Calculate the Frequency and Period of Oscillations: Bad Shock Absorbers in a Car

The frequency of the car’s oscillations will be that of a simple harmonic oscillator as given in the equation $f=\frac{1}{\mathrm{2\pi }}\sqrt{\frac{k}{m}}$. The mass and the force constant are both given.

1. Enter the known values of k and m:
   $$f=\frac{1}{\mathrm{2\pi }}\sqrt{\frac{k}{m}}=\frac{1}{\mathrm{2\pi }}\sqrt{\frac{6\text{.}\text{53}\times {\text{10}}^4\text{N/m}}{\text{900}\text{kg}}}.$$

   16.17
2. Calculate the frequency:
   $$\frac{1}{\mathrm{2\pi }}\sqrt{\text{72.}6/{\text{s}}^{\mathrm{–2}}}=1\text{.}{\text{3656}/\text{s}}^{\text{–1}}\approx 1\text{.}{\text{36}/\text{s}}^{\text{–1}}=\text{1.36 Hz}.$$

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3. You could use $T=\mathrm{2\pi }\sqrt{\frac{m}{k}}$ to calculate the period, but it is simpler to use the relationship $T=1/f$ and substitute the value just found for $f$:
   $$T=\frac{1}{f}=\frac{1}{1\text{.}\text{356}\text{Hz}}=0\text{.}\text{738}\text{s}.$$

   16.19

The values of $T$ and $f$ both seem about right for a bouncing car. You can observe these oscillations if you push down hard on the end of a car and let go.