# Resonance between spring and pendulum: Different frequencies

• greypilgrim
However, the pendulum mode loses amplitude quickly and only has a sporadic swing, which suggests that resonance does not occur.The energy transfer between modes is not determined by some resonance.

#### 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.

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.

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.

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").

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darth boozer said:
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.

## 1. What is resonance between a spring and pendulum?

Resonance between a spring and pendulum refers to the phenomenon where the two objects vibrate at the same frequency when the spring and pendulum are connected together. This results in a significant increase in the amplitude of the motion of both objects.

## 2. What causes resonance between a spring and pendulum?

Resonance between a spring and pendulum is caused by the transfer of energy between the two objects. When the spring and pendulum are connected, the energy from the pendulum is transferred to the spring, causing it to vibrate at the same frequency as the pendulum.

## 3. How does the frequency of the spring and pendulum affect resonance?

The frequency of the spring and pendulum has a direct impact on resonance. When the frequencies of the two objects are the same, resonance occurs and the amplitude of the vibrations increases. However, if the frequencies are different, the objects will not resonate and the amplitude will remain low.

## 4. Can resonance between a spring and pendulum be harmful?

In most cases, resonance between a spring and pendulum is not harmful. However, if the amplitude of the vibrations becomes too high, it can cause damage to the objects or surrounding structures. This is why engineers and scientists carefully study resonance and design structures to prevent it from occurring.

## 5. How is resonance between a spring and pendulum useful?

Resonance between a spring and pendulum has many practical applications. It is used in clocks and watches to keep time accurately, and in musical instruments to produce specific tones. It is also used in industrial settings to detect and measure vibrations and in earthquake-resistant building designs.