The unit of force is defined specifically to make the constant one.
Explanation
Newton’s Second Law of Motion states that the force applied to a body is directly proportional to the product of its mass and acceleration ($F \propto ma$). This can be written as an equation $F = kma$, where $k$ is a constant of proportionality. The numerical value of $k$ depends entirely on the units selected for measuring force, mass, and acceleration.
Detailed Analysis:
In coherent systems of units (such as SI or CGS), the units of derived quantities are defined to simplify mathematical formulations by making proportionality constants equal to unity (1).
- The unit of force is defined specifically to satisfy the condition $k=1$.
- In the SI system: The unit of force is the Newton. One Newton is defined as the force required to produce an acceleration of $1 \text{ m/s}^2$ in a body of mass $1 \text{ kg}$.
Substituting these values into the equation:
$1 \text{ N} = k \times (1 \text{ kg}) \times (1 \text{ m/s}^2)$
This implies $k = 1$. - In the CGS system: The unit of force is the Dyne, defined similarly to ensure $k=1$ when mass is in grams and acceleration is in $\text{cm/s}^2$.
Therefore, the constant becomes unity not because of a universal constant or empirical data, but because the unit of force is artificially defined to make it so.
Key Takeaway:
In physics, derived units (like the Newton or Dyne) are defined based on base units (like kilogram, meter, second) specifically to ensure that the proportionality constants in fundamental laws (like $F=ma$) resolve to unity.