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# You are riding in an automobile of mass 3000 kg .Assuming that you are examining the oscillation characteristics of its suspension system. The suspension sags 15 cm when the entire automobile is placed on it. (b) the damping constant b

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$\inline 750 \; kg.$.

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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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