A 86.0 kg person steps into a car of mass 2500.0 kg, causing it to sink 3.0 cm. Assuming no damping, with what frequency will the car and passenger vibrate on the springs?

Here is what I do. I take the mass and set it equal to k*x to get k, then divide by 4. Then I add the 2 masses to get the total mass, and I divide this by 4 as well. I plug these numbers into T = 2*pi*sqtroot of m/k to find T. Finally, I do 1/T to get the frequency.

This is obviously wrong, so can someone please tell me where I'm messing up?

To calculate the frequency at which the car and passenger vibrate on the springs, you need to use Hooke's Law and the concept of simple harmonic motion.

First, let's find the spring constant (k) of the car's suspension system using the given information. Hooke's Law states that the force exerted by a spring is directly proportional to the displacement from its equilibrium position. In equation form, it is expressed as F = -kx, where F is the force, k is the spring constant, and x is the displacement.

Given that the person's weight causes the car to sink 3.0 cm (or 0.03 m), we can write the equation as:
mg = kx
where m is the mass of the person (86.0 kg) and g is the acceleration due to gravity (approximately 9.8 m/s^2).

Rearranging this equation, we get:
k = mg/x = (86.0 kg × 9.8 m/s^2) / 0.03 m = 282666.7 N/m

Now, let's find the total mass (M) of the car and passenger. The total mass is simply the sum of the person's mass and the car's mass:
M = m + M_car = 86.0 kg + 2500.0 kg = 2586.0 kg

To calculate the period (T) of the vibrating system, we use the formula T = 2π√(M/k):
T = 2π√(2586.0 kg / 282666.7 N/m)

Finally, to find the frequency (f), we take the reciprocal of the period:
f = 1/T

You can now calculate the frequency using a calculator. Just be sure to properly convert the units to get the correct result.

Note: Damping is assumed to be absent in this problem, so the system will vibrate indefinitely at the calculated frequency. In reality, damping usually exists, causing the vibrations to slowly decay over time.