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. Also, the amplitude of oscillation decreases by 50% during one complete oscillation. Estimate the values of

**(a)** the spring constant k and

**(b)** the damping constant b for the spring and shock absorber system of one wheel, assuming that each wheel supports 750 kg.

Asked by Abhisek | 1 year ago | 141

**(a)** Mass of the automobile = 3000 kg

The suspension sags by 15 cm

Decrease in amplitude =50% during one complete oscillation

Let k be the spring constant of each spring, then the spring constant of the four springs in parallel is

K= 4k

Since F = 4kx

Mg = 4kx

⇒ k =\( \dfrac{Mg}{4x}\)

=\( \dfrac{ (3000 \times 10)}{(4 \times 0.15) }\)

= 5 x 10^{4} N

**(b)** Each wheel supports 750 kg weight

t = \( \dfrac{2π}{\dfrac{\sqrt{m}}{\sqrt{k}}}\)

= 2 x 3.14 x \( \dfrac{\sqrt{750}}{\sqrt{5}\times 104}\)

= 0.77 sec

Using, x =\( x_{0}e^{-\dfrac{bt}{2m}},\)

we get

\( \dfrac{50}{100}x_{0} \)

\( = x_{0}e^{-\dfrac{b\times 0.77}{2\times 750}}\)

\( log_{e}^2\dfrac{ (b\times 0.77)}{(1500)log_{e}^e}\)

b= \( \dfrac{(1500)log_{e}2 }{0.77}\)

b = \( \dfrac{(1500 \times 0.6931)}{0.77}\) = 1350.2 kg/s

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