A uniform solid cylinder with mass ?M? and radius 2?R? rests on a horizontal tabletop. A string is attached by a yoke to a frictionless axle through the center of the cylinder so that the cylinder can rotate about the axle. The string runs over a disk-shaped pulley with mass M and radius R that is mounted on a frictionless axle through its center. A block of mass M is suspended from the free end of the string (?Fig. P10.75?). The string doesn’t slip over the pulley surface, and the cylinder rolls without slipping on the tabletop. Find the magnitude of the acceleration of the block after the system is released from rest.
Solution 87P Introduction We have to calculate the acceleration of the mass hanging. Step 1 The following figure shows the free body diagram of the each component. Step 2 Let us first consider the cylinder at the table. Since the table is rotating without slipping, the frictional force will be equal to the tension of the string. Now we have 2 I 2= T (22)…….(1) Now the moment inertia about the axle of the cylinder is 1 2 Icm = M22R) Now the cylinder is actually rotating against point of contact between the cylinder and table. Hence the moment of inertia against the point will be I = I + M(2R) = 2MR + 4MR = 6MR ……..(2) 2 2 cm Now using this value in the equation (1) we have 6MR = T (2R2 Now let us consider mass is moving with acceleration a. Hene we can write that a = 2R Hence we have 2 a 6MR (2R = T2(2R) T = Ma3 2 2