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) M = 3000 kg
x = 15 cm = 0.15 m
k = spring constant of each spring
force constant of 4 springs connected in parallel
K = 4k
4kx = Mg
` k = (Mg)/(4x) = (3000 xx 10)/(4 xx 0.15) = 5 xx 10^(4) N//m`
`m = (3000)/(4) = 750 kg `
As `x = x_(0)e^(-bt//2m)`
`(1)/(2)x_(0) = x_(0)e^(-bt//2m)`
`2 = e^(+bt//2m)`
`log_(e)2 = (bt)/(2m)`
`b = (2m log_(e)2)/(t)`
As `t = 2pisqrt((M)/(K)) = 2 xx (22)/(7) xx sqrt((3000)/(4 xx 5 xx 10^(4))`
= 0.76 s
`therefore b = (2 xx 750 xx 0.693)/(0.75) = 1367.7` kg/s