Why do we need to add a mass to a spring to make a simple harmonic motion in it? Why can't only a spring 'without a mass' make a simple harmonic motion when we apply an external force?
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$\begingroup$ If you have the means to read a Mathematica notebook, you can check this out library.wolfram.com/infocenter/ID/7773 to see the additional complexities of having a massive spring. Low level physics courses do not assume the student has experience solving partial differential equations. $\endgroup$– Bill WattsCommented Apr 21, 2021 at 21:20
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3 Answers
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Because the mass of spring is distributed along the spring, the motion will not be harmonic, but more complicated.
answered Apr 21, 2021 at 19:08
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There's no reason, really. It's just to provide students with a simple, clear, metaphor.
In fact any physical system in which the strength applied varies at least in a fist approximation linearly with a distance or angle or generalised coordinate will follow the pattern:
$$m\ddot{x}+kx=0$$
So, basically, all mechanical systems enter into that category for small amplitude movements.
- A pendulum.
- A guitar String.
- A vibrating membrane.
The reason behind the ubiquity of this pattern is found in Taylors's development formula https://en.wikipedia.org/wiki/Taylor_series (for x small: $f (x) \sim k \cdot x)$
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Because the Newtonian equation of motion of a spring-mass system is:
$$m\ddot{x}+kx=0$$
which is the ODE of a harmonic oscillator, with solution:
$$x(t)=A\sin\left(\omega t+\varphi\right)$$
where $\omega^2=\frac{k}{m}$.
