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In $F=kma$, $k=1$ but in $F=kx$, $k$ is not equal to 1?So what are the conditions for the constant of proportionality to be set 1?

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  • $\begingroup$ If the coefficient is one, then the coefficient is one. If the coefficient is not one, the coefficient is not one. There obviously can't exist any other "general rule" that would say something about the value of a coefficient in all situations in physics or mathematics or any science. Some of the coefficients are totally physical, others may be changed by various changes of the conventions (and people usually try to make as many coefficients in their basic laws equal to one as possible), and many coefficients in sciences etc. depend on measurements, different levels of evidence, and so on. $\endgroup$ – Luboš Motl May 5 '16 at 18:09
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I think this question has been asked already several times, for example :

How do we know that $F = ma$, not $F = k \cdot ma$
Are Newton's "laws" of motion laws or definitions of force and mass?
Why isn't it $E \approx 27.642 \times mc^2$?


Constants of proportionality depend on the nature of the equation and the system of units. Their values can be chosen for convenience when a physical unit is being defined; otherwise the value is determined by other physical units or by nature.

$F=ma$ is being used to define the unit of force as a derived unit, based on the units for mass and acceleration. For convenience $k$ has been chosen as $1$, but any value could be used instead. (See Coulomb's Equation below.) Similar is $E={\frac Fq}$ which is being used to define electric field strength.

$F={\frac{GmM}{r^2}}$ is a universal law : it applies to every pair of masses in the universe. The value of the constant $G$ is decided by nature because all the other quantities ($F$, $m$, $r$) have defined units.

$F=kx$ is not being used to define anything : units for $F$ and $x$ are already defined. But it is not a universal law : it is an approximation which applies to many different materials and situations - eg the extension of springs and wires, the bending or sagging of beams, the deformation of pneumatic tyres. So there are really many different equations $F=kx$, each with a different value of $k$, which like $G$ can only be found by experiment. We cannot set $k = 1$ because we don't have a choice. We can make the equation less situation-dependent by writing it in terms of stress and strain : ${\frac FA}=Y{\frac xL}$. The constant of proportionality $Y$ (Young's Modulus) then applies to a particular material at a particular temperature - unlike $G$ it is not a universal constant.

$F={\frac{kqQ}{r^2}}$ is another universal law. In the gaussian/cgs/electrostatic system of units $k$ is chosen to be $1$ and this equation is used to define the unit of charge $Q$, the esu (also called statcoulomb or franklin). In SI, units are already defined for $F$, $Q$ and $r$. The value of $k$ (Coulomb's constant) is universal and determined by nature. However, this $k$ is related through Maxwell's equations of electromagnetism to the magnetic constant $\mu_0$ and the speed of light $c$. For convenience it is written as $k ={ \frac 1{4\pi \epsilon_0}}$ while the values of $\mu_0$ and $\epsilon_0$ are chosen to make the tidy equation $\mu_0\epsilon_0 c^2=1$. See the Wiki article for more about this.

https://en.wikipedia.org/wiki/Vacuum_permittivity#Historical_origin_of_the_parameter_.CE.B50

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