Jayvion Caldwell

2022-07-14

We choose a system to be a spring + a connected body. If we stretch the body and leave it, it will make a simple harmonic motion. The center of mass of this system is approximately the center of mass of the body because the spring has a negligible mass. Newton's second law states that: ${F}_{external}=M{a}_{cm}$ However, it is apparent that the center of mass has an acceleration though the total external force on the system is zero. What is the explanation?

Brenton Gay

Expert

When applying Newton's second law ${\stackrel{\to }{F}}_{\text{net,ext}}={m}_{\text{sys}}{\stackrel{\to }{a}}_{\text{CM}}$, you must be very diligent in keeping track what is internal and external to your system. The net force ${\stackrel{\to }{F}}_{\text{net,ext}}$ only and always deals with forces exerted by external objects, meaning those forces produced by objects you choose not to be in your system. The mass and acceleration, on the other hand, deal with internal objects, never external ones.
In your situation, it sounds like your system is the spring and the block. All other objects are external and should be interpreted as exerting an external force on your system. One such external object is the object holding the fixed end of the spring in place. Presumably a wall of some sort. This wall exerts a time varying force of magnitude $k\mathrm{\Delta }\stackrel{\to }{s}$ or $-k\mathrm{\Delta }\stackrel{\to }{s}$, depending on which convention you choose for the meaning of $\mathrm{\Delta }\stackrel{\to }{s}$

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