When four people with a combined mass of 181 kg sit down in a car, the car drops 1.5 cm lower on its springs. After they exit, they bounce the car up and down. What is the frequency of the car's vibration if its mass (when empty) is [tex]2.0 \times 10^3 \, \text{kg}[/tex]?

The frequency of the car's vibration, considering the four people's weight, is approximately 0.150 Hz, calculated using the change in weight on the springs and the car's spring constant.

1. Calculate the change in weight on the springs when the four people sit in the car:

- Change in weight on the springs = Change in height * (Mass of four people * Acceleration due to gravity)

- Given: Change in height = 1.5 cm = 0.015 m

- Given: Mass of four people = 181 kg

- Acceleration due to gravity = 9.8 m/s²

- Change in weight on the springs = 0.015 m * (181 kg * 9.8 m/s²)

- Change in weight on the springs ≈ 26.61 N

2. Calculate the spring constant of the car's suspension system:

- Spring Constant = Change in weight on the springs / Change in height

- Spring Constant = 26.61 N / 0.015 m

- Spring Constant ≈ 1774 N/m

3. Calculate the mass of the car (when empty) in kilograms:

- Given: Mass of the car = 2.0 × 10³ kg

4. Calculate the natural frequency of vibration of the car's suspension system:

- Frequency = 1 / (2π) * √(Spring Constant / Mass of the car)

- Frequency = 1 / (2π) * √(1774 N/m / 2000 kg)

- Frequency ≈ 1 / (2π) * √(0.887 N/m)

- Frequency ≈ 1 / (2π) * 0.942 Hz

- Frequency ≈ 0.150 Hz

So, the frequency of the car's vibration is approximately 0.150 Hz.