16.103  (Rigid Body F=ma, General Plane Motion)

A drum of 60 mm radius is riveted to a disk of 120 mm radius.  The disk and drum have a combined mass of 6 kg and radius of gyration, k_G, of 90 mm.  A cord is wrapped around the inner drum and is pulled with force P = 20 N in the direction shown.  (The cord unwraps from the drum as the spool rolls to the right).  If the disk rolls without slipping, please determine:
(a)  The angular acceleration, a, of the disk.
(b)  The acceleration, a_G, of the mass center at G.
(c)  The minimum value of the coefficient of friction, m, consistent with this motion.

Comments/Hints:
1.  Calculate I_G = .0486 kg-m².
2.  Draw a FBD and a KD.  On the KD are ma_G and I_Ga terms.
3.  Sum forces in the x and y directions and equate to ma terms.
4.  Sum moments about G and set equal to I_Ga.
5.  You need a kinematics equation:  For a wheel that is not slipping, a_G = ra = 0.12a.
6.  Solve for:  a = 26.7 rad/sec²;  a_G = 3.2 m/s²;  F = 0.8 N (very small!).
7.  The required m = (F/N)_calculated = 0.0136.