Astudent holds one end of a thread, which is wrapped around a cylindrical spool, as shown above. the student then drops the spool from a height h above the floor, and the thread unwinds as it falls. the spool has a mass m and a radius r, and the thread has negligible mass. the spool can be approximated as a solid cylinder of moment of inertia i = 1 mr2. express your answers in terms of m, r, h, and fundamental constants. at time t = 0, the spinning spool lands on the floor without bouncing and comes free from the thread. it continues to spin, but slips on the floor's surface while doing so. assume a constant coefficient of sliding friction m. (c) calculate the angular velocity of the spool as a function of time t.

Industrial processes often require the damping of vibrations. consider a workbench that uses four large ideal springs in place of traditional legs, with each spring supporting one corner of the workbench and exhibiting a force constant of 47500 n/m. this workbench is used when creating products that involve highly unstable chemicals. according to specifications, the acceleration of the manufacturing equipment must be held to less than 22.3 m/s2 for the manufacturing process to proceed safely. the amplitude of vibrations are expected to remain under 0.622 mm throughout. given that the workbench and maufacturing equipment combine to 6.05 kg, find the maximum acceleration expected of the equipment and workbench. (ignore the mass of the springs.)