$A$ man can move on a horizontal plank supported symmetrically as shown. The variation of normal reaction on support $A$ with distance $x$ of the man from the end of the plank is best represented by :

A mass $m$ hangs with the help of a string wrapped around a pulley on a frictionless bearing. The pulley has mass m and radius $R$. Assuming pulley to be a perfect uniform circular disc, the acceleration of the mass $m$, if the string does not slip on the pulley, is

Figure below shows a shampoo bottle in a perfect cylindrical shape. In a simple experiment, the stability of the bottle filled with different amount of shampoo volume is observed. The bottle is tilted from one side and then released. Let the angle $\theta$ depicts the critical angular displacement resulting, in the bottle losing its stability and tipping over. Choose the graph correctly depicting the fraction $f$ of shampoo filled $(f=1$ corresponds to completely filled) versus the tipping angle $\theta$

A small $100$ $g$ sleeve $B$ can slide on a smooth, circular and rigid wire frame $A$ of radius $5$ $m$ placed in vertical place. The wire frame is rotating about its vertical diameter at $2$ $rad/s$. When the sleeve is brought at a particular angular position other than the bottom and the top of the ring, the sleeve will not slide on the wire frame. ……… $N$ is force of interaction between the sleeve and the wire frame at this position.

$A$ body is in equilibrium under the influence of a number of forces. Each force has a different line of action. The minimum number of forces required is