Minval

Prompt

PROMPT 01 - Billiard balls rolling on a U-shaped trough (multi-statement) 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. - Finish by placing the final answer inside an unfilled outline box (SurroundingRectangle with fill_opacity=0). PROBLEM Two identical billiard balls (uniform solid spheres) each of radius $R$ and mass $m$ moving with a speed $V_0$ roll without slipping towards each other on a horizontal U-shaped trough that is sufficiently deep such that the balls are clear of its base. The width of the trough is $2\ell$. The resulting collision is perfectly elastic and during it each ball reverses its linear velocity though their angular velocities are not affected. The surface of trough is not smooth. Mark the CORRECT statement(s). OPTIONS (A) Initial magnitude of angular velocity of each ball is $\dfrac{V_0}{R}$. (B) Including the first collision, the balls will collide more than once if $\ell > \sqrt{\dfrac{3}{5}}R$. (C) Between the end of first collision to the start of second collision, the angular momentum of the system of each ball is constant about any point on the center line of the horizontal plane containing the top edges of the trough. (D) If second collision occurs for a given value of $\ell$, the value of linear speed of each ball just before second collision is independent of coefficient of friction between trough and sphere surfaces. VERIFIED SOLUTION 1. Each sphere rests only on the two top edges of the trough separated by width $2\ell$. The vertical distance from the center of the ball to the horizontal plane containing the contact edges is $r = \sqrt{R^2 - \ell^2}$. Pure rolling requires $\omega_0 = \dfrac{V_0}{r} = \dfrac{V_0}{\sqrt{R^2 - \ell^2}} \neq \dfrac{V_0}{R}$. Statement (A) is wrong. 2. Between collisions, the vertical components of the normal forces balance gravity and their horizontal components cancel, producing zero net torque about the center line of the horizontal plane containing the top edges. Symmetrical friction forces at $\pm \ell$ also produce equal and opposite torques about any point on this center line. Thus, angular momentum of each ball is conserved about any point on the center line. Statement (C) is correct. 3. Just after the first collision, a ball has linear velocity $-V_0$ and angular velocity $\omega_0 = V_0/r$. Conserving angular momentum about the center line until pure rolling is restored ($v_f = \omega_f r$) with $I_{\text{cm}} = \frac{2}{5}mR^2$ gives $-mV_0 r + I_{\text{cm}}\omega_0 = mv_f r + I_{\text{cm}}\omega_f$. 4. Solving for $v_f$ gives $v_f = V_0 \left(\dfrac{\frac{2}{5}R^2 - r^2}{\frac{2}{5}R^2 + r^2}\right)$. For a second collision to occur, each ball must reverse direction and move back towards the collision point ($v_f > 0$), which requires $r^2 < \frac{2}{5}R^2 \implies R^2 - \ell^2 < \frac{2}{5}R^2 \implies \ell > \sqrt{\dfrac{3}{5}}R$. Statement (B) is correct. 5. Because $v_f < V_0$, the average velocity during the slipping phase is $\frac{1}{2}(-V_0 + v_f) < 0$, so pure rolling is restored before the ball returns to the collision point. Thereafter it rolls without friction at constant speed $v_f$, which is independent of the coefficient of friction. Statement (D) is correct. 6. Correct statements: (B), (C) and (D). FIGURE - Draw the accompanying figure(s) as you like. 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.