%$ neutralization of the acid? [$K_a$ for $CH_3COOH = 2 \times 10^{-5}$]