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Define one newton force.
If a body of $1\ kg.$ is at rest. If a force acts upon it and acceleration of $1\ m/s^2$ has been produced in it, then
Force, $F=mass\times acceleration$
Or $F=1\ kg\times 1\ m/s^2$
Or $F=1\ kg-m/s^2$
Or $F=1\ Newton$
Thus, $1\ N$ is the required force that can accelerate a body having a mass of $1\ kg$ to $1\ m/s^2$.
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