Let us assume a liquid film over an horizontal solid plate. the flow, with constant thickness ℎ and subject to the gravity, is driven by a constant shear stress at the top surface: �[\(ℎ) = �. assume that the liquid film is thin enough and flat and that the flow is fully developed with zero net flow rate. determine the velocity profile �(�) and the pressure gradient ��/��.

Aloaded platform of total mass 500 kg is supported by a dashpot and by a set of springs of effective stiffness 72 kn/m. it is observed that when the platform is depressed through a distance x = 12.5 cm below its equilibrium position and then released without any initial velocity; it reaches its equilibrium position in the shortest possible time without overshoot. find the position and velocity of the loaded platform 0.10 sec. after its release. if a further load of 400 kg is added to the platform, find, i) the frequency of damped vibrations, and i) the amplitude of vibration after 2 complete oscillations, given that the initial amplitude is 15 cm.

Apiping systems consists of 6 m of 6-std type k and 12 m of 4-std type k, both drawn copper tubing. the system conveys ethylene glycol at a rate of 0.013 m3/s. the pressure drop across the system is to be calculated. there are two 90° elbows in the 6-in pipe, a reduction from the 6-in pipe to the 4-in pipe and four 90° elbows in the 4-in pipe. all fittings are soldered (same as flanged) and regular. the inlet and outlet of the system are at the same height.

Precipitation hardening can be achieved in many light alloys by a three-step heat treatment. what is carried out and what is the purpose at each step? why is such a treatment necessary based on the nucleation and growth theory of phase transformation. compare this treatment with tempering martensite for carbon steels (similarities and differences).