To solve this problem, we need to analyze the forces acting on the block and determine the minimum angular velocity required to keep it stationary.$\\$Given:$\\$• Mass of the block $m = 10$ kg$\\$• Radius of the cylinder $r = 1$ m$\\$• Coefficient of friction $\mu = 0.1 \\$• Gravitational acceleration $g = 10$ m/s$^2 \\$When the cylinder rotates, the block experiences a centripetal force that provides the necessary frictional force to counteract gravity.$\\$The frictional force $f_s$ must be equal to the gravitational force acting on the block:$\\ f_s = \mu \cdot N = m \cdot g \\$The normal force $N$ is provided by the centripetal force due to rotation:$\\ N = m \cdot \omega^2 \cdot r \\$Substitute $N$ into the frictional force equation:$\\ \mu \cdot (m \cdot \omega^2 \cdot r) = m \cdot g \\$Cancel $m$ from both sides:$\\ \mu \cdot \omega^2 \cdot r = g \\$Solve for $\omega^2: \\ \omega^2 = \frac{g}{\mu \cdot r} \\$Substitute the known values:$\\ \omega^2 = \frac{10}{0.1 \cdot 1} \\ \omega^2 = 100 \\$Take the square root to find $\omega: \\ \omega = \sqrt{100} \\ \omega = 10$ rad/s$\\$Therefore, the minimum angular velocity needed is $10$ rad/s. This corresponds to Option 3.