The upper half of an inclined plane with an angle of inclination $\phi$ is smooth,while the lower half is rough. $A$ body starting from rest at the top of the inclined plane comes to rest at the bottom of the inclined plane. Then the coefficient of friction for the lower half is

If $u=f(r)$,where $r^2=x^2+y^2$,then $\left(\frac{\partial^2 u}{\partial x^2}+\frac{\partial^2 u}{\partial y^2}\right)$ is equal to

Given in the figure are two blocks $A$ and $B$ of weight $20 \, N$ and $100 \, N$,respectively. These are being pressed against a wall by a force $F$ such that the system does not slide as shown. If the coefficient of friction between the blocks is $0.1$ and between block $B$ and the wall is $0.15$,the frictional force applied by the wall on block $B$ is ........ $N$.

$A$ concave mirror of focal length $f$ produces an image $n$ times the size of the object. If the image is real,then the distance of the object from the mirror is:

Two capacitors $C_1$ and $C_2$ in a circuit are joined as shown in the figure. The potential of point $A$ is $V_1$ and that of $B$ is $V_2$. The potential of point $D$ will be

$A$ thermodynamic system is taken through the cycle $PQRS P$ process. The net work done by the system is ....... $J$. (Note: Assuming a standard $P-V$ diagram where $P$ is in $kPa$ and $V$ is in $m^3$,with vertices $P(1, 10)$,$Q(3, 10)$,$R(3, 20)$,$S(1, 20)$)