When a conservative force does positive work on a body then

A machine, in an amusement park, consists of a cage of the end of one arm, hinged at $O$. The cage revolves along a vertical circle of radius $r$ $\left(ABCDEFGH\right)$ about its hinge $O$, at constant linear speed $v=\sqrt{gr}$. The cage is so attached that the man of weight ${}^{\text{'}}{W}^{\text{'}}$ standing on a weighing machine, inside the cage, is always vertical. Then which of the following is correct

A hollow vertical cylinder of radius $r$ and height $h$ has a smooth internal surface. A small particle is placed in contact with the inner side of the upper rim, at point $A$, and given a horizontal speed $u$, tangential to the rim. It leaves the lower rim at point $B$, vertically below $A$. If $n$ is an integer then

A particle is released from rest at the position $x=x_0$. The potential energy graph of the particle is shown below:

$U\left(x\right)=\left\{\begin{array}{l}-ax; x<0\\ b{x}^2; x>0\end{array}\right.$

Which of the following statements is/are correct? 

$K=$kinetic energy, $U=$potential energy, and $W=$work. Regarding expression $K_2+U_2=K_1+U_1+W_{\mathrm{other}}$

Which of the following is/are incorrect?