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I know that for a pendulum we need small amplitudes. But why is it necessary that a spring oscillator should have small amplitude of oscillation?

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  • $\begingroup$ The small angle approximation is only needed for pendulum-type SHO. $\endgroup$ – Gert Feb 8 '20 at 11:39
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    $\begingroup$ What I can think of is that Hooke's law works to an extent for spring. Beyond that, due to elasticity and elongation of the spring, the K value in Hooke's Law becomes variable. Therefore, to avoid this, the amplitude must be small. $\endgroup$ – Pratham Hullamballi Feb 8 '20 at 12:26
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Small amplitudes are only required for the pendulum.

However, it's true that we could expand it to all SHM. Why? Because a simple harmonic motion (SHM) is the motion caused by an ideal spring.

That is, a motion caused by a force $F=-k\cdot x$

The thing is that this is an idealization. As always, reality isn't that beautiful. There is friction, and there are many other effects. In particular, if you pull a spring too much, you might deformate it. Then, it won't obey $F=-k x$ any longer.

In conclusion: if amplitudes are too big, a REAL spring might cease to fulfill $F=-kx$, so it won't cause SHM any longer.

But that's for REAL spring. On paper, we always treat them as ideal springs, which satisfyf $F=-kx$ for ANY ampltiude. This is how we model reality. Reality is always more complex.

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The equation of motion of the simple pendulum is:

$$l\ddot{\theta}+g\sin \theta=0$$

This solves for small $\theta$ because then:

$$\sin \theta \approx \theta$$

So:

$$l\ddot{\theta}+g\theta=0$$

For the mass-spring SHO the EoM is:

$$m\ddot{x}+kx=0$$

As long as $k$ remains invariant of $x$, no approximation is needed. Amplitude now doesn't matter (within reason).

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  • $\begingroup$ In my textbook it has been asked that why should amplitude of an oscillating body be small $\endgroup$ – User Feb 8 '20 at 12:07
  • $\begingroup$ They refer to the pendulum. $\endgroup$ – Gert Feb 8 '20 at 12:09

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