# Q. 14.21 (b) You are riding in an automobile of mass $\inline 3000kg\;$. Assuming that you are examining the oscillation characteristics of its suspension system. The suspension sags $\inline 15\; cm$ when the entire automobile is placed on it. Also, the amplitude of oscillation decreases by $\inline 50^{o}/_{o}$ during one complete oscillation. Estimate the values of(b) the damping constant b for the spring and shock absorber system of one wheel, assuming that each wheel supports$\inline 750 \; kg.$.

The amplitude of oscillation decreases by 50 % in one oscillation i.e. in one time period.

$\\T=2\pi \sqrt{\frac{m}{k}}\\ T=2\pi \times \sqrt{\frac{3000}{4\times 4.9\times 10^{4}}}\\ T=0.77\ s$

For damping factor b we have

$x=x_{0}e^{\left ( -\frac{bt}{2m} \right )}$

x=x0/2

t=0.77s

m=750 kg

$\\e^{-\frac{0.77b}{2\times 750}}=0.5\\ ln(e^{-\frac{0.77b}{2\times 750}})=ln0.5\\ \frac{0.77b}{1500}=ln2\\ b=\frac{0.693\times 1500}{0.77}\\ b=1350.2287\ kg\ s^{-1}$

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