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In a simple pendulum system, how does the extensibility/elasticity of the string affect the time period of oscillation? Would it lead to a random or systematic error? Would the elasticity of the string result in changing lengths of the pendulum across the oscillation, hence altering the time period?

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For small oscillations the time period of a pendulum is

$$ T \approx 2\pi\sqrt\frac{L}{g} $$

where $L$ is the length of the string and $g$ is the gravitational constant. An elastic string would increase the length of the string the higher the velocity is (i.e. closer to the bottom), due to the centrifugal force. Therefore, such a string would increase the pediod length of the pendulum.

For small oscillations and not very elastic strings, the difference to a regular pendulum will not be too large and the deviations will be systematic. Depending on the initial conditions (especially large amplitudes and/or potential energy in the spring) it will rather be a chaotic system with no periodic behaviour anymore.

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  • $\begingroup$ The system has another vibration mode, oscillation along the length of the string, which has a constant frequency depending on the stiffness of the string and the size of the mass. The interaction between the two different modes of vibration can be complicated, especially if they have almost the same frequency. $\endgroup$ – alephzero Apr 13 at 11:04
  • $\begingroup$ Yes thanks for the input. What I meant by "not very elastic strings" is exactly that, but poorly worded. If the periods of the two systems are very different, the pendulum acts as a driving force far from resonance for the spring, giving the behaviour in the first paragraph. Otherwise it can be very chaotic. $\endgroup$ – noah Apr 13 at 11:10
  • $\begingroup$ Thanks! Do you have any references for the answer? $\endgroup$ – radishearrings777 Apr 14 at 5:40
  • $\begingroup$ These guys did some numerics on it. $\endgroup$ – noah Apr 14 at 9:55

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