Ncert Exercises & Solutions

14.21 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.

The mass of the automobile is supported by 4 springs in parallel .

The equation for the restoring force for the system :

F = - 4 kx = mg

where k is the spring constant of the suspension system .

Time period, T = 2 π \sqrt{\frac{m}{4 k}}

And spring constant, k = \frac{mg}{4 x} = \frac{3000 \times 10}{4 \times 0.15} = 5000 = 5 \times 10^{4} N / m

Each wheel supports a mass, M = \frac{3000}{4} = 750 kg

For damping factor b, the equation for displacement is written as :

x = x_{0} e^{- bt / 2 M}

The amplitude of oscillation decreases by 50 % .

\begin{array}{l} ∴ x = \frac{x_{0}}{2} \\ \frac{x_{0}}{2} = x_{0} e^{- bt / 2 M} \\ \log_{e} 2 = \frac{bt}{2 M} \\ ∴ b = \frac{2 {Mlog}_{e} 2}{t} \end{array}

where,

t = time period = T = 2 π \sqrt{\frac{m}{4 k}} = 2 π \sqrt{\frac{3000}{4 \times 5 \times 10^{4}}} = 0.7691 s

Therefore, the damping constant of the spring is 1351.58 kg / s .

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