A uniform disc of mass 5 kg and radius 1 m is placed vertically on a rough horizontal surface. A horizontal time varying force is applied on the center of disc and its variation with time is shown in the graph. The co–efficient of friction between disc and surface is 0.4. (Take g = 10 m / $s^{2}$ )
Then which of the following option is/are correct

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at t = 7 sec disc is rolling without slipping

at t = 4 sec disc is rolling without slipping

at t = 3 sec disc is rolling without slipping

at t = 3 second friction force is 10 N

answer is A, B, D.

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Detailed Solution

$\begin{array}{l} T r a n s l a t i o n : F - f = m a - - - (1) \\ \Rightarrow R o t a t i o n : τ = I α \\ \Rightarrow f R = \frac{m R^{2}}{2} \frac{a}{R} (∵ a = R α) \\ \Rightarrow f = \frac{m a}{2} - - - (3) \end{array}$

$\begin{array}{l} f r o m (1) & (3) \\ \Rightarrow f = \frac{F}{3} = \frac{10 t}{3} \\ (∵ \frac{F}{t} = \frac{80}{8} \Rightarrow F = 10 t) \end{array}$

$\begin{array}{l} F o r p u r e r o l l i n g \\ \Rightarrow f \leq μ m g \\ \Rightarrow \frac{10 t}{3} \leq 20 \\ \Rightarrow t \leq 6 s \end{array}$