ManimEval Hard

Prompt

Suspended compass-type rod structure (maximum elevation of pivot) TASK Convert the PROBLEM and VERIFIED SOLUTION below into ONE runnable Manim Community Edition animation. Do not re-solve or alter the physics. OUTPUT CONTRACT (TOKEN-SAFE) - Return ONLY one Python code block. No prose outside it. - Exactly one class: GeneratedScene(Scene). Imports only: from manim import * and import numpy as np. - Prioritise a complete, runnable final answer over extra polish. Try not to add extra derivations or alternative solution routes, though that is not a fixed rule. How the solution is staged on screen is your call. - Compact helper functions. No comments needed - this code is only compiled automatically, so skip them and spend the tokens on the scene itself. No external files or assets, no plugins, no ThreeDScene, no config changes. VISUAL AND TECHNICAL RULES - BACKGROUND: pure black (#000000) for the whole animation. Coloured strokes and fills only on the physical objects, arrows, highlights and labels themselves. - Text for prose, MathTex for maths. Every MathTex is ONE complete raw LaTeX string. Never split a fraction or radical across arguments and never colour a substring; colour whole objects. - Pre-create every label and equation. Never construct Text or MathTex inside an updater or inside always_redraw; use DecimalNumber for live values. - Use the same colour for the same physical quantity in the diagram and in the equations. - Wait about 2 s after each major equation or checkpoint. Finish by placing the final answer inside an unfilled outline box (SurroundingRectangle with fill_opacity=0) and wait(4). NOTE FOR CLAUDE MODELS: IF YOU ARE A CLAUDE MODEL KEEP THE CODE UNDER 400 lines. Do not overthink - focus on writing the Manim code, stay inside the token limit, and finish the whole code block instead of running out of tokens part-way through. PROBLEM The figure shows a compass type structure suspended from a string connected to the ceiling. The arms of this structure can be rotated about the pivot $P$ and then locked so that the angle at $P$ does not change after locking. The arms are uniform rods. If $2\theta$ is the angle at $P$ as shown (in radians) then the value of $\theta$ so that the point $P$ is as elevated as possible is given by $\theta = \sin^{-1}\left(\dfrac{1}{\sqrt{n}}\right)$. Find the value of $n$. VERIFIED SOLUTION 1. Let each uniform rod ($PA$ and $PB$, where $A$ is connected to the vertical suspension string) have length $L$ and mass $M$. By symmetry, the combined center of mass $C$ lies on the angle bisector of $\angle APB = 2\theta$, so $\angle APC = \theta$ and $PC = \dfrac{L}{2}\cos\theta$. 2. Since the structure is freely suspended from the string at $A$, in rotational equilibrium the center of mass $C$ must lie vertically below $A$, making the line $AC$ vertical. Using the Law of Cosines in $\triangle APC$ gives $AC = L\sqrt{1 - \dfrac{3}{4}\cos^2\theta}$. 3. The vertical depth $h$ of the pivot $P$ below the suspension point $A$ is the projection of rod $AP$ onto the vertical line $AC$: $h = AP\cos(\angle PAC) = \dfrac{AP^2 + AC^2 - PC^2}{2\,AC} = L\,\dfrac{1 - \frac{1}{2}\cos^2\theta}{\sqrt{1 - \frac{3}{4}\cos^2\theta}}$. 4. Substituting $u = \sqrt{1 - \dfrac{3}{4}\cos^2\theta}$ gives $\cos^2\theta = \dfrac{4}{3}(1 - u^2)$ and simplifies the depth to $h(u) = \dfrac{L}{3}\left(\dfrac{1}{u} + 2u\right)$. For $P$ to be as elevated as possible, $h(u)$ must be minimized: setting $\dfrac{dh}{du} = 0$ yields $u^2 = \dfrac{1}{2}$. 5. From $1 - \dfrac{3}{4}\cos^2\theta = \dfrac{1}{2}$, we obtain $\cos^2\theta = \dfrac{2}{3}$ and $\sin^2\theta = 1 - \dfrac{2}{3} = \dfrac{1}{3}$, so $\theta = \sin^{-1}\left(\dfrac{1}{\sqrt{3}}\right)$. Comparing with $\theta = \sin^{-1}\left(\dfrac{1}{\sqrt{n}}\right)$ gives $n = 3$. FIGURE - Draw the accompanying figure(s) as you like (a horizontal ceiling with a vertical string attached to the lower end $A$ of the first rod $PA$, which slopes upward to the pivot $P$, while the second rod $PB$ slopes downward from $P$ making an interior angle $2\theta$ at $P$). STAGING (decide how you want it) - Decide the staging yourself: build the figure, then reveal the solution steps, preferably in the order they are listed. Number of beats, grouping, camera moves and transitions are your call. - Stay inside the token limit and do not overthink the staging: a clean, complete, runnable scene beats an elaborate one. Keep the code compact. - Keep the background black, and put the diagram beside the working rather than on top of it.