You are riding in an automatic vehicle of mass 3000 kg . Assume t… - Vidyadip

Question

You are riding in an automatic vehicle of mass 3000 kg . Assume that you are testing the oscillatory characteristics of the suspension system of this vechicle when the entire vehicle is placed on it, the suspended is inclined by 15 cm . Also, the amplitude of oscillation decreases by $50 \%$ in the period of one complete oscillation. Estimate the value of the following
(a) spring constant, and
(b) for the shock absorber system of a spring and $a$ wheel damping constant $b$.
Assume that each wheel carries a mass of 750 kg is.

✓

Answer

Given :
(a) Mass of the vehicle
$\begin{aligned}M & =3000 kg \\m & =\text { Mass carried by each wheel } \\m & =750 kg \\y & =15 cm=15 \times 10^{-2} m=0.15 m \\a & =\text { Acceleration }=g \\k & =\text { Spring constant }=? \\\frac{m}{k} & =\frac{y}{a}=\frac{y}{g} \\m g & =k y \\k & =\frac{m g}{y} \\\text {Putting value}\quad k & =\frac{750 \times 9.8}{0.15} \\& =4.9 \times 10^4 N / m \\& \simeq 5 \times 10^4 N / m\end{aligned}$
(b) Recession constant

Time period $T =2 \pi \sqrt{\frac{m}{k}}$
$ \begin{aligned} & =2 \times 3.14 \times \sqrt{\frac{750}{4.9 \times 10^4}} \\ & =\frac{2 \times 3.14 \times 86.60}{700} \\ T & =0.776 \text { second } \end{aligned} $
Amplitude is halved in one cycle
$ A=A_0 e^{\frac{-b t}{2 m}} $
$\begin{aligned} t= T \text { in } A & =\frac{ A _0}{2} \text { putting value } \\ \frac{ A _0}{2} & = A _0 e ^{\frac{-b T}{2 m}} \\ 2 & = e ^{\frac{b T}{2 m}} \\ \frac{b T}{2 m} & =\log _{ e } 2=0.693\end{aligned}$
Damping constant $b=\frac{2 m \times 0.693}{T}$
$\begin{aligned} & =\frac{2 \times 750 \times 0.693}{0.77} \\ & =1350 kg / sec . \\ b & =1350 kg / s \end{aligned}$

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