In the figure, a ladder of mass $m$ is shown leaning against a wall. It is in static equilibrium making an angle $\theta$ with the horizontal floor. The coefficient of friction between the wall and the ladder is $\mu_1$ and that between the floor and the ladder is $\mu_2$. The normal reaction of the wall on the ladder is $N_1$ and that of the floor is $N_2$. If the ladder is about to slip, then

$Image$

$(A)$ $\mu_1=0 \mu_2 \neq 0$ and $N _2 \tan \theta=\frac{ mg }{2}$

$(B)$ $\mu_1 \neq 0 \mu_2=0$ and $N_1 \tan \theta=\frac{m g}{2}$

$(C)$ $\mu_1 \neq 0 \mu_2 \neq 0$ and $N _2 \tan \theta=\frac{ mg }{1+\mu_1 \mu_2}$

$(D)$ $\mu_1=0 \mu_2 \neq 0$ and $N _1 \tan \theta=\frac{ mg }{2}$

A block of mass $1\,kg$ lies on a horizontal surface in a truck. The coefficient of static friction between the block and the surface is $0.6$ . If the acceleration of the truck is $5\,m\,s^{-2}$ . The frictional force acting on the block is ........ $N$ 

A body of mass $10$ kg slides along a rough horizontal surface. The coefficient of friction is $1/\sqrt 3 $. Taking $g = 10\,m/{s^2}$, the least force which acts at an angle of $30^o $ to the horizontal is ...... $N$

Block $B$ of mass $100 kg$ rests on a rough surface of friction coefficient $\mu = 1/3$. $A$ rope is tied to block $B$ as shown in figure. The maximum acceleration with which boy $A$ of $25 kg$ can climbs on rope without making block move is:

A stone weighing $1$ kg and sliding on ice with a velocity of $2$ m/s is stopped by friction in $10$ sec. The force of friction (assuming it to be constant) will be ......... $N$