Question

You are riding in an automobile of mass 3000 kg. Assuming that you are examining the oscillation characteristics of its suspension system, the suspension hangs 15 cm when the entire automobile is placed on it. Also, the amplitude of oscillation decreases by $50\%$ during one complete oscillation. The values of the spring constant k and the damping constant b for the spring and shock absorber system of one wheel (assuming that each wheel supports 750 kg) are:

• A. $2\times {10}^{5}N{m}^{-1},1.8kg{s}^{-1}$
• B. $5\times {10}^{4}N{m}^{-1},8.1kg{s}^{-1}$
• C. $2\times {10}^{5}N{m}^{-1},1344kg{s}^{-1}$
• D. $5\times {10}^{4}N{m}^{-1},1344kg{s}^{-1}$

Answer 1

The correct option is D

$5\times {10}^{4}N{m}^{-1},1344kg{s}^{-1}$

Give, mass of the automobile m = 3000 kg

Since the suspension sags by 15 cm when the entire automobile is placed on it, we must remember there are four wheels hence four suspensions.

We have weight of entire vehicle = spring force due to four suspensions

$3000\times g=k\times \left(\frac{15}{100}\right)\times 4$

$\Rightarrow 3000\times 10=k\times \frac{15}{100}\times 4\Rightarrow k=5\times {10}^{4}N{m}^{-1}$

We know the time dependence of amplitude during oscillation

$A\left(t\right)=A_0{e}^{\frac{-bt}{2m}}$

In one time period $A\left(T\right)=\frac{A_0}{2}$ (is what is given)

$\frac{A_0}{2}=A_0{e}^{\frac{-bt}{2m}}$

$\Rightarrow -In2=\frac{-bt}{2m}$

Also ${\omega }^{\prime }=\sqrt{\frac{k}{m}-\frac{b^2}{4m^2}}\Rightarrow T=\frac{2\pi }{\sqrt{frackm-\frac{b^2}{4m^2}}}$

$\Rightarrow In2=\frac{b\times 2\pi }{2m\sqrt{\frac{k}{m}-\frac{b^2}{4m^2}}}$

$m^2(ln2{)}^2=\frac{b^2{\pi }^2}{\frac{k}{m}-\frac{b^2}{4m^2}}$

$(ln2{)}^2{m}^2(\frac{k}{m}-\frac{b^2}{4m^2})=b^2{\pi }^2$

$(ln2{)}^{2}mk\frac{(ln2{)}^2}{4}{b}^2=b^2{\pi }^2$

$\frac{(ln2{)}^{2}mk}{{\pi }^2+\frac{\left(ln2\right)2}{4}}=b^2$

$b\approx 1344kg/s$