7.5 Two masses on a string

Source: Ideas from Computational Physics (Landau, Páez, Bordeianu), §7.5. Rewritten in original wording — self-contained reduced model, not a paste of a missing §7.1 listing. Continues 7.4b Tests-before-use labs. Labs: 7.5b String-problem labs.

From statics to $\mathbf f(\mathbf x)=\mathbf 0$

Two masses hanging from three string segments give a classic statics puzzle: horizontal and vertical force balance at each mass, plus geometric constraints on the angles. The unknowns are tensions $T_1,T_2,T_3$ and angles $\theta_1,\theta_2,\theta_3$ — six equations, nonlinear in the trig functions.

That full system is a fine ultimate target, but it is too many unknowns for a first nonlinear lab. The computational point of the chapter is smaller and sharper:

• Physics produces a nonlinear system $\mathbf f(\mathbf x)=\mathbf 0$.
• Identities like $\cos\theta=\pm\sqrt{1-\sin^{2}\theta}$ reduce the dimension — and the sign selects a geometric branch.
• Multidimensional Newton (Jacobian $J$, solve $J\Delta\mathbf x=-\mathbf f$) finds a root from a fair guess.
• Physical filters reject algebraic roots: $T>0$, $|\sin|,|\cos|\le 1$, and a sketch that matches the apparatus.

Sign of the cosine

Write $s=\sin\theta$ and $c=\cos\theta$. Replacing $\cos$ by $\pm\sqrt{1-s^{2}}$ drops an unknown — but the $\pm$ encodes which side of vertical the string leans. The guided walk below makes that identity and the toy $f_1,f_2$ system pickable among lookalikes.

A self-contained Newton toy

To practice the same algebra without six coupled statics equations, use a two-unknown nonlinear system (unit circle ∩ line). Pick the residuals and the Newton update among lookalikes.

One intersection is $(0.6,0.8)$. From a start near $(0.5,0.5)$, Newton with an analytic or finite-difference Jacobian locks onto that root. The circle also meets the line elsewhere — a bad start or flipped cos sign can send you there. Always re-check residuals and geometry after convergence.

Physical checklist (full string problem)

When you return to tensions and angles:

• All $T_i>0$ (strings do not push).
• $|s_i|,|c_i|\le 1$ after every identity replacement.
• Redraw the masses: horizontal components must cancel; verticals must support the weights.
• Prefer $c=+\sqrt{1-s^{2}}$ when the sketch needs $\cos>0$.

Quiz

Two masses on a string

1 / 4

Replacing cos θ by ±√(1−sin²θ)…

Pick an answer to see if you are right.

Next: 7.5b String-problem labs.