A uniform cube of side $a$and mass $m$ rests on a rough horizontal table. A horizontal force $F$ is applied normal to one of the faces at a point directly above the centre of the face at a height $\frac{3a}{4}$ above the base. (i) What is the minimum value of $F$ for which the cube begins to tip about an edge?

(ii) What is the minimum value of ${\mu }_{\mathrm{s}}$ so that toppling occurs.

(iii) If $\mu ={\mu }_{\min }$, find minimum force for topping.

(iv) Minimum ${\mu }_{\mathrm{s}}$ so that $F_{\min }$(as in part-(i)) can cause toppling.

In the given figures find the accelerations and the friction forces involved,

A wooden block of mass $m$ resting on a rough horizontal table (coefficient of friction = $\mathrm{\mu }$) is pulled by a force $F$ as shown in the figure. The acceleration of the block moving horizontally is:

Two blocks $m_1=4 \mathrm{kg}$ and $m_2=2 \mathrm{kg}$, connected by a weightless rod on a plane having inclination of $37°$ as shown in the figure. The coefficients of dynamic friction of $m_1$ and $m_2$ with the inclined plane is $\mathrm{\mu }=0.25$. Then the common acceleration of the two blocks and the tension in the rod are: (Take $g=10 \mathrm{m} {\mathrm{s}}^{-2}$)

A cart of mass $M$ has a block of mass $m$ attached to it as shown in the figure. Co-efficient of friction between the block and cart is $\mu$. What is the minimum acceleration of the cart so that the block m does not fall?