# Simple harmonic motion amplitude of oscillation

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?

• The small angle approximation is only needed for pendulum-type SHO. – Gert Feb 8 '20 at 11:39
• 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. – Pratham Hullamballi Feb 8 '20 at 12:26

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.

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).

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