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For simply harmonic motion, acceleration $= -\omega^2 x$, where $\omega$ is the angular frequency.

Within limits of Hooke's law, the restoring force on the spring is given by

$$F= -k \cdot x$$

This force fits the simple harmonic motion condition with $k=\omega^2m$.

But, we have $F = 0$ for $x=0$.

If I displace block (of some mass) attached to a spring (massless) rightward (on horizontal plane), and then release it, it would accelerate leftward because of above spring force.

But at the moment when $x=0$, $F=0$ and thus $a=0$.

Then why does the spring block system experience simple harmonic motion?

Or why would the block accelerate farther leftward if there is no force on it?

I'm confused.

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    $\begingroup$ A=0, doesn't Imply V=0, the object will continue to move and then subsequently experience a force $\endgroup$ Mar 16, 2022 at 13:26
  • $\begingroup$ You might be interested in this related question and its answers which cover a related topic. It might help in your general understanding of simple harmonic motion. $\endgroup$
    – Floris
    Mar 16, 2022 at 22:54
  • $\begingroup$ Simple harmonic motion : Wikipedia $\endgroup$
    – J...
    Mar 17, 2022 at 0:57

2 Answers 2

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But at the moment when $x=0,\, F=0$ and thus $a=0$, but the velocity $v\ne 0$ so the block overshoots the $x=0$ position and then has a force acting on it in the opposite direction to its direction of travel.

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Or why would the block accelerate farther leftward if there is no force on it?

Assuming the spring-mass system is horizontal then you are right: at $x = 0$ there is no net force on the block which means net acceleration is also zero. But you are forgetting that the block (mass) has certain velocity at that point and it keeps moving until negative acceleration provided by the spring restoring force stops it completely.

Then why does the spring block system experience simple harmonic motion?

Because of inertia of the block (mass) connected to the spring.

For simply harmonic motion, acceleration $= -\omega^2 x$, where $\omega$ is the angular frequency.

This is correct. From $F = -kx$ it follows that acceleration depends on the spring elongation $x$ which is defined as a sine function (harmonic oscillation). Therefore, the acceleration of the block itself is also a sine function.

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    $\begingroup$ -1. Yes, the acceleration is a sine function of time. It is also true that $a = -\omega^2 x$ is a correct expression for the acceleration as a function of $x$. $\endgroup$
    – Jojo
    Mar 16, 2022 at 21:53
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    $\begingroup$ @Joe Well done, SE Physics needs more users like you! $\endgroup$ Mar 16, 2022 at 22:28

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