Slight rearrangement of Newton's Law

[Timothy S]

Instead of F = ma, is it valid to write λxa = F, (in the case of a linear, one dimensional object). Does this make sense, I am just curious. Also, would this lead to any real insights about a system?

Edit: For clarity's sake, I replaced mass with linear density times length, which in this case is in the x direction.

 

[mfb]

Depends on how you use it. It can be meaningful.
As an example issue, does every part of your line have the same acceleration?

 

[Philip Wood]

One case for which we use the law in more or less this form is for studying the propagation of mechanical waves through a medium such as a spring. We'd consider a small length, \Delta x, of the spring, and the resultant, F_res, of the slightly different pulls on each 'end'of this piece of the spring, from neighbouring parts of the spring.

Then \lambda \Delta x\ a = F_res

F_res is then found, essentially by applying Hooke's law, but it's quite tricky as F_res arises because the displacements of particles of the spring from their equilibrium position vary as we go along the spring.