What is the relation between speed and angular velocity and radius in the given problem? 
Can anyone help me in solving the above question ? 
I have considered the bottom-most point, and used the conservation of angular momentum at that point. I am confused that what should be the value of moment of inertia to be taken ? $I$=$MR^2$ or $I= MR^2 + MR^2$?
 A: There are usually 2 phases in the motion. First the object slides while possibly also rotating. Kinetic friction reduces linear velocity and may reduce or increase angular velocity. This phase continues until the no slip condition is reached. In the second phase there is pure rolling motion.
There is a kinetic friction force $F=\mu mg$ acting on the disk, which causes linear deceleration $a=-\frac{F}{m}$. This force also exerts a torque $Fr=J\alpha$ where $J=\frac12 mr^2$ is the moment of inertia about the centre of the disk. The torque causes angular deceleration $\alpha=-\frac{Fr}{J}$. 
The linear and angular velocities at time $t$ after the disk is released are $v=v_0-at$ and $\omega=\omega_0-\alpha t$. 
If the disk stops altogether before pure rolling motion commences, then the linear and angular velocities become zero at the same time $t$. This allows you to find the relation between $v_0, \omega_0$.
Reference : Sliding and Rolling - The Physics of a Rolling Ball.
A: You don't need to use the moment of inertia at all. You need to know the relationship between an angle, the radius of a circle, and the arc-length of the piece of circumference corresponding to the angle. I'm not going to give you the answer, but think of this in a kind of three-step process: 


*

*What is the relationship between the angular velocity, $\omega_0$, of a rotating disk and how far a point on the circumference of said disk moves after a certain amount of time, t?

*Imagine if the outside of the disk were covered in ink and rolling on a piece of paper. If the disk is rolling at $\omega_0$ for time $t$, how long would the strip of ink be?

*What does the length of the strip of ink tell you about $v_0$?


Picture these things (or draw them out) and make some equations.
A: As we know, when a body is set rolling on a rough surface, eventually one of two states will be achieved: 


*

*the object will achieve pure rolling , wherein the velocity and the angular velocity are such that the point of contact with the ground is always at rest relative to the ground, and so equilibrium is achieved.

*All the energy of the object is dissipated as it comes to a halt, this you will notice is just a special case of the circumstance mentioned above , as this also satisfies the pure rolling condition.
Now, the secind situation can only occur with special initial conditions . 
What these conditions are can be derived as follows:

*assume constant value of friction coefficient

*write down the retarding force and torque on the body at any instant , and use newtons laws to find the value of linear and angular acceleration

*according to the problem, when v reaches zero, so does the rotation speed , so use that to get the condition.

