# Spring, resonance frequency and a bit of fun

1. Feb 6, 2010

### magwas

1. The problem statement, all variables and given/known data

The Hungarian national highway M7 is made of concrete blocks 4m long. The joints have a small gap.
Józsi have a mass of 80 kg.
He owns a Polski Fiat, which is lowered by 10 cm when he sits in.

At what speed would Józsi's car disintegrate?

2. Relevant equations
v = l/t
F = -k*dx
F = m*a
g = 9.81 kg*m/s

3. The attempt at a solution

I remember I have solved it once (maybe with other numbers), but the road got some asphalt coating since.

2. Feb 7, 2010

### kuruman

I have never heard of a car disintegrating on the highway, but then again I have never driven a Polski Fiat. You need to find the resonant frequency of the car, but for that I think that you will need the mass of the car in addition to Józsi's mass.

3. Feb 7, 2010

### magwas

hooke's law:
$$F = - dy k$$
resonance frequency:
$$\omega = \sqrt{\frac{k}{m}}$$

solve hooke's law for k and substitue gravitational force of Józsi

$$k = - \frac{g m_{j}}{dy}$$

substituing k for the resonance frequency

$$\omega = \sqrt{- \frac{g m_{j}}{dy m}}$$

m here is the combined mass of Józsi and the car:

$$\omega = \sqrt{- \frac{g m_{j}}{dy \left(m_{c} + m_{j}\right)}}$$

substituing the given numbers and 800 kg for the car:
$$\omega = 2.98633250295104 \ \ \frac{1}{s}$$

now $$v = l \omega = 11.9453300118042 \frac{m}{s} = 43.003188042495 \frac{km}{h}$$

If you would have driven on M7 at that time, you would not be surprised when your car disintegrated :)

4. Feb 7, 2010

### kuruman

One slight problem. The speed should be v/T where T is the period, i.e. the time elapsed from bump to bump. How is ω related to T?

5. Feb 8, 2010

### magwas

I would think $$\omega=\frac{1}{T}$$. Each bumps should reach the car at the same phase to increase the amplitude, I guess.

6. Feb 8, 2010

### ehild

$$\omega=\frac{2\pi}{T}$$

ehild

7. Feb 8, 2010

### magwas

I see. $$\omega$$ is the angular velocity, given in $$\frac{rad}{s}$$, and not the frequency.
So $$v = l \frac{\omega}{2\pi} = 1.9 \frac{m}{s} = 6.8 \frac{km}{h}$$

8. Feb 8, 2010

### kuruman

That speed is a bit low for a car to disintegrate, but you know M7 and the Polski Fiat (and what they can do to each other) better than I.

9. Feb 8, 2010

### kuruman

This is where the problem is

If m is the combined mass supported by the spring equivalent of the car when the car is loaded, then

$$k=\frac{mg}{dy}$$
$$\omega=\sqrt{\frac{k}{m}}=\sqrt{\frac{g}{dy}}$$

10. Feb 8, 2010

### magwas

Well, our teacher back then have given numbers which lead to some 80 km/s, presumably dy has been very different, and the mass of the car a bit (I have looked it up, 905 kg is given by manufacturer).
I always thought of it as a funny example which raises the spirit in the classroom, making understanding resonance frequency easier. Now it also serves an example of how much one can forget about highschool physics if it isn't used:)
Tried to come up with a dy to get 80 km/h. I got something around 0.6 mm.

dy is the displacement of the car as Józsi sits in, so I believe that one should use only mass of Józsi in the numerator. However to find the resonance frequency of the whole system, one should use the combined mass, which is in the denominator.

11. Feb 8, 2010

### kuruman

You are correct, of course. I don't know what I was thinking.

12. Feb 8, 2010

### ideasrule

I guess what they say about Hungarians is true...

13. Feb 8, 2010

### magwas

I am curious. What do they say?