Gform rod of density p, cross-sectional area a, mod ulus of elasticity e1 and equilibrium length is clamped at one end; and at the other end, it is welded to another uniform rod of corresponding properties p2, a2, e2 and l2. the longitudinal vibration of the composite rod system is considered (a) find the boundary conditions and identify them as "geometric" or "natural" boundary conditions (b) find the equation for the natural frequencies of the system.

Apump is used to circulate hot water in a home heating system. water enters the well-insulated pump operating at steady state at a rate of 0.42 gal/min. the inlet pressure and temperature are 14.7 lbf/in.2, and 180°f, respectively; at the exit the pressure is 60 lbf/in.2 the pump requires 1/15 hp of power input. water can be modeled as an incompressible substance with constant density of 60.58 lb/ft3 and constant specific heat of 1 btu/lb or. neglecting kinetic and potential energy effects, determine the temperature change, in °r, as the water flows through the pump.