How do you incorporate friction into kinematics? A small disc of mass $m$ is pushed to $v_0$ on a surface with kinetic coefficient of $\mu$.
Is $v_f = v_0 -{\mu mg} \text{ t} $     ?
(Just based on the idea that increasing $\mu$ , $m$ or $g$ would increase the slow down.)
 A: Since disc is a continous body the magnitude of friction causing on each part will be different just like in a rope having mass tension force is different at different points
You can solve this by considering torque of friction
$f_r = {\mu } (dm) g$
$f_r = \frac{M}{\pi R^2}2pirdr $
$\int T.dt= f_rr$
After finding expression for torque you can equate it to $I{\omega}$
To find final velocity I leave the calculations to you
A: Sliding friction force is −mg so the acceleration is just −g which is now dimensionally correct as the friction coefficient is dimensionless. Giving a fixed deceleration until stopping when that friction goes to zero suddenly.
In this approximate model of friction the mass has no effect on deceleration .. clearly two discs side by side would slow down the same even if they had an irrelevant join! And also even if one disc was glued on top of the other then the combined mass would still slow down at the same rate since the friction gets doubled because of the extra downforce meanwhile the doubled inertia requires more force for the same deceleration. It’s a bit like with free fall under gravity g ...the acceleration does not depend on the mass. You can try these experiments at home. Try sliding two coins of different mass together across a clean table top and let go... they should move together and slow down and stop side by side.. in theory! You may need to repeat several times and consider an average as friction experiments are very sensitive to dirt and minor irregularities or patchiness on the table or coins. This would be an interesting experiment to do and upload to YouTube .. let us know when you’ve done it!
