The Force Law for Simple Harmonic Motion — Question 10

Question 10

Pre-tension can create a transverse linear restoring force

A mass $m$ is attached at the midpoint of a straight horizontal elastic arrangement. Two identical springs run to anchors a distance $\ell$ to either side. Each spring has stiffness $k$ and tension $T_0\ge0$ when the mass is centered. The mass is constrained to move transversely by distance $y$, and other forces balance at $y=0$.

1. Derive the leading small-$y$ transverse restoring force when $T_0>0$ and the frequency.

2. If $T_0=0$ and each natural length is $\ell$, determine the leading force order. Why is assigning stiffness $2k$ to transverse motion incorrect?

Question 10 - Solution

Pre-tension can create a transverse linear restoring force

(a) Derivation and calculation

$$s=\sqrt{\ell^2+y^2},\qquad T=T_0+k(s-\ell),\qquad F_y=-2T\frac ys.$$$$s=\ell+\frac{y^2}{2\ell}+O(y^4)\Rightarrow F_y=-\frac{2T_0}{\ell}y+O(y^3).$$$$\boxed{m\ddot y+\frac{2T_0}{\ell}y\simeq0},\qquad\boxed{\omega=\sqrt{\frac{2T_0}{m\ell}}}\quad(T_0>0).$$

(b) Result and interpretation

For $T_0=0$, the axial extension is second order:$$T=k(s-\ell)\simeq\frac{ky^2}{2\ell},\qquad\frac ys\simeq\frac y\ell.$$$$\boxed{F_y\simeq-\frac{k}{\ell^2}y^3},\qquad-\left.\frac{dF_y}{dy}\right|_{0}=0.$$There is no nonzero linear frequency at the center. The axial constants do not sum to a transverse stiffness $2k$.

Physical check

The small parameter is $|y|/\ell$. Finite-amplitude motion can still be restoring when the leading force is cubic, but it is not linear SHM.