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  • You are riding in a automobile of Mass 4000 Kg  Assuming that you are examining the oscillation characteristics of its suspension system. The suspension says  18 cm  when the en

You are riding in a automobile of Mass 4000 Kg  Assuming that you are examining the oscillation characteristics of its suspension system. The suspension says  18 cm  when the entire automobile is placed on it. Also. the amplitude of oscillation decreases by 20% during one complete oscillation. Estimate the values of                                                         

(a) The spring constant K

(b) The damping constant  b for the spring and shock absorber system of one wheel assuming that each wheel supports  600 Kg


 

Option: 1

K=\frac{1}{18} \times 10^6 \mathrm{~N},  b=\frac{2000}{0.843} \log e^5


Option: 2

K=\frac{1}{19} \times 10^6 \mathrm{~N},b=\frac{3000}{0.843} \log e^5


Option: 3

  K=\frac{1}{20} \times 10^6 \mathrm{~N},b=\frac{4000}{0.843} \log e^5


Option: 4

K=\frac{1}{24} \times 10^6 \mathrm{~N},b=\frac{5000}{0.843} \log e^5


Answers (1)

best_answer

(a) Here Mass M = 4000 Kg
displacement  x=18 cm =0.18 m   g=10 m / s2

There are four spring systems. If  k  is the spring constant of each spring then total spring constant of all the four springs in parallel is

K_p =4 K \quad \therefore \mathrm{Mg}=K_p x=4 \mathrm{K} x\\

K =\frac{\mathrm{Mg}}{4 x}=\frac{4000 \times 10}{4 \times 0.18}=\frac{4000 \times 1000}{4 \times 18} \\

       =\frac{1}{18} \times 10^6 \mathrm{~N}

(b) For each spring system supporting

 =\frac{4000}{4}=1000 \mathrm{~kg} \text { of weight } \\
t=2 \pi \sqrt{\frac{M}{K} }=2 \times 3.14 \times \sqrt{\frac{1000}{\frac{1}{18}\times10^{6}}\ }

t=\sqrt{180\times10^{-4}}\,\times \,6.28=6.28\times 13.42\times 10^{-2}

t=0.843 \,\,sec

\therefore \,using\,\,x=x_{0}e^{\frac{-bt}{2M}}\,\,\,\,\,we \,\, get\,\,\,

\frac{20}{100}\,x_{0}= x_{0}e^{\frac{-b\times 0.843}{2\times 1000}}=x_{0}e^{\frac{-b\times 0.843}{2000}}

\text { or } e^{\frac{0.843 \,b}{2000}}=5 \\

Taking the natural logarithm of both sides

=\frac{0.843\,b}{2000}=\log e^5 \\

b=\frac{2000}{0.843} \log e^5

Posted by

Ritika Kankaria

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