### Q.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.

**Answer:-**(a) Mass of the automobile, *m* = 3000 kg

Displacement in the suspension system, *x* = 15 cm = 0.15 m

There are 4 springs in parallel to the support of the mass of the automobile.

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* = 2π √*m*/4*k*

and *k* = *mg*/4*x* = 3000 × 10/ 4 × 0.15 = 5000 = 5 × 10^{4} Nm

Spring Constant, *k *= 5 × 10^{4} Nm

(b) Each wheel supports a mass, *M* = 3000/4 = 750 kg

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

*x* = *x*_{0}*e*^{–bt/2M}

The amplitude of oscilliation decreases by 50 %.

∴* x* = *x*_{0}/2

*x*_{0}/2 = *x*_{0}*e*^{–bt/2M}

*
*log

*2 = bt/2*

_{e}*M*

∴

*b*= 2

*M*log

*2 /*

_{e}*t*

where,