A solid cylinder $A$ of mass $12 \mathrm{kg}$ is being pulled by another weight $B$ ($3 \mathrm{kg}$) along the horizontal plane as shown in the figure (pulley is massless). The weight $B$ is attached to the end of a string wound around the circumference of the cylinder. If there is no slipping anywhere and the string is inextensible, then find acceleration (in $\mathrm{m} {\mathrm{s}}^{-2}$) of $A$.

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A small sphere of mass $1 \mathrm{kg}$ is rolling without slipping on a stationary base with linear speed, $v=\sqrt{\frac{200}{7}} \mathrm{m} {\mathrm{s}}^{-1}$. Find its linear speed at point $C$. ( Take $g=10 \mathrm{m} {\mathrm{s}}^{-2}$.)

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 that is directly above the centre of the face at a height $\frac{3a}{4}$ above the base. The minimum value of $F$ for which the cube begins to topple about an edge is (assume that cube does not slide),

An equilateral prism of mass $m$ rests on a rough horizontal surface with coefficient of friction $\mu$. A horizontal force $F$ is applied on the prism as shown in the figure. If the coefficient of friction is sufficiently high so that the prism does not slide before toppling, then the minimum force required to topple the prism is,

A uniform cube of mass $m$ and side $a$ is placed on a frictionless horizontal surface. A vertical force, $F=\frac{mg}{4}$ is applied to the edge as shown in figure. The normal reaction effectively acts at a distance,

A force $F$ is applied on the top of a cube as shown in the figure. The coefficient of friction between the cube and ground is $\mu$. If $F$ is gradually increased, the cube will topple before sliding. Then, the range of $\mu$ is,