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

Mass of the automobile, m=3000kg
Displacement in the suspension system, x=15cm=0.15m
(i)There are 4 springs in parallel to the support of the mass of the automobile.The equation for the restoring force for the system:
F=4kx=mg
Where, k is the spring constant of the suspension system
Time period,` T=2pisqrt(m/(4k)) ` and `k=(mg)/(4x) =(3000xx10)/4xx0.15)=5000=5xx10^(4)N//m`
Spring constant, `k=5xx10^(4)N//m`
Each wheel supports a mass, `M=(3000)/(4)=750kg`
(ii)For damping factor b, the equation for displacement is written as:`x=x_(0)e^(-btau//2M)`
The amplitude of oscillation decreases by 50% `x=x_(0)/2`
`(x_(0))/2=x_(0)e(-btau//2M) `
` log_(e)2=(btau)/(2M)`
`b=(2Mlog_(e)2)/(tau)`
where
Time period ,`tau=2pisqrt(m/(4k))=2pisqrt((3000)/(4xx5xx10^(4)))=0.791s`
`b=(2xx750xx0.693)/(0.7691)=1351.58kg//s`
Therefor, the damping constant of the spring is `1351.58kg//5`