Question

Find
i) Time of pure rolling
ii) velocity of plank
iii) velocity of disc
iv) angular velocity of disc
v) work done by friction

Answer 1

Sliding and Rolling disc m on top of a rough plank M resting on a smooth surface...

Mass of Disc = m = 1kg.   Mass of Plank = M = 2kg.  Radius = R = 1m. 
Coefficient of kinetic Friction = μ = 0.50.  This is a high value really.
Friction force = f, acts in the backward direction to slow down the disc.  Friction f acts on the plank in the forward direction.

Initial angular velocity = 0.  Initial speed of disc = u = 10m/s.
Angular velocity at time t = ω.  Angular acceleration = α. 
Linear acceleration of the disc = a.
Linear acceleration of the plank = a2. Initial speed of plank = 0.
Friction between the plank and floor = 0.  Smooth floor.

Plank is a moving non inertial frame of reference. So a pseudo force (linear) = - m *a2 is to be added to the disc m.

For the plank:  f = M * a2  => a2 = f/M    --- (1)

Torque on the disc about its center O = T = I α = f R
So  m R²/2 * α =  f R      =>  α = 2f/(m R)   --- (2)

For the disc, in the frame of reference of the accelerating plank:
       Total force =  - f - m * a2 = m * a 
       a = - f [ 1/m + 1/M] = - f (m+M)/(mM)    --- (3)

       v = u + a t = u - f (m+M) t /(mM)       .....  v decreases  ...
       ω = ω₀ + α t = 0 + 2 f t/(m R)   --- (4)      .... ω increases ..

For Rolling to start,  v = R ω  --- (5)
Solving equations (4) and (5), we get the time t  to start rolling.
        t = m M u / (m + 2M) f       --- (6)

Substituting this value in (4) we get:
        v_r = 2 M u /(m+3M).        ω_r = 2 M u /[ (m+3M) R ]     --- (7)

        velocity of the plank = v2 = a2 * t = f t /M = m u/(m+3M)    --- (8)

v_r is relative to the plank.  So absolute velocity v of the disc m is:
        v = v_r + v2 = (m+2M) u /(m+3M)    --- (9)

Initial KE = 1/2  m u²
Final KE = 1/2  m v² + 1/2  I ω_r² +  1/2 M v2²
               = 1/2 m² u² /(m+3M)²  * [ (m+2M)² + 2 m² + m M ] 
               = 1/2 m² u² /(m+3M)² * [m²+ 6M² + 5 m M ]

Work done by friction = Loss in KE
  W_f = 1/2 m u² / (m+3M)² * [ (m+3M)² - m² - 6M² - 5 m M ]
         = 1/2 m u² * [ M /(m + 3M) ]