For each of the following loop transfer function f(s), find the portion of the root locus on the real axis, asymptotes, the arrival and departure angles at any complex zero or pole, and the frequency of any imaginary-axis crossing. plot the sketch of the root locus based on above findings. verify your results using matlab. submit plots of your hand sketches and the matlab results.
a. f(s)=- s(s+1)(s +5)(8 +10)
b. f(s)=; s +2 s(s+1)(3+5)(8+10) (s+2)(s+6)
c. f(s)= (+1)(3+5)(s+10)

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.

The filament of an incandescent lamp has a temperature of 2000k. calculate the fraction of radiation emitted in the visible light band if the filament is approximated as blackbody

The thermal expansion or contraction of a given metal is a function of the f a)-density b)-initial temperature c)- temperature difference d)- linear coefficient of thermal expansion e)- final temperature f)- original length