A cylinder of diameter $2R$ and mass $M$ rests on two rough pegs (coefficient of static friction $\mu$ ), the distance $2kR$ apart. A gradually increasing torque $\tau$ is applied as shown. The cylinder will topple before it slips if

A uniform rod $AB$ of weight $100 N$ rests on a rough peg at $C$ and a force $F$ acts at $A$ as shown in figure. If $BC=CM$ and $\tan \alpha =\frac{4}{3}.$ The minimum coefficient of friction at $C$ is : (where $M$ is mid point of rod)

One stick leans on another as shown in figure. A right angle is formed where they meet, and the right stick makes an angle $\mathrm{\theta }$ with the horizontal. The left stick extends infinitesimally beyond the end of the right stick. The coefficient of friction between the two sticks is $\mathrm{\mu }$. The sticks have the same mass density per unit length and are both hinged at the ground. What is the minimum value of $\mathrm{tan\theta }$ for which the sticks do not fall? (Take $\mathrm{\mu }=0.04)$

A person supports a book between finger and thumb as shown (the point of grip is assumed to be at the corner of the book). If the book has a weight of $W$ then the person is producing a torque on the book of

A string is wrapped around the rim of a wheel of moment of inertia $0.20 \mathrm{kg}-{\mathrm{m}}^2$ and radius $20\mathrm{cm}.$ The wheel is free to rotate about its axis and initially the wheel is rest. The string is now pulled by a force of $20 N.$ The angular velocity of the string after $5$ seconds will be :-

In the figure $\left(A\right)$ half of the meter scale is made of wood while the other half of steel. The wooden part is pivoted at $O.$ A force $F$ is applied at the end of steel part. In figure $\left(B\right)$ the steel part is pivoted at $\text{'}O\text{'}$ and the same force is applied at the wooden end:-

In the following figure $r_1$ and $r_2$ are $5\mathrm{cm}$ and $30\mathrm{cm}$ respectively. If the moment of inertia of the wheel is $5100 \mathrm{kg}-{\mathrm{m}}^2$ then its angular acceleration will be :-