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

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$(A)$ $\mu_1=0 \mu_2 \neq 0$ 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 $m$ is in contact with the cart $C$ as shown in the figure. The coefficient of static friction between the block and the cart is $\mu .$ The acceleration $\alpha$ of the cart that will prevent the block from falling satisfies

A block of mass $m$ is pressed against a vertical surface by a spring of unstretched length $l$ . If the coefficient of friction between the block and the surface is $\mu $. Choose the correct statement.

A $\vec F\,\, = \,\,\hat i\, + \,4\hat j\,$ acts on block shown. The force of friction acting on the block is :

The limiting value of static friction between two contact surfaces is ........... 

A block of mass $10\, kg$ moving at $10\,m/s$ is released to slide on rough surface having  coefficient of friction $0.2.$ It will stop by travelling distance ........ $m$