Resonance between spring and pendulum: Different frequencies

[greypilgrim]

Hi.

I have a spring with spring constant 30 N/m and a mass of 0.5 kg. With the mass at the bottom, the spring has the length 58 cm at rest. If I now pull down the mass and release it, it starts with a vertical oscillation, then the spring also starts to swing sideways like a pendulum and the vertical oscillation loses amplitude until for a short moment there's only the sideways motion. Then the whole process reverses and starts over.

Apparently the energy moves from the "spring mode" to the "pendulum mode" and back. However, from above values I calculate frequencies of 1.23 Hz for the spring mode and only 0.65 Hz for the pendulum. Wouldn't resonance require them to be the same? There seems to be a factor 2.

 

[darth boozer]

Resonant frequency of a mass on a spring (spring mode) is

f = 1/2π√(m/k)

where f is frequency, m is mass and k is spring constant.

For your setup this is approximately 1.23 Hz.

Resonant frequency of mass on a spring (pendulum mode) is

f = 1/2π√(L/g)

where L is the length of the pendulum and g is the local acceleration of gravity.

For your setup this is approximately 0.65 Hz.The frequencies depend on totally different variables and the apparent relationship is coincidental.

While it appears to be a factor of two, it is only close, probably close enough for the exchange of modes through harmonics.

If any of the parameters, length, spring constant or mass, are changed, the apparent relationship disappears.

Not having tried the actual experiment, I don't know if the changed values would affect the interchange of modes.

 

[nasu]

The transfer of energy between modes is not determined by some resonance. So you don't need to look for a special relationship between the two frequencies.

 

[olivermsun]

I thought resonance is the situation when one mode is able to transfer energy to another efficiently...
In the spring-pendulum system described in the OP, the spring mode being at nearly double the frequency of the pendulum mode is probably quite important for the energy transfer (this sounds like an example of "parametric excitation").

 

[greypilgrim]

While it appears to be a factor of two, it is only close, probably close enough for the exchange of modes through harmonics.

My measurements are far from precise. For example I neglected the mass of the spring itself, and I only estimated the c.o.m. of the weight.

When both modes are active, the down position of the spring mode seems to coincide with the left and right positions of the pendulum mode, which supports the 2:1 ratio theory of the frequencies.