91Ó°ÊÓ

The given is the mass of the car $m_{car}=1200Kg$.The objective is to find

the spring constant of each spring if the empty car bounces up and down $2.0$times each second and the car's oscillation frequency while carrying four $70Kg$passengers. Frequency formula is used to solve.

(a) Since the mass of the car is equally distributed over the four springs, the mass on each spring is

m =$\frac{1200Kg}{4}=300Kg$

The frequency of oscillation of the car is found from the equation

$f=\frac{1}{2\mathrm{Ï€}}\sqrt{\frac{k}{m}}$

Rearrange and solve for k

$f^2=\frac{1}{4{\mathrm{Ï€}}^2}\frac{k}{m}$

$k=4{\mathrm{Ï€}}^{2}m{f}^2$

Substitute the numerical values

$k=4×{\mathrm{π}}^2×(300\mathrm{Kg})×{(2.0\mathrm{Hz})}^2$

$k=4.74×{10}^4\raisebox{1ex}{$N$}\!\left/ \!\raisebox{-1ex}{$m$}\right.$

(b) The car is now carrying four passengers each of mass $70Kg$. Again, the mass of the car and the passengers are equally distributed over the four springs. So, the mass on each spring is

$m=\frac{1200Kg+4(70Kg)}{4}$

$m=370Kg$

The frequency of oscillation of the car while carrying the passengers is found from the equation

$f=\frac{1}{2\mathrm{Ï€}}\sqrt{\frac{k}{m}}$

Substitute the numerical value

$f=\frac{1}{2\mathrm{π}}\sqrt{\frac{4.74×{10}^4{\displaystyle \raisebox{1ex}{$N$}\!\left/ \!\raisebox{-1ex}{$m$}\right.}}{370Kg}}$