Suppose that a digraph d = (v; a) with edge length function ` has no negative circuit but has a 0-length circuit c. let p be an arbitrary potential function, and ` be the edge length function obtained by reweighting ` with p. prove that for any edge a 2 c, ` (a) = 0

Abrake has a normal braking torque of 2.8 kip in and heat-dissipating cast-iron surfaces whose mass is 40 lbm. suppose a load is brought to rest in 8.0 s from an initial angular speed of 1600 rev/min using the normal braking torque; estimate the temperature rise of the heat dissipating surfaces.

Along 8-cm diameter steam pipe whose external surface temperature is 900c connects two buildings. the pipe is exposed to ambient air at 70c with a wind speed of 50 km/hr blowing across the pipe. determine the heat loss from the pipe per unit length. (b) air at 500c enters a section of a rectangular duct (15 cm x 20 cm) at an average velocity of 7 m/s. if the walls of the duct are maintained at 100c. a) the length of the tube for an exit temperature of the air to be 40 0c. b)the rate of heat transfer from the air. c) the fan power needed to overcome the pressure drop in this section of the duct.

Aplate of dimensions 3 m x 3 m is placed 0.37 mm apart from a fixed plate. the plate requires a force of 2n to move at speed of 45 cm/s. evaluate the viscosity of the fluid in between the plates