$A$ ring of mass $M$ and radius $R$ sliding with a velocity $v_0$ suddenly enters a rough surface where the coefficient of friction is $\mu$,as shown in the figure. Choose the correct alternative$(s)$.

$A$ rod is hinged at its centre and rotated by applying a constant torque starting from rest. The power developed by the external torque as a function of time is:

Which of the following statements are correct?

$A$ flywheel rotates about an axis. Due to friction at the axis,it experiences an angular retardation proportional to its angular velocity. If its angular velocity falls to half while it makes $n$ rotations,how many more rotations will it make before coming to rest?

$A$ particle of mass $M=0.2 \ kg$ is initially at rest in the $xy$-plane at a point $(x=-l, y=-h)$,where $l=10 \ m$ and $h=1 \ m$. The particle is accelerated at time $t=0$ with a constant acceleration $a=10 \ m/s^2$ along the positive $x$-direction. Its angular momentum and torque with respect to the origin,in $SI$ units,are represented by $\vec{L}$ and $\vec{\tau}$,respectively. $\hat{i}, \hat{j}$ and $\hat{k}$ are unit vectors along the positive $x, y$ and $z$-directions,respectively. If $\hat{k}=\hat{i} \times \hat{j}$,then which of the following statement$(s)$ is(are) correct?
$(A)$ The particle arrives at the point $(x=l, y=-h)$ at time $t=2 \ s$.
$(B)$ $\vec{\tau}=2 \hat{k}$ when the particle passes through the point $(x=l, y=-h)$.
$(C)$ $\vec{L}=4 \hat{k}$ when the particle passes through the point $(x=l, y=-h)$.
$(D)$ $\vec{\tau}=\hat{k}$ when the particle passes through the point $(x=0, y=-h)$.

$A$ ring of mass $m = 1 \ kg$ and radius $R = 1.25 \ m$ is kept on a rough horizontal ground. $A$ small body of same mass $m = 1 \ kg$ is stuck to the top of the ring. When it is given a slight push forward,the ring starts rolling purely on the ground. What is the maximum speed of the centre of the ring (in $m/s$)?