Attwood's machine A uniform circular pulley of mass 2m can rotate t...

Attwood's machine A uniform circular pulley of mass 2m can rotate treely about Its axis 12.2 of symmetry which is fixed in a horizontal position. Two masses m, 3m are connected by light inextensible string which passes over the pulley without slipping. The whole system undergoes a planar motion with the masses moving vertically. Take the rotation angle of the pulley as generalised coordinate and obtain Lagrange's equation for the motion. Deduce the upward acceleration of the mass m. 362 12.5 A uniform solid cylinder cylinder of radius b. between the the two C with mass m and In radius the motion, a rolls the on axes the rough of outer cylinders surface of remain a fixed parallel horizontal to each upward other. vertical. Let a be Taking the angle 0 as generalised coordinate, plane con- the cylinder axes and the taining is equivalent to the Lagrange's equation and verify it energy that conservation obtain equation. the cylinder C is at rest on top of the fixed cylinder when reaction it is given force a very small C. disturbance. Deduce that C will as a leave the fixed cylinder when 0 = cos (4/7). Is the assumption that Initially Find, function of 0. the normal component of the exerted on rolling persists up to this moment realistic? 12.7 A uniform ball of mass m rolls down a rough wedge of mass M and angle a, which itself can slide on a smooth horizontal table. The whole system undergoes planar motion. How many degrees of freedom has this system? Obtain Lagrange's equations. For the special case in which M = 3m/2, find (i) the acceleration of the wedge, and (ii) the acceleration of the ball relative to the wedge.