A block is projected with a speed $v_0$ such that it strikes the point of projection after describing the path as shown by the dotted line. If friction exists for the path of length $d$ and the vertical circular path is smooth, assuming $\mu =$ coefficient of friction

(a) Find $v_0$.

(b) What is the minimum value of $v_1$.

A small ball is suspended from point $O$ by a thread of length $l$. A nail is driven into the wall at a distance of $1/2$ below $O$, at $A$. The ball is drawn aside so that the thread takes up a horizontal position at the level of point $O$ and then released.

(a) At what angle from the vertical of the ball’s trajectory, will the tension in the thread disappear?

(b) What will be the highest point from the lowermost point of circular track, to which it will rise?

Figure shows a real string on a real pulley. The string runs on the pulley without slipping as shown in the figure. The direction of the resultant force on an element shown in the figure on the string maybe