Cars have shock absorbers to damp the oscillations that would otherwise occur when the springs that attach the wheels to the car's frame are compressed or stretched. Ideally, the shock absorbers provide critical damping. If the shock absorbers fail, they provide less damping, resulting in an underdamped motion. You can perform a simple test of your shock absorbers by pushing down on one corner of your car and then quickly releasing it If this results in an up-and- down oscillation of the car, you know that your shock absorbers need changing. The spring on each wheel of a car has a spring constant of \(4005 \mathrm{~N} / \mathrm{m}\), and the car has a mass of \(851 \mathrm{~kg}\), equally distributed over all four wheels. Its shock absorbers have gone bad and provide only \(60.7 \%\) of the damping they were initially designed to provide. What will the period of the underdamped oscillation of this car be if the pushing-down shock absorber test is performed?

Step by step solution

01

Get the effective spring constant for one wheel

Since the car's mass is equally distributed over all four wheels, we can calculate the effective spring constant for one wheel by dividing the car's total spring constant by 4. \(k_{eff} = \frac{4005 \mathrm{~N/m}}{4} = 1001.25 \mathrm{~N/m}\)

02

Calculate the critical damping coefficient

The critical damping coefficient can be calculated by using the following formula: \(c_{crit} = 2 \sqrt{m k_{eff}}\) Plug in the mass m = 851 kg and effective spring constant \(k_{eff} = 1001.25 \mathrm{~N/m}\): \(c_{crit} = 2 \sqrt{(851 \mathrm{~kg})(1001.25 \mathrm{~N/m})} = 1841.90 \mathrm{~N s/m}\)

03

Find the actual damping coefficient

As the shock absorbers provide only 60.7% of the damping, multiply the critical damping coefficient by this percentage to find the actual damping coefficient: \(c = 0.607 \cdot c_{crit} = 0.607 \cdot 1841.90 \mathrm{~N s/m} = 1117.62 \mathrm{~N s/m}\)

04

Calculate the angular frequency of the underdamped oscillation

The angular frequency of the underdamped oscillation can be calculated using the following formula: \(\omega_d = \sqrt{\frac{k_{eff}}{m} - \frac{c^2}{4m^2}}\) Plug in the effective spring constant \(k_{eff} = 1001.25 \mathrm{~N/m}\), mass m = 851 kg, and actual damping coefficient \(c = 1117.62 \mathrm{~N s/m}\): \(\omega_d = \sqrt{\frac{1001.25 \mathrm{~N/m}}{851 \mathrm{~kg}} - \frac{(1117.62 \mathrm{~N s/m})^2}{4(851 \mathrm{~kg})^2}} = 0.4847 \mathrm{~rad/s}\)

05

Find the period of the underdamped oscillation

Finally, now that we have the angular frequency, we can calculate the period of the underdamped oscillation using the following formula: \(T = \frac{2 \pi}{\omega_d}\) Plug in the angular frequency \(\omega_d = 0.4847 \mathrm{~rad/s}\): \(T = \frac{2 \pi}{0.4847 \mathrm{~rad/s}} = 12.97 \mathrm{~s}\) Thus, the period of the underdamped oscillation of this car will be approximately 12.97 seconds if the pushing-down shock absorber test is performed.

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