I understand Hooke's Law to be
F = kX
Where F is the tension applied to an object, X is the extension/change in length, and k is the spring constant, whose units are N/m.
I'm learning Mechanics 3 under the A Level Edexcel Maths specification (the M3 stuff is on page 70), and I am being told that Hooke's Law can also be defined as:
F = (λx)/l
Where x is the change in length, F is the tension and λ is the Modulus of Elasticity/ the Young's Modulus.
However, this would imply that the spring constant is equal to the Modulus of Elasticity, divided by the length. Instantly, if one already knows the units for the Modulus of Elasticity (Nm^-2), we can see that that would imply the spring constant is equal to Nm^-3, while the formula F = kX implies its units is N/metre.
I investigated further, and we know that modulus of elasticity is stress/strain, which is
With a new variable being introduced as A, which is the cross-sectional area of the object.
We know that k = F/X, and do we can express the modulus of elasticity as
Which would therefore mean that the spring constant k is (λA)/l
and not (λ/l)
We can check the units too, and (λA)/l gives N/metre.
So my question is: Why am I seeing two incompatible versions of Hooke's Law? The spring constant cannot be equal to F/x and λ/l, as the units aren't the same for one, and my above calculations also show that we need to include the cross-sectional area.
However, this is in the M3 syllabus and it's not anything new (although I haven't actually seen the formula anywhere else for k being λ/l) , so it's likely that there's something I'm not understanding.
Could anyone shed any light?