You are riding an automobile of mass 3000kg. Assuming that you are examining the oscillation characteristics of its suspension system. The suspension sags 15cm when the entire automobile is placed on it. Also, the amplitude of oscillation decreases by` 50%` during one complete oscillation. Estimate the values of (a) the spring constant k and (b) damping constant b for the spring and shock absorber system of one wheel, assuming that each wheel supports `750kg.g=10m//s^(2)`.

Explanation

(a) Here , mass M=300 kg, displacement , x=15 cm , g=10 m/`s^(2)`. There are four spring system. If k is the spring constant of each spring, then total spring constant of all the four springs in parallel is
`K_(p)=4k :. M_(g)=k_(p)x=4kx`
`implies K=(Mg)/(4x) =(3000xx10)/(4xx0.15)=5xx10^(4)N`
(b) for each spring system supporting 750 kg of weight,
`t=2pisqrt(m/k) =2xx3.14 xxsqrt(750/(5xx10^(4))) =0.77 sec`
`:. ` Using `x=x_(0)e^(-M/(2m)),` we get
`50/100 x_(0) =x_(0)e^(-(bxx0.77)/(2xx750))` or `e^((0.77b)/(1500))=2`
Taking logrithm of both sides,
`(0.77b)/0.77xx2.303xx0.3010`
`=1350 . 4 kg s^(-1)`