Thermodynamics of a Block Sliding Across a Frictional Surface

The following idealized problem is intended to illustrate some basic thermodynamic concepts involved in kinetic friction. A block of mass m is sliding on top of a frictional,1 flat-topped table of mass M. The table is magnetically levitated, so that it can move without thermal contact and friction across a horizontal floor. The table is initially stationary, while the block has initial speed vi and slides to rest relative to the table. The block and table are inside a large vacuum tank, so there is no air resistance, buoyancy, nor thermal losses to the atmosphere. Furthermore the inner surface of the vacuum tank is a perfect mirror so that the tank does not radiatively exchange heat with the block and table. The block and table are homogeneous, both initially have temperature Ti, and they each have large thermal conductivities so that they rapidly attain a common final temperature Tf after the block has come to rest. The specific heat capacity of the block is cb and that of the table is ct, and these heat capacities are assumed to be temperature independent over the range of temperatures that arises in this problem. (a) Find the common final speed vf and temperature Tf of the block and table. (b) Find the changes in the bulk kinetic energies K (in the center-of-mass frame of the isolated block-table system), the internal energies U, and the entropies S of the block and table. (c) Discuss the first and second laws of thermodynamics in connection with these results.