question_answer 21)

You are riding 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. g = 10 \[\text{k=E}{{\text{A}}^{\text{2}}}\text{/V}\].

Answer:

(a) Here, M = 3000 kg ; x = \[\text{T=2 }\!\!\pi\!\!\text{ }\sqrt{\frac{\text{inertia factor}}{\text{spring factor}}}\] m ; If k is the spring constant of each spring, then spring constant of 4 springs in parallel to support the whole mass is, K = 4 k \[\text{=2 }\!\!\pi\!\!\text{ }\sqrt{\frac{\text{m}}{\text{E}{{\text{A}}^{\text{2}}}\text{/V}}}\text{=}\frac{\text{2 }\!\!\pi\!\!\text{ }}{\text{A}}\sqrt{\frac{\text{mV}}{\text{E}}}\] 4 kx = mg or \[\therefore \] \[\text{v=}\frac{\text{1}}{\text{T}}\text{=}\frac{\text{A}}{\text{2 }\!\!\pi\!\!\text{ }}\sqrt{\frac{\text{E}}{\text{mV}}}\] (b) If m is the mass supported by each spring, then \[E=\gamma P,\] From \[\text{ }\!\!\gamma\!\!\text{ =}{{\text{C}}_{\text{p}}}\text{/}{{\text{C}}_{\text{ }\!\!\upsilon\!\!\text{ }}}\text{.}\]\[\text{m/}{{\text{s}}^{\text{2}}}\]or \[0\cdot 15\] or \[\therefore \] or \[k=\frac{Mg}{4x}\] Now, \[=\frac{3000\times 10}{4\times 0\cdot 15}=5\times {{10}^{4}}N/m\] From (i) \[m=\frac{3000}{4}=750kg\]