A car of mass 1300 kg is carrying two occupants, each of mass 80 kg. The car is supported by four springs, each with a spring constant of 20,000 N/m. Calculate the frequency of vibration of the car.

The mass of the car is $1300 \text{ kg}$.
The combined mass of two people is $160 \text{ kg}$.

$$m = m_{car} + m_{people}$$ $$m = 1300 \text{ kg} + 160 \text{ kg}$$ $$m = 1460 \text{ kg}$$

The car is supported by four springs, and each spring has a force constant of $20,000 \text{ N/m}$. Since the springs act in parallel to support the car, their effective spring constant is the sum of their individual constants.

$$k_{total} = 4 \times k_{each}$$ $$k_{total} = 4 \times 20,000 \text{ N/m}$$ $$k_{total} = 80,000 \text{ N/m}$$

The frequency of vibration for a mass-spring system is given by the formula:

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

Substitute the calculated values for $m$ and $k_{total}$:

$$f = \frac{1}{2\pi} \sqrt{\frac{80,000 \text{ N/m}}{1460 \text{ kg}}}$$ $$f = \frac{1}{2\pi} \sqrt{54.7945 \text{ s}^{-2}}$$ $$f = \frac{1}{2\pi} \times 7.4023 \text{ s}^{-1}$$ $$f \approx \frac{7.4023}{6.28318} \text{ Hz}$$ $$f \approx 1.178 \text{ Hz}$$

Rounding to three significant figures:

$$f \approx 1.18 \text{ Hz}$$

The frequency of vibration of the car is $1.18 \text{ Hz}$.