ManimMixed Eval

Prompt

You are an expert physics educator and Manim Community Edition specialist. Convert the PROBLEM and VERIFIED SOLUTION below into ONE runnable Manim 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. - Prioritize a complete, runnable final answer over extra polish. Follow the supplied SCENE PLAN in order; do not add unrequested sections, sweeps, or alternative derivations. - Use compact helper functions and at most 6 short comments. No external files/assets, plugins, ThreeDScene, or config changes. VISUAL + TECHNICAL RULES - BACKGROUND (CONSISTENT STYLE): Keep the full scene background pure black (#000000) for the entire animation. Do not use colored background rectangles, panels, gradients, BackgroundRectangle, or backdrop fills behind the diagram, text, or equations. Colored strokes/fills are allowed only for the physical objects, arrows, highlights, bars, and labels themselves. - Persistent diagram on LEFT (x=-6.8..-0.6); derivation on RIGHT (x=0.2..6.8). Keep everything in frame. - Show at most 3 right-side items at once; fade/replace the prior group at each new step. - Text for prose; MathTex for math. Every MathTex is ONE complete raw LaTeX string. Never split fractions/radicals across arguments or color substrings; color whole objects. - Pre-create all labels/equations. Never instantiate Text/MathTex inside an updater or always_redraw; use DecimalNumber for live values. - Prefer Create, Write, FadeIn/FadeOut, Transform, ReplacementTransform, Indicate, Circumscribe. Avoid TransformMatchingTex. - Use consistent colors for the same physical quantity in diagram and equations. Wait about 2 s after each major equation/checkpoint. End with the supplied final answer in an unfilled outline box and wait(4). PROBLEM A simplification of a kind of interlock is shown in the figure. All surfaces are smooth and frictionless. The body m has mass m = 1 kg and the block M = 15 kg. Find the time taken by m to reach the base if it is released at height h = 4 m above the base of M. Use g = 10 m/s^2. Options: (a) 1 s (b) sqrt(3) s (c) 2 s (d) 2 sqrt(2) s FIGURE TO DRAW - A fixed vertical wall is at the far right, with one small fixed pulley near its top and one near its bottom. - Block M is a movable L-shaped frame: a tall vertical limb on the left joined to a horizontal foot extending right along the bottom. - A movable pulley B is attached to the top of M's vertical limb. - The small rectangular body m hangs immediately to the right of that vertical limb, touching its smooth right face. The bottom of m is h above the horizontal foot. - One inextensible rope runs from the top of m vertically up over B, horizontally right to the upper fixed pulley, vertically down to the lower fixed pulley, then horizontally left to the right end of M's foot. - During the motion, M and B move right; m has the same horizontal motion as M and slides downward relative to M. VERIFIED SOLUTION (show in this order) Let a be the rightward acceleration of M (and the horizontal acceleration of m). Let a_rel be the downward acceleration of m relative to M. 1. Rope constraint: if x_M is M's position and y_rel is the vertical rope length from B to m, then L_rope = y_rel + 2(W - x_M) + constant, because both horizontal rope runs shorten when M moves right. Hence a_rel = 2a. 2. Horizontal forces: the smooth face of M pushes m right with normal force N = m a = a. On M, the rope pulls right with T at pulley B and T at the foot attachment, while m pushes back left with N. Thus 2T - N = M a, so 2T = (M+m)a = 16a and T = 8a. 3. Vertical motion of m: m g - T = m a_rel. With m=1, g=10, T=8a and a_rel=2a: 10 - 8a = 2a, so a = 1 m/s^2 and a_rel = 2 m/s^2. 4. The relative fall is h = (1/2) a_rel t^2. Hence 4 = (1/2)(2)t^2 = t^2, giving t = 2 s. SCENE PLAN (only these beats) 1. Build the wall, fixed pulleys, L-shaped M, body m, pulley B and complete rope path on the left. Label M=15 kg, m=1 kg and h=4 m. 2. Highlight the two horizontal rope runs in orange and the vertical run in blue. Animate small matching rightward displacement markers on M and show the rope-length equation, ending with a_rel = 2a. 3. Add force arrows: mg down and T up on m; N right on m; two rightward T arrows and one leftward N arrow on M. Show the horizontal equations on the right. 4. Replace the right side with the vertical equation and substitutions leading to a=1 m/s^2 and a_rel=2 m/s^2. 5. Replay the motion schematically for t=0 to 2 s: M shifts right by one visual unit while m drops two visual units relative to M (state "distances compressed; ratio exact"). Show a DecimalNumber timer and h = (1/2)a_rel t^2 -> t=2 s. FINAL ANSWER TO BOX: t = 2 s, option (c).