Question

A A compact car has a mass of $1200\mathrm{kg}$. Assume that the car has one spring on each wheel, that the springs are identical, and that the mass is equally distributed over the four springs.

a. What is the spring constant of each spring if the empty car bounces up and down $2.0$times each second?

b. What will be the car's oscillation frequency while carrying four$70\mathrm{kg}$ passengers?

Short answer

(a) the spring constant of each spring if the empty car bounces up and down $2.0$times each second is k = $4.74*{10}^4\raisebox{1ex}{$N$}\!\left/ \!\raisebox{-1ex}{$m$}\right.$

(b) the car's oscillation frequency while carrying four $70Kg$passengers is f =$1.8Hz$

Step-by-step solution

Introduction 

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

$f=\frac{1}{2\pi }\sqrt{\frac{k}{m}}$

Rearrange and solve for k

$f^2=\frac{1}{4{\pi }^2}\frac{k}{m}$

$k=4{\pi }^{2}m{f}^2$

Substitute the numerical values

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

$k=4.74\times {10}^4\raisebox{1ex}{$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

$f=\frac{1}{2\pi }\sqrt{\frac{k}{m}}$

Substitute the numerical value

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

$f=1.8Hz$