{"id":331,"date":"2026-05-08T02:16:46","date_gmt":"2026-05-07T17:16:46","guid":{"rendered":"https:\/\/wuhanqing.cn\/wordpress\/?p=331"},"modified":"2026-05-08T02:16:46","modified_gmt":"2026-05-07T17:16:46","slug":"%e5%ae%9e%e9%aa%8c%e6%8a%a5%e5%91%8a%e8%bd%ac%e5%8a%a8%e6%83%af%e9%87%8f%e6%b5%8b%e5%ae%9a%e4%b8%8e%e8%a7%92%e5%8a%a8%e9%87%8f%e5%ae%88%e6%81%92","status":"publish","type":"post","link":"https:\/\/wuhanqing.cn\/wordpress\/2026\/05\/08\/%e5%ae%9e%e9%aa%8c%e6%8a%a5%e5%91%8a%e8%bd%ac%e5%8a%a8%e6%83%af%e9%87%8f%e6%b5%8b%e5%ae%9a%e4%b8%8e%e8%a7%92%e5%8a%a8%e9%87%8f%e5%ae%88%e6%81%92\/","title":{"rendered":"[\u5b9e\u9a8c\u62a5\u544a]\u8f6c\u52a8\u60ef\u91cf\u6d4b\u5b9a\u4e0e\u89d2\u52a8\u91cf\u5b88\u6052"},"content":{"rendered":"

# 1.\u5b9e\u9a8c\u6807\u9898<\/p>\n

\u672c\u6b21\u5b9e\u9a8c\u7684\u4e3b\u9898\u662f**\u8f6c\u52a8\u60ef\u91cf\u6d4b\u5b9a\u4e0e\u89d2\u52a8\u91cf\u5b88\u6052**\u3002<\/p>\n

# 2. \u5b9e\u9a8c\u76ee\u7684<\/p>\n

\u901a\u8fc7\u6d4b\u91cf\u65cb\u8f6c\u521a\u4f53(Rigid Body)\u7684\u89d2\u52a0\u901f\u5ea6(Angular Acceleration)\uff0c\u4ee5\u5b9e\u9a8c\u65b9\u5f0f\u786e\u5b9a\u8be5\u7269\u4f53\u7684**\u8f6c\u52a8\u60ef\u91cf(Moment of Inertia)**\uff0c\u5e76\u4e0e\u57fa\u4e8e\u51e0\u4f55\u7ed3\u6784\u8ba1\u7b97\u7684\u7406\u8bba\u503c\u8fdb\u884c\u6bd4\u8f83\uff0c\u4ee5\u7406\u89e3\u65cb\u8f6c\u8fd0\u52a8\u7684\u529b\u5b66\u539f\u7406\u3002\u540c\u65f6\uff0c\u4ee5\u5b9e\u9a8c\u65b9\u5f0f\u9a8c\u8bc1\u5f53\u5916\u90e8\u626d\u77e9\u4e0d\u4f5c\u7528\u4e8e\u65cb\u8f6c\u7cfb\u7edf\u65f6\uff0c**\u89d2\u52a8\u91cf(Angular Momentum)**\u5f97\u4ee5\u5b88\u6052\uff0c\u5e76\u5728\u6b64\u8fc7\u7a0b\u4e2d\u8003\u5bdf\u80fd\u91cf\u7684\u53d8\u5316\u3002<\/p>\n

# 3. \u76f8\u5173\u7406\u8bba<\/p>\n

## 3.1 \u8f6c\u52a8\u60ef\u91cf (Moment of Inertia)<\/p>\n

\u5728\u65cb\u8f6c\u8fd0\u52a8\u7684\u7269\u4f53\u4e2d\uff0c\u5bf9\u5e94\u4e8e\u76f4\u7ebf\u8fd0\u52a8\u4e2d\u7684\"\u8d28\u91cf(Mass)\"\u7684\u7269\u7406\u91cf\uff0c\u8868\u793a\u7269\u4f53\u4fdd\u6301\u5176\u65cb\u8f6c\u72b6\u6001\u7684\u6027\u8d28\u5927\u5c0f\u3002\u5f53\u8d28\u91cf\u4e3a$m_i$\u7684\u7c92\u5b50\u8ddd\u65cb\u8f6c\u8f74\u4e3a$r_i$\u65f6\uff0c\u8f6c\u52a8\u60ef\u91cf$I$\u5b9a\u4e49\u5982\u4e0b\uff1a<\/p>\n

$$I = \\sum m_i r_i^2$$<\/p>\n

\u5bf9\u4e8e\u5177\u6709\u8fde\u7eed\u8d28\u91cf\u5206\u5e03\u7684\u521a\u4f53\uff0c\u901a\u8fc7\u5bf9\u5fae\u5c0f\u8d28\u91cf$dm$\u8fdb\u884c\u79ef\u5206(Integral)\u6765\u6c42\u5f97\u3002<\/p>\n

$$I = \\int r^2 dm$$<\/p>\n

### 3.1.1 \u5706\u76d8(Disk)\u4e0e\u5706\u73af(Ring)\u7684\u8f6c\u52a8\u60ef\u91cf\u63a8\u5bfc (\u8bc1\u660e)<\/p>\n

#### 3.1.1.1 \u5747\u5300\u5706\u76d8 (Solid Disk)<\/p>\n

\u6c42\u534a\u5f84\u4e3a$R$\u3001\u603b\u8d28\u91cf\u4e3a$M$\u7684\u5747\u5300\u5706\u76d8\u5bf9\u5176\u4e2d\u5fc3\u8f74\u7684\u8f6c\u52a8\u60ef\u91cf\u3002\u5706\u76d8\u7684\u9762\u5bc6\u5ea6(Surface density)\u4e3a$\\sigma = \\frac{M}{\\pi R^2}$\u3002<\/p>\n

\u8bbe\u534a\u5f84\u4e3a$r$\u3001\u539a\u5ea6\u4e3a$dr$\u7684\u5fae\u5c0f\u5706\u73af\uff0c\u5219\u5fae\u5c0f\u9762\u79ef$dA = 2\\pi r dr$\uff0c\u5fae\u5c0f\u8d28\u91cf$dm = \\sigma dA = \\frac{M}{\\pi R^2} \\cdot 2\\pi r dr$\u3002<\/p>\n

$$I_{disk} = \\int_0^R r^2 dm = \\int_0^R r^2 \\left( \\frac{2Mr}{R^2} \\right) dr = \\frac{2M}{R^2} \\int_0^R r^3 dr$$<\/p>\n

$$I_{disk} = \\frac{2M}{R^2} \\left[ \\frac{r^4}{4} \\right]_0^R = \\frac{2M}{R^2} \\cdot \\frac{R^4}{4} = \\frac{1}{2}MR^2$$<\/p>\n

#### 3.1.1.2 \u539a\u5706\u73af (Thick Ring)<\/p>\n

\u6c42\u5185\u534a\u5f84(Inner radius)\u4e3a$R_1$\u3001\u5916\u534a\u5f84(Outer radius)\u4e3a$R_2$\u3001\u8d28\u91cf\u4e3a$M$\u7684\u5706\u73af\u7684\u8f6c\u52a8\u60ef\u91cf\u3002\u9762\u5bc6\u5ea6(Surface density)\u4e3a$\\sigma = \\frac{M}{\\pi(R_2^2 - R_1^2)}$\u3002<\/p>\n

$$I_{ring} = \\int_{R_1}^{R_2} r^2 \\left( \\frac{2Mr}{R_2^2 - R_1^2} \\right) dr = \\frac{2M}{R_2^2 - R_1^2} \\left[ \\frac{r^4}{4} \\right]_{R_1}^{R_2}$$<\/p>\n

$$I_{ring} = \\frac{2M}{R_2^2 - R_1^2} \\cdot \\frac{R_2^4 - R_1^4}{4} = \\frac{M}{2(R_2^2 - R_1^2)}(R_2^2 - R_1^2)(R_2^2 + R_1^2)$$<\/p>\n

$$I_{ring} = \\frac{1}{2}M(R_1^2 + R_2^2)$$<\/p>\n

## 3.2 \u89d2\u52a8\u91cf\u4e0e\u626d\u77e9\u7684\u5173\u7cfb\u63a8\u5bfc (\u725b\u987f\u7b2c\u4e8c\u5b9a\u5f8b\u7684\u65cb\u8f6c\u53d8\u6362)<\/p>\n

\u4e3a\u4e86\u4ece\u529b\u5b66\u4e0a\u89e3\u91ca\u65cb\u8f6c\u8fd0\u52a8\uff0c\u5c06\u5e73\u79fb\u8fd0\u52a8\u7684\u57fa\u672c\u89c4\u5f8b\u725b\u987f\u7b2c\u4e8c\u5b9a\u5f8b($F = ma$)\u8f6c\u6362\u6210\u65cb\u8f6c\u8fd0\u52a8\u7684\u5f62\u5f0f\u3002<\/p>\n

\u5bf9\u4e8e\u5355\u4e2a\u8d28\u91cf\u4e3a$m$\u7684\u7c92\u5b50\uff0c\u725b\u987f\u7b2c\u4e8c\u5b9a\u5f8b\u7684\u5fae\u5206\u5f62\u5f0f\u53ef\u7528\u7ebf\u52a8\u91cf($p = mv$)\u7684\u65f6\u95f4\u53d8\u5316\u7387\u5b9a\u4e49\u5982\u4e0b\uff1a<\/p>\n

$$F = \\frac{dp}{dt} = m\\frac{dv}{dt}$$<\/p>\n

\u5f53\u8be5\u7c92\u5b50\u4ece\u539f\u70b9\u5904\u4f4d\u7f6e\u77e2\u91cf\u4e3a$r$\u65f6\uff0c\u4f5c\u7528\u5728\u7c92\u5b50\u4e0a\u7684\u626d\u77e9(Torque, $\\tau$)\u5b9a\u4e49\u4e3a\u4f4d\u7f6e\u77e2\u91cf\u4e0e\u529b\u77e2\u91cf\u7684\u53c9\u79ef(Cross Product, Outer Product)\uff1a<\/p>\n

$$\\tau = r \\times F$$<\/p>\n

\u5c06\u4e0a\u5f0f\u4e2d\u7684$F$\u4ee3\u5165\u725b\u987f\u7b2c\u4e8c\u5b9a\u5f8b\u7684\u5fae\u5206\u5f62\u5f0f\uff0c\u5f97\uff1a<\/p>\n

$$\\tau = r \\times \\frac{dp}{dt} \\tag{1}$$<\/p>\n

\u53e6\u4e00\u65b9\u9762\uff0c\u7c92\u5b50\u7684\u89d2\u52a8\u91cf(Angular Momentum, $L$)\u5b9a\u4e49\u4e3a\u4f4d\u7f6e\u77e2\u91cf\u4e0e\u7ebf\u52a8\u91cf(Linear Momentum)\u7684\u53c9\u79ef\uff1a<\/p>\n

$$L = r \\times p$$<\/p>\n

\u5bf9\u8be5\u89d2\u52a8\u91cf\u5173\u4e8e\u65f6\u95f4$t$\u8fdb\u884c\u5fae\u5206\u3002\u6839\u636e\u5fae\u5206\u7684\u4e58\u79ef\u6cd5\u5219\uff0c\u53ef\u5f97\uff1a<\/p>\n

$$\\frac{dL}{dt} = \\frac{d}{dt}(r \\times p) = \\left( \\frac{dr}{dt} \\times p \\right) + \\left( r \\times \\frac{dp}{dt} \\right)$$<\/p>\n

\u5176\u4e2d$\\frac{dr}{dt}$\u4e3a\u7c92\u5b50\u7684\u901f\u5ea6$v$\uff0c\u7ebf\u52a8\u91cf$p = mv$\u3002\u901f\u5ea6\u77e2\u91cf$v$\u4e0e\u5e73\u884c\u4e8e\u81ea\u8eab\u7684$mv$\u7684\u53c9\u79ef\u4e3a0($v \\times mv = 0$)\u3002\u56e0\u6b64\u7b2c\u4e00\u9879\u88ab\u6d88\u53bb\uff0c\u4ec5\u4fdd\u7559\u7b2c\u4e8c\u9879\u3002<\/p>\n

$$\\frac{dL}{dt} = r \\times \\frac{dp}{dt} \\tag{2}$$<\/p>\n

\u6bd4\u8f83\u5f0f(1)\u4e0e\u5f0f(2)\uff0c\u63a8\u5bfc\u51fa\u626d\u77e9\u4e0e\u89d2\u52a8\u91cf\u4e4b\u95f4\u7684\u57fa\u672c\u5173\u7cfb\u3002\u5373\uff0c\u7cfb\u7edf\u6240\u53d7\u7684\u5408\u626d\u77e9\u7b49\u4e8e\u89d2\u52a8\u91cf\u7684\u65f6\u95f4\u53d8\u5316\u7387\u3002<\/p>\n

$$\\tau = \\frac{dL}{dt} \\tag{3}$$<\/p>\n

\u73b0\u5728\u5c06\u6b64\u5173\u7cfb\u6269\u5c55\u5230\u7ed5\u56fa\u5b9a\u8f74\u65cb\u8f6c\u7684\u521a\u4f53\u3002\u5f53\u521a\u4f53\u4ee5\u89d2\u901f\u5ea6$\\omega$\u65cb\u8f6c\u65f6\uff0c\u6bcf\u4e2a\u7c92\u5b50\u7684\u901f\u5ea6\u4e3a$v = r\\omega$(\u6807\u91cf\u5f62\u5f0f)\uff0c\u56e0\u6b64\u89d2\u52a8\u91cf$L$\u53ef\u8868\u793a\u4e3a\uff1a<\/p>\n

$$L = r \\cdot p = r(mv) = mr^2\\omega$$<\/p>\n

\u5176\u4e2d$mr^2$\u4e3a\u7c92\u5b50\u7684\u8f6c\u52a8\u60ef\u91cf$I$\uff0c\u56e0\u6b64\u6210\u7acb$L = I\\omega$\u3002\u5c06\u5176\u4ee3\u5165\u5f0f(3)\uff0c\u5173\u4e8e\u65f6\u95f4\u6c42\u5fae\u5206\uff0c\u5f97\uff1a<\/p>\n

$$\\tau = \\frac{d}{dt}(I\\omega) = I\\frac{d\\omega}{dt} = I\\alpha \\tag{4}$$<\/p>\n

\u6700\u7ec8\u63a8\u5bfc\u51fa\u4e0e\u5e73\u79fb\u8fd0\u52a8\u7684$F = m\\frac{dv}{dt}$\u5b8c\u5168\u5bf9\u5e94\u7684\u65cb\u8f6c\u8fd0\u52a8\u7684\u8fd0\u52a8\u65b9\u7a0b\uff1a<\/p>\n

$$\\tau = r \\times F = I\\alpha = \\frac{dL}{dt}$$<\/p>\n

\u6b64\u65b9\u7a0b\u662f\u672c\u5b9e\u9a8c\u4e2d\u901a\u8fc7\u6d4b\u91cf\u539f\u76d8\u548c\u73af\u7684\u65cb\u8f6c\u52a0\u901f\u5ea6($\\alpha$)\u6765\u53cd\u63a8\u8f6c\u52a8\u60ef\u91cf($I$)\u7684\u6838\u5fc3\u6570\u5b66\u4f9d\u636e\u3002<\/p>\n

## 3.3 \u65cb\u8f6c\u8fd0\u52a8\u52a8\u80fd (Rotational Kinetic Energy) \u53ca\u63a8\u5bfc<\/p>\n

\u5c31\u5982\u540c\u5e73\u79fb\u8fd0\u52a8(Translational motion)\u7269\u4f53\u7684\u52a8\u80fd\u7531\u8d28\u91cf\u548c\u901f\u5ea6\u51b3\u5b9a\u4e00\u6837\uff0c\u56f4\u7ed5\u56fa\u5b9a\u8f74\u65cb\u8f6c\u7684\u521a\u4f53\u7684\u52a8\u80fd\u4e5f\u53ef\u901a\u8fc7\u8f6c\u52a8\u60ef\u91cf\u548c\u89d2\u901f\u5ea6\u6765\u5b9a\u4e49\u3002\u8fd9\u79f0\u4e3a\u65cb\u8f6c\u8fd0\u52a8\u52a8\u80fd(Rotational Kinetic Energy)\uff0c\u53ef\u901a\u8fc7\u5c06\u5e73\u79fb\u8fd0\u52a8\u52a8\u80fd\u7684\u57fa\u672c\u5b9a\u4e49\u5e94\u7528\u4e8e\u65cb\u8f6c\u521a\u4f53\u7684\u6bcf\u4e2a\u5fae\u5c0f\u8d28\u91cf\u6765\u63a8\u5bfc\u3002<\/p>\n

### 3.3.1 \u63a8\u5bfc\u8fc7\u7a0b<\/p>\n

\u8bbe\u67d0\u521a\u4f53\u56f4\u7ed5\u56fa\u5b9a\u65cb\u8f6c\u8f74\u4ee5\u89d2\u901f\u5ea6$\\omega$\u65cb\u8f6c\u3002\u8be5\u521a\u4f53\u53ef\u89c6\u4e3a\u7531\u8bb8\u591a\u5fae\u5c0f\u7c92\u5b50\u7ec4\u6210\u3002<\/p>\n

\u8bbe\u8d28\u91cf\u4e3a$m_i$\u7684\u7b2c$i$\u4e2a\u7c92\u5b50\u8ddd\u65cb\u8f6c\u8f74\u7684\u5782\u76f4\u8ddd\u79bb\u4e3a$r_i$\uff0c\u5176\u7ebf\u901f\u5ea6$v_i$\u4e0e\u89d2\u901f\u5ea6$\\omega$\u7684\u5173\u7cfb\u4e3a\uff1a<\/p>\n

$$v_i = r_i \\omega$$<\/p>\n

\u8be5\u7b2c$i$\u4e2a\u7c92\u5b50\u7684\u5e73\u79fb\u8fd0\u52a8\u52a8\u80fd$K_i$\u6839\u636e\u725b\u987f\u529b\u5b66\u7684\u57fa\u672c\u5b9a\u4e49\u4e3a\uff1a<\/p>\n

$$K_i = \\frac{1}{2}m_i v_i^2$$<\/p>\n

\u5c06$v_i = r_i \\omega$\u4ee3\u5165\u5e76\u6574\u7406\uff0c\u5f97\uff1a<\/p>\n

$$K_i = \\frac{1}{2}m_i (r_i \\omega)^2 = \\frac{1}{2} m_i r_i^2 \\omega^2$$<\/p>\n

\u521a\u4f53\u6574\u4f53\u7684\u65cb\u8f6c\u8fd0\u52a8\u52a8\u80fd$K_{rot}$\u7b49\u4e8e\u7ec4\u6210\u8be5\u521a\u4f53\u6240\u6709\u7c92\u5b50\u7684\u52a8\u80fd\u4e4b\u548c\u3002\u82e5\u521a\u4f53\u4fdd\u6301\u5b8c\u6574\u5f62\u6001\u7684\u521a\u4f53(Rigid body)\uff0c\u65cb\u8f6c\u65f6\u6240\u6709\u7c92\u5b50\u5171\u4eab\u540c\u4e00\u89d2\u901f\u5ea6$\\omega$\uff0c\u56e0\u6b64$\\omega$\u53ef\u4ece\u6c42\u548c\u7b26\u53f7($\\sum$)\u5916\u63d0\u51fa\u3002<\/p>\n

$$K_{rot} = \\sum_{i} K_i = \\sum_{i} \\left( \\frac{1}{2} m_i r_i^2 \\omega^2 \\right)$$<\/p>\n

$$K_{rot} = \\frac{1}{2} \\left( \\sum_{i} m_i r_i^2 \\right) \\omega^2$$<\/p>\n

\u6b64\u65f6\u62ec\u53f7\u5185\u7684\u5f0f\u5b50$\\sum_{i} m_i r_i^2$\u4e0e\u7b2c3.1\u8282\u4e2d\u5b9a\u4e49\u7684\u8f6c\u52a8\u60ef\u91cf(Moment of Inertia, $I$)\u76f8\u540c\u3002<\/p>\n

\u56e0\u6b64\uff0c\u5c06\u62ec\u53f7\u90e8\u5206\u4ee3\u6362\u4e3a$I$\uff0c\u5b8c\u6210\u6700\u7ec8\u7684\u65cb\u8f6c\u8fd0\u52a8\u52a8\u80fd\u516c\u5f0f\uff1a<\/p>\n

$$K_{rot} = \\frac{1}{2}I\\omega^2$$<\/p>\n

\u8fd9\u4e2a\u7ed3\u679c\u4e0e\u5e73\u79fb\u8fd0\u52a8\u52a8\u80fd\u516c\u5f0f$K = \\frac{1}{2}mv^2$\u5f62\u6210\u5b8c\u7f8e\u7684\u6570\u5b66\u5bf9\u79f0\u3002\u5373\uff0c\u5728\u65cb\u8f6c\u8fd0\u52a8\u4e2d\uff0c\u8d28\u91cf$m$\u7531\u8f6c\u52a8\u60ef\u91cf$I$\u53d6\u4ee3\uff0c\u7ebf\u901f\u5ea6$v$\u7531\u89d2\u901f\u5ea6$\\omega$\u53d6\u4ee3\uff0c\u5176\u4f5c\u7528\u5b8c\u5168\u76f8\u540c\u3002<\/p>\n

### 3.3.2 \u672c\u5b9e\u9a8c\u4e2d\u7684\u7269\u7406\u610f\u4e49<\/p>\n

\u5728\u672c\u5b9e\u9a8c\u7684\"\u5b9e\u9a8cB(\u89d2\u52a8\u91cf\u5b88\u6052)\"\u4e2d\uff0c\u5c06\u8d28\u91cf\u73af\u6254\u5230\u65cb\u8f6c\u5706\u76d8\u4e0a\u7684\u8fc7\u7a0b\uff0c\u7531\u4e8e\u4e0d\u5b58\u5728\u5916\u90e8\u626d\u77e9\uff0c\u89d2\u52a8\u91cf($L$)\u5f97\u4ee5\u5b88\u6052\uff0c\u4f46\u7531\u4e8e\u5185\u90e8\u6469\u64e6\u529b\u4f7f\u4e24\u4e2a\u7269\u4f53\u6700\u7ec8\u4ee5\u76f8\u540c\u7684\u89d2\u901f\u5ea6\u65cb\u8f6c\uff0c\u8fd9\u5728\u529b\u5b66\u4e0a\u7b49\u540c\u4e8e\u5b8c\u5168\u975e\u5f39\u6027\u78b0\u649e(Perfectly inelastic collision)\u3002\u56e0\u6b64\uff0c\u5229\u7528\u6b64\u516c\u5f0f\u53ef\u8ba1\u7b97\u78b0\u649e\u524d\u540e\u7684\u65cb\u8f6c\u8fd0\u52a8\u52a8\u80fd\uff0c\u8bc1\u660e\u5373\u4f7f\u89d2\u52a8\u91cf\u5f97\u4ee5\u5b88\u6052\uff0c\u8fd0\u52a8\u52a8\u80fd\u4e5f\u4f1a\u56e0\u6469\u64e6\u70ed\u7b49\u800c\u635f\u5931($\\Delta K_{rot} < 0$)\u3002\n\n## 3.4 \u89d2\u52a8\u91cf\u5b88\u6052\u5b9a\u5f8b (Conservation of Angular Momentum)\n\n\u5f53\u7cfb\u7edf\u6240\u53d7\u5916\u529b\u4ea7\u751f\u7684\u5408\u626d\u77e9\u4e3a0\u65f6($\\tau_{ext} = 0$)\uff0c\u7cfb\u7edf\u7684\u603b\u89d2\u52a8\u91cf\u4fdd\u6301\u4e0d\u53d8\u3002\n\n$$\\frac{dL}{dt} = 0 \\implies L = I_i \\omega_i = I_f \\omega_f = \\text{Constant}$$\n\n\u5728\u672c\u5b9e\u9a8c\u4e2d\uff0c\u5c06\u8d28\u91cf\u73af\u653e\u5728\u65cb\u8f6c\u5706\u76d8\u4e0a\uff0c\u4f7f\u8f6c\u52a8\u60ef\u91cf\u4ece$I_i \\to I_f$\u53d8\u5316\u65f6\uff0c\u901a\u8fc7\u89c2\u5bdf\u89d2\u901f\u5ea6\u4ece$\\omega_i \\to \\omega_f$\u7684\u53d8\u5316\u8fc7\u7a0b\u6765\u9a8c\u8bc1\u6b64\u539f\u7406\u3002\n\n## 3.5 \u5b9e\u9a8c\u6027\u8f6c\u52a8\u60ef\u91cf\u6d4b\u5b9a\u539f\u7406\u53ca\u516c\u5f0f\u63a8\u5bfc\n\n\u672c\u5b9e\u9a8c\u91c7\u7528\u5728\u65cb\u8f6c\u8f74(\u534a\u5f84$r$)\u4e0a\u7f20\u7ed5\u7ef3\u5b50\uff0c\u7ef3\u5b50\u672b\u7aef\u60ac\u6302\u8d28\u91cf\u4e3a$m$\u7684\u781d\u7801\u5e76\u4f7f\u5176\u81ea\u7531\u4e0b\u843d\u7684\u65b9\u5f0f\u3002\u5f53\u781d\u7801\u5728\u91cd\u529b\u4f5c\u7528\u4e0b\u52a0\u901f\u4e0b\u843d\u65f6\uff0c\u4f1a\u7275\u62c9\u7ef3\u5b50\uff0c\u6b64\u5f20\u529b(Tension, $T$)\u4f1a\u5728\u65cb\u8f6c\u8f74\u5904\u4ea7\u751f\u626d\u77e9\uff0c\u5f15\u53d1\u6574\u4e2a\u7cfb\u7edf\u7684\u65cb\u8f6c\u8fd0\u52a8\u3002\u5c06\u6b64\u529b\u5b66\u8fc7\u7a0b\u7528\u6570\u5f0f\u5206\u89e3\u5982\u4e0b\u3002\n\n### 3.5.1 \u781d\u7801\u7684\u5e73\u79fb\u8fd0\u52a8\u65b9\u7a0b (Translational Equation of Motion)\n\n\u4f5c\u7528\u5728\u4e0b\u843d\u8d28\u91cf$m$\u781d\u7801\u4e0a\u7684\u5408\u529b\u4e3a\u7ad6\u76f4\u5411\u4e0b\u7684\u91cd\u529b$mg$\u4e0e\u7ad6\u76f4\u5411\u4e0a\u7684\u7ef3\u5b50\u5f20\u529b$T$\u3002\u8bbe\u781d\u7801\u4e0b\u843d\u65b9\u5411\u4e3a\u6b63(+)\uff0c\u5219\u6839\u636e\u725b\u987f\u7b2c\u4e8c\u5b9a\u5f8b\uff0c\u7ebf\u52a0\u901f\u5ea6$a$\u7684\u8fd0\u52a8\u65b9\u7a0b\u5982\u4e0b\uff1a\n\n$$mg - T = ma \\tag{1}$$\n\n### 3.5.2 \u521a\u4f53\u7684\u65cb\u8f6c\u8fd0\u52a8\u65b9\u7a0b (Rotational Equation of Motion)\n\n\u62c9\u52a8\u65cb\u8f6c\u8f74\u7684\u7ef3\u5b50\u5f20\u529b$T$\u6cbf\u534a\u5f84\u4e3a$r$\u7684\u65cb\u8f6c\u8f74\u5207\u7ebf\u65b9\u5411\u4f5c\u7528\uff0c\u5f62\u6210\u5408\u626d\u77e9$\\tau$\u3002\u8bbe\u65cb\u8f6c\u4f53\u7684\u603b\u8f6c\u52a8\u60ef\u91cf\u4e3a$I$\uff0c\u89d2\u52a0\u901f\u5ea6\u4e3a$\\alpha$\uff0c\u5219\u626d\u77e9\u65b9\u7a0b\u5982\u4e0b\u3002(\u56e0\u4e3a\u5f20\u529b\u4f5c\u7528\u7ebf\u4e0e\u534a\u5f84\u77e2\u91cf\u4e92\u76f8\u5782\u76f4\uff0c\u6545$\\sin 90^\\circ = 1$\u3002)\n\n$$\\tau = r \\times T = rT = I\\alpha \\tag{2}$$\n\n### 3.5.3 \u7ebf\u52a0\u901f\u5ea6\u4e0e\u89d2\u52a0\u901f\u5ea6\u7684\u7ea6\u675f\u6761\u4ef6 (Kinematic Constraint)\n\n\u5047\u8bbe\u5728\u7406\u60f3\u72b6\u6001\u4e0b\u7ef3\u5b50\u4e0d\u4f38\u957f\u6216\u4e0d\u5728\u65cb\u8f6c\u8f74\u4e0a\u6ed1\u52a8\uff0c\u5219\u781d\u7801\u7684\u7ebf\u52a0\u901f\u5ea6$a$\u4e0e\u65cb\u8f6c\u8f74\u8868\u9762\u7684\u5207\u7ebf\u52a0\u901f\u5ea6\u5b8c\u5168\u76f8\u540c\u3002\u56e0\u6b64\u7ebf\u8fd0\u52a8\u4e0e\u65cb\u8f6c\u8fd0\u52a8\u4e4b\u95f4\u5b58\u5728\u4ee5\u4e0b\u51e0\u4f55\u5173\u7cfb\uff1a\n\n$$a = r\\alpha \\tag{3}$$\n\n### 3.5.4 \u516c\u5f0f\u63a8\u5bfc\u5c55\u5f00\u8fc7\u7a0b\n\n\u901a\u8fc7\u8054\u7acb\u4e0a\u8ff0\u4e09\u4e2a\u65b9\u7a0b\uff0c\u53ef\u63a8\u5bfc\u51fa\u6211\u4eec\u5b9e\u9a8c\u4e2d\u6b32\u6c42\u7684\u8f6c\u52a8\u60ef\u91cf$I$\u7684\u8868\u8fbe\u5f0f\u3002\n\n\u9996\u5148\uff0c\u5c06\u5f0f(3)\u7684\u7ea6\u675f\u6761\u4ef6$a = r\\alpha$\u4ee3\u5165\u5f0f(1)\uff0c\u6574\u7406\u5173\u4e8e\u5f20\u529b$T$\u7684\u5f0f\u5b50\uff1a\n\n$$mg - T = m(r\\alpha)$$\n\n$$T = m(g - r\\alpha) \\tag{4}$$\n\n\u5c06\u63a8\u5bfc\u51fa\u7684\u5f20\u529b$T$\u4ee3\u5165\u5f0f(2)\u7684\u65cb\u8f6c\u8fd0\u52a8\u65b9\u7a0b\uff1a\n\n$$r \\cdot \\left[ m(g - r\\alpha) \\right] = I\\alpha$$\n\n\u5c55\u5f00\u5de6\u8fb9\u7684\u62ec\u53f7\uff0c\u5f97\uff1a\n\n$$mgr - mr^2\\alpha = I\\alpha$$\n\n\u4e3a\u4e86\u4ee5\u6211\u4eec\u7684\u6700\u7ec8\u76ee\u6807\u8f6c\u52a8\u60ef\u91cf$I$\u6765\u6574\u7406\u5f0f\u5b50\uff0c\u4e24\u8fb9\u540c\u9664\u4ee5\u89d2\u52a0\u901f\u5ea6$\\alpha$\uff1a\n\n$$I = \\frac{mgr - mr^2\\alpha}{\\alpha} = \\frac{mgr}{\\alpha} - mr^2$$\n\n\u6700\u540e\uff0c\u7528\u5171\u540c\u9879$mr^2$\u5c06\u53f3\u8fb9\u63d0\u53d6\uff0c\u5b8c\u6210\u6700\u7ec8\u7684\u8f6c\u52a8\u60ef\u91cf\u5b9e\u9a8c\u5f0f\uff1a\n\n$$I = mr^2 \\left( \\frac{g}{r\\alpha} - 1 \\right)$$\n\n\u901a\u8fc7\u6b64\u63a8\u5bfc\u516c\u5f0f\uff0c\u6211\u4eec\u53ef\u4ec5\u4ee3\u5165\u5b9e\u9a8c\u5ba4\u4e2d\u76f4\u63a5\u6d4b\u5b9a\u7684\u51e0\u4f55\u5e38\u6570(\u781d\u7801\u8d28\u91cf$m$\u3001\u65cb\u8f6c\u8f74\u534a\u5f84$r$)\u548c\u91cd\u529b\u52a0\u901f\u5ea6$g$\uff0c\u4ee5\u53ca\u901a\u8fc7SPARKvue\u6570\u636e\u91c7\u96c6\u8f6f\u4ef6\u901a\u8fc7\u7ebf\u6027\u56de\u5f52(Linear Fit)\u83b7\u5f97\u7684**\u89d2\u52a0\u901f\u5ea6(Angular Acceleration)**$\\alpha$\u503c\uff0c\u5c31\u53ef\u5b9a\u91cf\u786e\u5b9a\u590d\u6742\u5f62\u5f0f\u7684\u521a\u4f53\u8f6c\u52a8\u60ef\u91cf$I$\u3002\n\n# 4. \u5b9e\u9a8c\u65b9\u6cd5\n\n\u672c\u5b9e\u9a8c\u5206\u4e3a\u4e24\u4e2a\u4e3b\u8981\u90e8\u5206\u8fdb\u884c\u3002**\u5b9e\u9a8cA**\u4e2d\u6d4b\u5b9a\u781d\u7801\u4e0b\u843d\u5f15\u8d77\u7684\u5706\u76d8\u548c\u5706\u73af\u7684\u8f6c\u52a8\u60ef\u91cf\uff0c**\u5b9e\u9a8cB**\u4e2d\u901a\u8fc7\u5411\u65cb\u8f6c\u5706\u76d8\u4e0a\u653e\u7f6e\u5706\u73af\u6765\u9a8c\u8bc1\u89d2\u52a8\u91cf\u662f\u5426\u5b88\u6052\u3002\n\n## 4.1 \u5b9e\u9a8cA \u8f6c\u52a8\u60ef\u91cf\u6d4b\u5b9a(Measurement of Moment of Inertia)\n\n\"null\"<\/p>\n

1. **\u65cb\u8f6c\u88c5\u7f6e\u5b89\u88c5**\uff1a\u5982[\u56fe3]\u6240\u793a\uff0c\u5b89\u88c5\u65cb\u8f6c\u7acb\u67b6\uff0c\u5e76\u5c06\u5706\u76d8(Rotational Disk)\u8fde\u63a5\u5230\u65cb\u8f6c\u8f74\u3002\u4f7f\u7528\u6c34\u5e73\u4eea(Leveler)\u7cbe\u786e\u8c03\u6574\u65cb\u8f6c\u7acb\u67b6\u548c\u5706\u76d8\u7684\u6c34\u5e73\u3002<\/p>\n

2. **\u667a\u80fd\u95e8\u8bbe\u7f6e**\uff1a\u5c06\u667a\u80fd\u95e8(Smart Gate)\u56fa\u5b9a\u5230\u65cb\u8f6c\u7acb\u67b6\u3002\u6b64\u65f6\u8c03\u6574\u4f4d\u7f6e\uff0c\u4f7f\u667a\u80fd\u95e8\u7684\u4f20\u611f\u5668\u80fd\u7cbe\u786e\u611f\u77e5\u8fde\u63a5\u5230\u65cb\u8f6c\u8f74\u7684\u6ed1\u8f6e(Pulley)\u7684\u6c9f\u69fd(Spoke)\u3002<\/p>\n

3. **\u57fa\u7840\u6570\u636e\u6d4b\u5b9a**\uff1a\u7528\u7535\u5b50\u79e4\u6d4b\u5b9a\u781d\u7801\u548c\u781d\u7801\u67b6\u7684\u603b\u8d28\u91cf$m$\uff0c\u7528\u6e38\u6807\u5361\u5c3a(Vernier Calipers)\u7cbe\u5bc6\u6d4b\u5b9a\u7ef3\u5b50\u7f20\u7ed5\u7684\u65cb\u8f6c\u8f74\u534a\u5f84$r$\u3002<\/p>\n

4. **\u8f6f\u4ef6\u51c6\u5907**\uff1a\u8fd0\u884cSPARKvue\u5e94\u7528\uff0c\u9009\u62e9**[Smart Gate Only]** -> [Smart Pulley (Rotational)]\u3002\u5c06**Spoke Angle**\u8bbe\u7f6e\u4e3a$36^\\circ$(\u6216\u7b26\u5408\u88c5\u7f6e\u7684\u503c)\uff0c\u9009\u62e9**Velocity**\u548c**Acceleration**\u4f5c\u4e3a\u6d4b\u91cf\u7269\u7406\u91cf\u3002<\/p>\n

5. **\u5706\u76d8\u89d2\u52a0\u901f\u5ea6\u6d4b\u5b9a**\uff1a\u7f20\u7ed5\u7ef3\u5b50\u5e76\u4f7f\u781d\u7801\u4e0b\u843d\u7684\u540c\u65f6\u6d4b\u5b9a\u89d2\u901f\u5ea6\u3002\u5728\u6570\u636e\u56fe\u8868\u4e2d\u9009\u62e9\u89d2\u52a0\u901f\u5ea6($\\alpha$)\u4e3a\u5e38\u6570\u7684\u533a\u95f4\uff0c\u8fdb\u884c\u7ebf\u6027\u56de\u5f52(Linear Fit)\uff0c\u5e76\u8bb0\u5f55\u5176\u659c\u7387\u503c\u3002<\/p>\n

\"\"<\/p>\n

6. **\u91cd\u590d\u6d4b\u5b9a**\uff1a\u5c06\u6b65\u9aa45\u91cd\u590d\u603b\u51715\u6b21\uff0c\u6c42\u51fa\u5706\u76d8\u7684\u5e73\u5747\u89d2\u52a0\u901f\u5ea6\uff0c\u636e\u6b64\u8ba1\u7b97$I_{disk}$\u3002<\/p>\n

7. **\u5706\u73af(Mass Ring)\u6dfb\u52a0**\uff1a\u5c06\u5706\u73af\u653e\u5230\u5706\u76d8\u4e0a\uff0c\u91cd\u590d\u6b65\u9aa45~6\u3002\u6b64\u65f6\u6d4b\u5b9a\u7684\u503c\u4e3a\u5706\u76d8\u4e0e\u5706\u73af\u7684\u5408\u6210\u8f6c\u52a8\u60ef\u91cf($I_{total}$)\uff0c\u4ece\u4e2d\u51cf\u53bb\u4e4b\u524d\u6c42\u5f97\u7684$I_{disk}$\uff0c\u63a8\u5bfc$I_{ring}$\u7684\u5b9e\u9a8c\u503c\u3002<\/p>\n

## 4.2 \u5b9e\u9a8cB \u89d2\u52a8\u91cf\u5b88\u6052(Conservation of Angular Momentum)<\/p>\n

1. **\u88c5\u7f6e\u91cd\u65b0\u6574\u5217**\uff1a\u79fb\u9664\u5728\u5b9e\u9a8cA\u4e2d\u4f7f\u7528\u7684\u7ef3\u5b50\u548c\u781d\u7801\u3002\u7cfb\u7edf\u5e94\u5904\u4e8e\u65e0\u5916\u90e8\u626d\u77e9\u7684\u81ea\u7531\u65cb\u8f6c\u72b6\u6001\u3002<\/p>\n

2. **\u521d\u59cb\u65cb\u8f6c\u53ca\u6d4b\u5b9a\u5f00\u59cb**\uff1a\u7528\u624b\u8f7b\u8f7b\u8f6c\u52a8\u5706\u76d8\u4f7f\u5176\u65cb\u8f6c\uff0c\u7136\u540e\u6309\u4e0bSPARKvue\u7684\u6d4b\u5b9a(Start)\u6309\u94ae\u3002<\/p>\n

3. **\u521d\u59cb\u89d2\u901f\u5ea6($\\omega_1$)\u6d4b\u5b9a**\uff1a\u5f53\u5706\u76d8\u7a33\u5b9a\u65cb\u8f6c\u65f6\uff0c\u8bb0\u5f55\u5706\u73af\u653e\u4e0b\u524d\u7684\u89d2\u901f\u5ea6$\\omega_1$\u3002<\/p>\n

4. **\u5706\u73af\u6295\u653e**\uff1a\u6cbf\u5706\u76d8\u7684\u65cb\u8f6c\u8f74\u65b9\u5411\u5c0f\u5fc3\u653e\u4e0b\u5706\u73af\u3002\u6b64\u65f6\u6ce8\u610f\u4f7f\u5706\u73af\u7cbe\u786e\u5750\u843d\u5728\u5706\u76d8\u7684\u4e2d\u5fc3\u3002<\/p>\n

5. **\u4e4b\u540e\u89d2\u901f\u5ea6($\\omega_2$)\u6d4b\u5b9a**\uff1a\u5706\u73af\u5750\u843d\u540e\uff0c\u8bb0\u5f55\u5706\u76d8\u4e0e\u5706\u73af\u5171\u540c\u7a33\u5b9a\u65cb\u8f6c\u65f6\u7684\u89d2\u901f\u5ea6$\\omega_2$\u3002<\/p>\n

6. **\u6570\u636e\u5206\u6790**\uff1a\u4f7f\u7528\u6d4b\u5b9a\u7684$\\omega_1, \\omega_2$\u548c\u5b9e\u9a8cA\u4e2d\u83b7\u5f97\u7684$I_{disk}, I_{total}$\u8ba1\u7b97\u78b0\u649e\u524d\u540e\u7684\u89d2\u52a8\u91cf$L_1, L_2$\uff0c\u786e\u8ba4\u662f\u5426\u5b88\u6052\u3002<\/p>\n

7. **\u91cd\u590d\u6d4b\u5b9a**\uff1a\u5c06\u6b65\u9aa42~6\u91cd\u590d\u603b\u51715\u6b21\uff0c\u786e\u4fdd\u6570\u636e\u7684\u53ef\u4fe1\u5ea6\u3002<\/p>\n

# 5. \u5b9e\u9a8c\u7ed3\u679c<\/p>\n

## 5.1 \u5b9e\u9a8c\u57fa\u672c\u53c2\u6570 (Constants)<\/p>\n

\u5728\u5b9e\u9a8cA\u548cB\u7684\u8ba1\u7b97\u4e2d\u5171\u540c\u4f7f\u7528\u7684\u51e0\u4f55\u5e38\u6570\u53ca\u8d28\u91cf\u6d4b\u5b9a\u503c\u5982\u4e0b\u3002<\/p>\n\n\n\n\n\n\n\n\n\n\n\n\n
\u53c2\u6570 (Parameter)<\/strong><\/th>\n\u7b26\u53f7 (Symbol)<\/strong><\/th>\n\u6d4b\u5b9a\u503c (Value)<\/strong><\/th>\n\u5355\u4f4d (Unit)<\/strong><\/th>\n<\/tr>\n<\/thead>\n
\u781d\u7801\u8d28\u91cf<\/strong> (Hanging Mass)<\/td>\n$m$<\/td>\n0.145<\/td>\n$\\text{kg}$<\/td>\n<\/tr>\n
\u65cb\u8f6c\u8f74\u534a\u5f84<\/strong> (Axle Radius)<\/td>\n$r$<\/td>\n0.0115<\/td>\n$\\text{m}$<\/td>\n<\/tr>\n
\u5706\u76d8\u8d28\u91cf<\/strong> (Disk Mass)<\/td>\n$M_{disk}$<\/td>\n1.427<\/td>\n$\\text{kg}$<\/td>\n<\/tr>\n
\u5706\u76d8\u534a\u5f84<\/strong> (Disk Radius)<\/td>\n$R_{disk}$<\/td>\n0.1145<\/td>\n$\\text{m}$<\/td>\n<\/tr>\n
\u5706\u73af\u8d28\u91cf<\/strong> (Ring Mass)<\/td>\n$M_{ring}$<\/td>\n1.441<\/td>\n$\\text{kg}$<\/td>\n<\/tr>\n
\u5706\u73af\u5185\u5f84<\/strong> (Ring Inner Radius)<\/td>\n$R_1$<\/td>\n0.054<\/td>\n$\\text{m}$<\/td>\n<\/tr>\n
\u5706\u73af\u5916\u5f84<\/strong> (Ring Outer Radius)<\/td>\n$R_2$<\/td>\n0.0635<\/td>\n$\\text{m}$<\/td>\n<\/tr>\n
\u91cd\u529b\u52a0\u901f\u5ea6<\/strong> (Gravity)<\/td>\n$g$<\/td>\n9.8<\/td>\n$\\text{m\/s}^2$<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

## 5.2 \u5b9e\u9a8cA \u8f6c\u52a8\u60ef\u91cf\u6d4b\u5b9a (Rotational Inertia)<\/p>\n

### 5.2.1 \u6d4b\u5b9a\u6570\u636e\u53ca\u5b9e\u9a8c\u503c\u8ba1\u7b97<\/p>\n

\u5bf9\u5706\u76d8\u5355\u72ec\u65cb\u8f6c\u548c\u5706\u76d8+\u5706\u73af\u91cd\u53e0\u65cb\u8f6c\u5206\u522b\u8fdb\u884c5\u6b21\u781d\u7801\u4e0b\u843d\uff0c\u6d4b\u5b9a\u89d2\u52a0\u901f\u5ea6($\\alpha$)\u3002\u5b9e\u9a8c\u8f6c\u52a8\u60ef\u91cf\u5229\u7528\u5f0f$I = mr^2 (\\frac{g}{r\\alpha} - 1)$\u63a8\u5bfc\u3002\u5706\u73af\u5355\u72ec\u7684\u8f6c\u52a8\u60ef\u91cf\u5b9e\u9a8c\u503c\u901a\u8fc7\u4ece\u603b\u8f6c\u52a8\u60ef\u91cf\u4e2d\u51cf\u53bb\u5706\u76d8\u7684\u8f6c\u52a8\u60ef\u91cf($I_{ring} = I_{total} - I_{disk}$)\u8ba1\u7b97\u3002<\/p>\n\n\n\n\n\n\n\n\n\n\n
\u6d4b\u5b9a\u6b21\u6570 $(Trial)$<\/strong><\/th>\n\u5706\u76d8 $\u03b1 (rad\/s^2)$<\/strong><\/th>\n\u5706\u76d8 $I_{disk}\u200b (kg \\cdot m^2)$<\/strong><\/th>\n\u5706\u76d8+\u5706\u73af $\u03b1 (rad\/s^2)$<\/strong><\/th>\n\u603b\u8f6c\u52a8\u60ef\u91cf $I_{total}\u200b (kg \\cdot m^2)$<\/strong><\/th>\n<\/tr>\n<\/thead>\n
Trial 1<\/strong><\/td>\n1.860<\/td>\n0.008767<\/td>\n1.210<\/td>\n0.013486<\/td>\n<\/tr>\n
Trial 2<\/strong><\/td>\n1.850<\/td>\n0.008814<\/td>\n1.210<\/td>\n0.013486<\/td>\n<\/tr>\n
Trial 3<\/strong><\/td>\n1.850<\/td>\n0.008814<\/td>\n1.220<\/td>\n0.013375<\/td>\n<\/tr>\n
Trial 4<\/strong><\/td>\n1.850<\/td>\n0.008814<\/td>\n1.210<\/td>\n0.013486<\/td>\n<\/tr>\n
Trial 5<\/strong><\/td>\n1.850<\/td>\n0.008767<\/td>\n1.210<\/td>\n0.013486<\/td>\n<\/tr>\n
\u5e73\u5747 (AVERAGE)<\/strong><\/td>\n1.8525<\/strong><\/td>\n0.008795<\/strong><\/td>\n1.2120<\/strong><\/td>\n0.013464<\/strong><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

### 5.2.2 \u8bef\u5dee\u5206\u6790 (Error Analysis Summary)<\/p>\n

\u5706\u76d8\u548c\u5706\u73af\u7684\u7406\u8bba\u8f6c\u52a8\u60ef\u91cf\u5206\u522b\u4f7f\u7528$I_{disk} = \\frac{1}{2}MR^2$\u3001$I_{ring} = \\frac{1}{2}M(R_1^2 + R_2^2)$\u516c\u5f0f\u8ba1\u7b97\uff0c\u4e0e\u5b9e\u9a8c\u5e73\u5747\u503c\u8fdb\u884c\u6bd4\u8f83\u3002<\/p>\n\n\n\n\n\n\n
\u9879\u76ee $(Item)$<\/strong><\/th>\n\u5b9e\u9a8c\u503c $I_{exp\u200b} (kg \\cdot m^2)$<\/strong><\/th>\n\u7406\u8bba\u503c $I_{theo}\u200b (kg \\cdot m^2)$<\/strong><\/th>\n\u76f8\u5bf9\u8bef\u5dee\u7387 $(\\%)$<\/strong><\/th>\n<\/tr>\n<\/thead>\n
\u5706\u76d8 (Disk Only)<\/strong><\/td>\n0.008795<\/td>\n0.009354<\/td>\n5.977<\/strong><\/td>\n<\/tr>\n
\u5706\u73af (Mass Ring)<\/strong><\/td>\n0.004669<\/td>\n0.005006<\/td>\n6.736<\/strong><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

_\u203b \u5706\u73af\u7684\u5b9e\u9a8c\u503c$I_{ring}$\u901a\u8fc7(\u603b\u8f6c\u52a8\u60ef\u91cf\u5e73\u57470.013464) - (\u5706\u76d8\u8f6c\u52a8\u60ef\u91cf\u5e73\u57470.008795) = 0.004669 \u8ba1\u7b97\u3002_<\/p>\n

## 5.3 \u5b9e\u9a8cB \u89d2\u52a8\u91cf\u5b88\u6052 (Conservation of Angular Momentum)<\/p>\n

\u5bf9\u65cb\u8f6c\u5706\u76d8\u4e0a\u653e\u7f6e\u5706\u73af\u7684\u5b8c\u5168\u975e\u5f39\u6027\u78b0\u649e\u5b9e\u9a8c\u91cd\u590d\u8fdb\u884c5\u6b21\u3002\u8ba1\u7b97\u78b0\u649e\u524d\u7684\u521d\u59cb\u89d2\u52a8\u91cf$L_1 = I_{disk} \\cdot \\omega_1$\u548c\u78b0\u649e\u540e\u7684\u4e4b\u540e\u89d2\u52a8\u91cf$L_2 = I_{total} \\cdot \\omega_2$\uff0c\u786e\u8ba4\u4e24\u503c\u7684\u5dee\u5f02\u3002<\/p>\n

_(\u6ce8\u610f\uff1a\u89d2\u52a8\u91cf\u8ba1\u7b97\u65f6\u4f7f\u7528\u7684\u8f6c\u52a8\u60ef\u91cf$I$\u91c7\u7528\u5b9e\u9a8cA\u4e2d\u83b7\u5f97\u7684\u5b9e\u9a8c\u5e73\u5747\u503c\u3002)_<\/p>\n\n\n\n\n\n\n\n\n\n\n
\u6d4b\u5b9a\u6b21\u6570 $(Trial)$<\/strong><\/th>\n\u521d\u59cb $\u03c9_1\u200b (rad\/s)$<\/strong><\/th>\n\u4e4b\u540e $\u03c9_2\u200b (rad\/s)$<\/strong><\/th>\n\u521d\u59cb $L_1\u200b (kg \\cdot m^2\/s)$<\/strong><\/th>\n\u4e4b\u540e $L_2\u200b (kg\u22c5m2\/s)$<\/strong><\/th>\n\u8bef\u5dee\u7387 $(\\%)$<\/strong><\/th>\n<\/tr>\n<\/thead>\n
Trial 1<\/strong><\/td>\n8.730<\/td>\n5.620<\/td>\n0.076781<\/td>\n0.075668<\/td>\n1.450<\/td>\n<\/tr>\n
Trial 2<\/strong><\/td>\n12.200<\/td>\n7.920<\/td>\n0.107300<\/td>\n0.106635<\/td>\n0.619<\/td>\n<\/tr>\n
Trial 3<\/strong><\/td>\n13.900<\/td>\n9.120<\/td>\n0.122251<\/td>\n0.122792<\/td>\n0.442<\/td>\n<\/tr>\n
Trial 4<\/strong><\/td>\n14.700<\/td>\n9.460<\/td>\n0.129288<\/td>\n0.127370<\/td>\n1.483<\/td>\n<\/tr>\n
Trial 5<\/strong><\/td>\n15.500<\/td>\n10.400<\/td>\n0.136324<\/td>\n0.140026<\/td>\n2.716<\/td>\n<\/tr>\n
\u5e73\u5747 (AVERAGE)<\/strong><\/td>\n13.006<\/strong><\/td>\n8.504<\/strong><\/td>\n0.114389<\/strong><\/td>\n0.114498<\/strong><\/td>\n1.342<\/strong><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

_\u203b \u8bef\u5dee\u7387\u8ba1\u7b97\u5f0f\uff1a$\\delta = \\frac{\\|L_1 - L_2\\|}{L_1} \\times 100 \\%$_<\/p>\n

# 6.\u5206\u6790\u53ca\u8ba8\u8bba<\/p>\n

\"\"<\/p>\n

\u5728\u672c\u5206\u6790\u4e2d\uff0c\u4e3a\u4e86\u5bf9\u5b9e\u9a8cA(\u8f6c\u52a8\u60ef\u91cf\u6d4b\u5b9a)\u548c\u5b9e\u9a8cB(\u89d2\u52a8\u91cf\u5b88\u6052)\u7684\u6d4b\u5b9a\u503c\u53ca\u7406\u8bba\u503c\u8fdb\u884c\u5b9a\u91cf\u548c\u53ef\u89c6\u5316\u6bd4\u8f83\uff0c\u91c7\u7528\u4e86\u57fa\u4e8ePython\u7684\u6570\u636e\u5206\u6790\u53ca\u53ef\u89c6\u5316\u7b97\u6cd5\u3002\u901a\u8fc7\u8fd9\u4e00\u65b9\u5f0f\uff0c\u7528\u53cc\u8f74(Dual-axis)\u7ec4\u5408\u56fe\u8868\u5b9e\u73b0\u4e86\u7edd\u5bf9\u7269\u7406\u91cf\u7684\u6bd4\u8f83(\u6761\u5f62\u56fe)\u548c\u76f8\u5bf9\u8bef\u5dee\u7684\u53d8\u52a8\u8d8b\u52bf(\u6298\u7ebf\u56fe)\u7684\u76f4\u89c2\u5206\u6790\uff0c\u8fdb\u800c\u5bf9\u6570\u636e\u7684\u53ef\u4fe1\u5ea6\u548c\u7cfb\u7edf\u8bef\u5dee\u8fdb\u884c\u4e86\u7efc\u5408\u8bc4\u4f30\u3002<\/p>\n

\u8f6c\u52a8\u60ef\u91cf\u5b9e\u9a8c\u503c$I_{exp}$\u4e0e\u7406\u8bba\u503c$I_{theo}$\u4e4b\u95f4\u7684\u76f8\u5bf9\u8bef\u5dee\u7387$\\delta_I$\uff0c\u4ee5\u53ca\u78b0\u649e\u524d\u540e\u7684\u89d2\u52a8\u91cf\u8bef\u5dee\u7387$\\delta_L$\u5b9a\u4e49\u5982\u4e0b\uff0c\u5e94\u7528\u4e8e\u5206\u6790\u3002<\/p>\n

$$\\delta_I(\\%) = \\frac{|I_{exp} - I_{theo}|}{I_{theo}} \\times 100\\%, \\qquad \\delta_L(\\%) = \\frac{|L_1 - L_2|}{L_1} \\times 100\\%$$<\/p>\n

## 6.1 \u5b9e\u9a8cA\uff1a\u5706\u76d8\u8f6c\u52a8\u60ef\u91cf\u7cbe\u5bc6\u6bd4\u8f83 (Disk Only)<\/p>\n

\u7b2c\u4e00\u5e45\u56fe\u5c55\u793a\u4e865\u6b21\u5706\u76d8(Disk)\u5355\u72ec\u65cb\u8f6c\u5b9e\u9a8c\u4e2d\u6d4b\u5b9a\u7684\u8f6c\u52a8\u60ef\u91cf\u5b9e\u9a8c\u503c\u4e0e\u57fa\u4e8e\u51e0\u4f55\u53c2\u6570\u7684\u7406\u8bba\u503c\u7684\u6bd4\u8f83\u3002<\/p>\n

\"\"<\/p>\n

\u4ece\u56fe\u8868\u5206\u6790\u6765\u770b\uff0c5\u6b21Trial\u4e2d\u6d4b\u5b9a\u7684\u5b9e\u9a8c\u503c(\u84dd\u8272\u6761\u5f62)\u59cb\u7ec8\u7565\u4f4e\u4e8e\u7406\u8bba\u503c(\u6a59\u8272\u6761\u5f62)\u3002\u56e0\u6b64\uff0c\u7528\u7ea2\u8272\u865a\u7ebf\u8868\u793a\u7684\u76f8\u5bf9\u8bef\u5dee\u7387\u4fdd\u6301\u5728\u7ea6**5.97%**\u7684\u975e\u5e38\u7a33\u5b9a\u6c34\u5e73\u3002<\/p>\n

\u6570\u636e\u7684\u5206\u6563\u5ea6(\u53d8\u52a8\u6027)\u6781\u4f4e\uff0c\u8bf4\u660e\u5b9e\u9a8c\u8005\u7684\u4e0b\u843d\u64cd\u4f5c\u3001\u901a\u8fc7SPARKvue\u7684\u89d2\u52a0\u901f\u5ea6($\\alpha$)\u6d4b\u5b9a\u4ee5\u53ca\u7ebf\u6027\u56de\u5f52(Linear fit)\u8fc7\u7a0b\u90fd\u662f\u6781\u4e3a\u7cbe\u5bc6\u8fdb\u884c\u7684\u3002\u5373\uff0c\u8fd9\u7ea66%\u7684\u8bef\u5dee\u5e94\u7406\u89e3\u4e3a\u7531\u5b9e\u9a8c\u88c5\u7f6e\u81ea\u8eab\u7684\u7ed3\u6784\u56e0\u7d20\u5bfc\u81f4\u7684**\u7cfb\u7edf\u8bef\u5dee(Systematic Error)**\uff0c\u800c\u975e\u968f\u673a\u8bef\u5dee(Random Error)\u3002<\/p>\n

## 6.2 \u5b9e\u9a8cA\uff1a\u5706\u73af\u8f6c\u52a8\u60ef\u91cf\u7cbe\u5bc6\u6bd4\u8f83 (Mass Ring)<\/p>\n

\u7b2c\u4e8c\u5e45\u56fe\u5c55\u793a\u4e86\u5706\u76d8\u548c\u5706\u73af\u7ed3\u5408\u72b6\u6001\u4e0b\u6d4b\u5b9a\u7684\u603b\u8f6c\u52a8\u60ef\u91cf($I_{total}$)\u51cf\u53bb\u5706\u76d8\u8f6c\u52a8\u60ef\u91cf($I_{disk}$)\u540e\u5f97\u51fa\u7684\u5706\u73af(Ring)\u5355\u72ec\u5b9e\u9a8c\u503c\u4e0e\u7406\u8bba\u503c\u7684\u6bd4\u8f83\u7ed3\u679c\u3002<\/p>\n

\"\"<\/p>\n

\u5706\u73af\u7684\u8f6c\u52a8\u60ef\u91cf\u8bef\u5dee\u7387\u540c\u6837\u7ea6\u4e3a**6.73%**\uff0c\u5c55\u73b0\u4e86\u4e0e\u5706\u76d8\u5b9e\u9a8c\u76f8\u4f3c\u7684\u8d8b\u52bf\u6027\u548c\u7a33\u5b9a\u7684\u8bef\u5dee\u8303\u56f4\u3002\u5b9e\u9a8cA\u5168\u7a0b\u4e2d\u5b9e\u9a8c\u503c\u6bd4\u7406\u8bba\u503c\u5c0f\u7ea66%(\u8ba1\u7b97\u4e3a$I_{exp} < I_{theo}$)\u7684\u539f\u56e0\u53ef\u901a\u8fc7\u4ee5\u4e0b\u4e24\u4e2a\u65b9\u9762\u8fdb\u884c\u6df1\u5165\u5206\u6790\u3002\n\n1. **\u7ef3\u5b50\u539a\u5ea6\u5bfc\u81f4\u7684\u6709\u6548\u534a\u5f84(Effective Radius)\u589e\u52a0\uff1a**\n\n\u5b9e\u9a8c\u516c\u5f0f$I = mr^2 (\\frac{g}{r\\alpha} - 1)$\u4e2d\uff0c\u65cb\u8f6c\u8f74\u7684\u534a\u5f84$r$\u4ee5\u5e73\u65b9\u7684\u5f62\u5f0f\u5f71\u54cd\u7ed3\u679c\u503c\u3002\u6211\u4eec\u4ee3\u5165\u7684$r = 0.0115\\text{ m}$\u662f\u7ef3\u5b50\u672a\u7f20\u7ed5\u65f6\u7684\u8f74\u534a\u5f84\u3002\u4f46\u5b9e\u9645\u4e0a\uff0c\u7531\u4e8e\u8f74\u4e0a\u7f20\u7ed5\u7684\u7ef3\u5b50\u539a\u5ea6($r_{string}$)\u548c\u91cd\u53e0\u73b0\u8c61\uff0c\u5b9e\u9645\u4f5c\u7528\u626d\u77e9\u7684\u6709\u6548\u534a\u5f84\u4e3a$r_{eff} = r + r_{string}\/2$\uff0c\u5fae\u5c0f\u5730\u66f4\u5927\u3002\u56e0\u4e3a\u4ee3\u5165\u4e86\u6bd4\u5b9e\u9645\u66f4\u5c0f\u7684$r$\u503c\uff0c\u63a8\u5bfc\u51fa\u7684\u8f6c\u52a8\u60ef\u91cf\u5b9e\u9a8c\u503c\u6bd4\u7406\u8bba\u503c\u5c0f\u3002\n\n2. **\u521a\u4f53\u7406\u60f3\u8d28\u91cf\u5206\u5e03\u5047\u8bbe\u7684\u5c40\u9650\u6027\uff1a**\n\n\u7406\u8bba\u503c\u8ba1\u7b97\u4e2d\u4f7f\u7528\u7684$0.5MR^2$\u7b49\u516c\u5f0f\u5047\u5b9a\u521a\u4f53\u7684\u5bc6\u5ea6\u5b8c\u5168\u5747\u5300\u7684\"\u7406\u60f3\u8fde\u7eed\u4f53\"\u3002\u7136\u800c\u5b9e\u9645\u7684\u5b9e\u9a8c\u7528\u5706\u76d8\u548c\u5706\u73af\u7531\u4e8e\u4e2d\u592e\u7528\u4e8e\u4f53\u7ed3\u7684\u69fd\u6216\u9500\u9489\u3001\u6750\u6599\u78e8\u635f\u7b49\u539f\u56e0\uff0c\u8d28\u91cf\u5206\u5e03\u5e76\u4e0d\u5b8c\u5168\u5747\u5300\uff0c\u8fd9\u5bfc\u81f4\u4e86\u4e0e\u7406\u8bba\u503c\u7684\u5dee\u5f02\u3002\n\n## 6.3 \u5b9e\u9a8cB\uff1a\u89d2\u52a8\u91cf\u5b88\u6052\u7efc\u5408\u5206\u6790 (Conservation of Angular Momentum)\n\n\u7b2c\u4e09\u5e45\u56fe\u5c55\u793a\u4e86\u5728\u5916\u529b\u88ab\u963b\u9694\u7684\u72b6\u6001\u4e0b\uff0c\u65cb\u8f6c\u5706\u76d8\u4e0a\u653e\u7f6e\u5706\u73af\u7684\u5b8c\u5168\u975e\u5f39\u6027\u78b0\u649e\u5b9e\u9a8c\u524d\u540e\u89d2\u52a8\u91cf($L_1$\u3001$L_2$)\u7684\u53d8\u5316\u3002\n\n\"\"<\/p>\n

\u8fd9\u4e2a\u56fe\u8868\u662f\u672c\u5b9e\u9a8c\u7684\u4eae\u70b9\uff0c\u4ece\u6761\u5f62\u56fe\u53ef\u4ee5\u770b\u51fa5\u6b21Trial\u4e2d\u521d\u59cb\u89d2\u52a8\u91cf$L_1$\u548c\u4e4b\u540e\u89d2\u52a8\u91cf$L_2$\u7684\u9ad8\u5ea6\u51e0\u4e4e\u5b8c\u7f8e\u4e00\u81f4\u3002\u4ee5\u53f3\u4fa7\u8f74\u4e3a\u57fa\u51c6\u7684\u8bef\u5dee\u7387(Difference)\u6298\u7ebf\u5728\u6700\u5c0f0.44%\u5230\u6700\u59272.71%\u4e4b\u95f4\u8bb0\u5f55\uff0c\u663e\u73b0\u4e86**\u5e73\u57471.34%**\u7684\u60ca\u4eba\u7cbe\u786e\u5ea6\u3002<\/p>\n

\u8fd9\u8868\u660e\u5c3d\u7ba1\u89d2\u901f\u5ea6\u4ece$\\omega_1$\u6025\u5267\u4e0b\u964d\u5230$\\omega_2$\uff0c\u4f46\u8f6c\u52a8\u60ef\u91cf\u7684\u589e\u52a0\u5e45\u5ea6($I_{disk} \\to I_{total}$)\u7cbe\u786e\u5730\u62b5\u6d88\u4e86\u8fd9\u4e00\u53d8\u5316\uff0c\u4f7f\u5f97$I_{disk}\\omega_1 = I_{total}\\omega_2$\u5728\u6570\u503c\u4e0a\u5b8c\u7f8e\u6210\u7acb\uff0c\u4ece\u800c\u4ee5\u6570\u636e\u65b9\u5f0f\u5b8c\u5168\u8bc1\u5b9e\u4e86\u8fd9\u4e00\u70b9\u3002<\/p>\n

\u4f46\u5fae\u5c0f\u7684\u6b8b\u4f59\u8bef\u5dee(\u7ea61.3%)\u4ea7\u751f\u7684\u539f\u56e0\u5982\u4e0b\u3002<\/p>\n

- **\u6469\u64e6\u626d\u77e9(Frictional Torque)\uff1a** \u5706\u73af\u88ab\u653e\u4e0b\u5230\u901f\u5ea6\u518d\u6b21\u7a33\u5b9a\u5316($\\omega_2$)\u4e3a\u6b62\u7684\u77ed\u6682\u65f6\u95f4\u5185\uff0c\u8f74\u627f\u6469\u64e6\u548c\u7a7a\u6c14\u963b\u529b\u5bf9\u7cfb\u7edf\u4ea7\u751f\u8d1f(-)\u7684\u5408\u626d\u77e9\uff0c\u5fae\u5c0f\u5730\u51cf\u5c11\u4e86\u89d2\u52a8\u91cf\u3002<\/p>\n

- **\u6295\u653e\u65f6\u4e2d\u5fc3\u8f74\u504f\u5dee\uff1a** \u5f53\u5706\u73af\u88ab\u653e\u4e0b\u65f6\u5982\u679c\u4e0d\u662f\u5b8c\u5168\u7f6e\u4e8e\u6b63\u4e2d\u592e\uff0c\u800c\u662f\u504f\u79bb\u4e2d\u5fc3\u8f74$d$\uff0c\u5219\u6839\u636e\u5e73\u884c\u8f74\u5b9a\u7406($I = I_{cm} + Md^2$)\uff0c\u4e4b\u540e\u7684\u8f6c\u52a8\u60ef\u91cf\u4f1a\u6bd4\u9884\u671f\u66f4\u5927\uff0c\u5bfc\u81f4\u89d2\u901f\u5ea6\u53d8\u5316\u4e2d\u4ea7\u751f\u5fae\u5c0f\u8bef\u5dee\u3002<\/p>\n

## 6.4 \u7efc\u5408\u7ed3\u8bba<\/p>\n

\u901a\u8fc7\u672c\u5b9e\u9a8c\uff0c\u6211\u4eec\u6210\u529f\u9a8c\u8bc1\u4e86\u4ee5\u4e0b\u529b\u5b66\u539f\u7406\u3002<\/p>\n

1. \u901a\u8fc7\u5b9e\u9a8c\u5f0f$I = mr^2 (\\frac{g}{r\\alpha} - 1)$\u5b9a\u91cf\u786e\u8ba4\u4e86\u521a\u4f53\u7684\u8d28\u91cf\u548c\u51e0\u4f55\u5206\u5e03(\u534a\u5f84)\u662f\u51b3\u5b9a\u8f6c\u52a8\u60ef\u91cf\u7684\u6838\u5fc3\u8981\u7d20\u3002(\u8bef\u5dee\u7387\u57285~6%\u5de6\u53f3)<\/p>\n

2. \u5728\u5916\u90e8\u5408\u626d\u77e9\u4e3a0\u7684\u9694\u79bb\u7cfb\u7edf\u4e2d\uff0c\u7cfb\u7edf\u5185\u90e8\u8d28\u91cf\u5206\u5e03\u53d8\u5316(\u5706\u73af\u6295\u653e)\u5bfc\u81f4\u89d2\u901f\u5ea6\u53d8\u5316\u65f6\uff0c\u7cfb\u7edf\u7684\u603b\u89d2\u52a8\u91cf\u4fdd\u6301\u4e0d\u53d8\uff0c\u8fd9\u4e00\u70b9\u5f97\u5230\u4e86\u5e73\u5747\u8bef\u5dee\u7387**1.34%**\u8fd9\u4e00\u9ad8\u5ea6\u53ef\u4fe1\u7684\u8bc1\u5b9e\u3002<\/p>\n

3. \u5728\u5b9e\u9a8c\u6570\u636e\u521d\u671f\u5206\u6790\u8fc7\u7a0b\u4e2d\uff0c\u53d1\u73b0\u4e86\u5c06\u65cb\u8f6c\u8f74\u7684\"\u76f4\u5f84\"\u8bef\u8ba4\u4e3a\"\u534a\u5f84\"\u800c\u4ee3\u5165\u7684\u81f4\u547d\u4eba\u4e3a\u9519\u8bef(\u521d\u671f\u8bef\u5dee\u738780%\u4ee5\u4e0a)\uff0c\u5e76\u8fdb\u884c\u4e86\u4fee\u6b63\u3002\u901a\u8fc7\u8fd9\u4e00\u8fc7\u7a0b\uff0c\u4f53\u9a8c\u4e86\u6570\u636e\u9a8c\u8bc1\u7684\u91cd\u8981\u6027\u4ee5\u53ca\u516c\u5f0f\u4e2d\u53d8\u91cf(\u7279\u522b\u662f\u5e73\u65b9\u9879$r$)\u7684\u654f\u611f\u5ea6\u3002<\/p>\n

## 6.5 Python\u6e90\u4ee3\u7801 (\u6570\u636e\u53ef\u89c6\u5316\u7b97\u6cd5)<\/p>\n

\u672c\u62a5\u544a\u4e2d\u6570\u636e\u5206\u6790\u4f7f\u7528\u7684Python\u7b97\u6cd5\u7684\u6838\u5fc3\u53ef\u89c6\u5316\u51fd\u6570(\u751f\u6210\u53cc\u8f74\u7ec4\u5408\u56fe\u8868)\u5982\u4e0b\u3002<\/p>\n

import csv\r\nimport matplotlib.pyplot as plt\r\nimport numpy as np\r\nimport os\r\n\r\nplt.rcParams['font.family'] = 'serif'\r\nplt.rcParams['font.serif'] = ['Times New Roman', 'DejaVu Serif']\r\nplt.rcParams['mathtext.fontset'] = 'stix' \r\nplt.rcParams['axes.labelsize'] = 12\r\nplt.rcParams['axes.titlesize'] = 14\r\nplt.rcParams['xtick.labelsize'] = 11\r\nplt.rcParams['ytick.labelsize'] = 11\r\nplt.rcParams['legend.fontsize'] = 10\r\n\r\ndef plot_combo_chart(x_labels, bar1, label1, bar2, label2, line_data, line_label, title, y_left, y_right, filename):\r\n    \"\"\"\r\n    Helper function to generate a dual-axis combination chart.\r\n    Bars: 6 decimal places (smaller font).\r\n    Line: 3 decimal places (slightly larger, bolder font).\r\n    \"\"\"\r\n    fig, ax1 = plt.subplots(figsize=(11, 6.5))\r\n    x = np.arange(len(x_labels))\r\n    width = 0.35\r\n\r\n    rects1 = ax1.bar(x - width\/2, bar1, width, label=label1, color='#4C72B0', alpha=0.85)\r\n    rects2 = ax1.bar(x + width\/2, bar2, width, label=label2, color='#DD8452', alpha=0.85)\r\n    \r\n    ax1.set_ylabel(y_left, fontweight='bold')\r\n    ax1.set_xticks(x)\r\n    ax1.set_xticklabels(x_labels)\r\n    \r\n    ax1.bar_label(rects1, fmt='%.6f', padding=4, fontsize=8, color='#1b2a49')\r\n    ax1.bar_label(rects2, fmt='%.6f', padding=4, fontsize=8, color='#5c2a11')\r\n    \r\n    max_bar_val = max(max(bar1), max(bar2))\r\n    if max_bar_val > 0:\r\n        ax1.set_ylim(0, max_bar_val * 1.25)\r\n    \r\n    ax2 = ax1.twinx()\r\n    line = ax2.plot(x, line_data, color='#C44E52', marker='D', linestyle='--', linewidth=2.5, markersize=8, label=line_label)\r\n    \r\n    ax2.set_ylabel(y_right, color='#C44E52', fontweight='bold')\r\n    ax2.tick_params(axis='y', labelcolor='#C44E52')\r\n    \r\n    for i, val in enumerate(line_data):\r\n        if not np.isnan(val):\r\n            ax2.annotate(f'{val:.3f}', \r\n                         (x[i], val), \r\n                         textcoords=\"offset points\", \r\n                         xytext=(0, 10), \r\n                         ha='center', \r\n                         fontsize=10.5, \r\n                         color='#C44E52', \r\n                         fontweight='bold')\r\n    \r\n    max_line_val = max([v for v in line_data if not np.isnan(v)] or [0])\r\n    if max_line_val > 0:\r\n        ax2.set_ylim(0, max_line_val * 1.35)<\/pre>\n
> \u5305\u542b\u5b8c\u6574\u6570\u636e\u89e3\u6790\u53ca\u9884\u5904\u7406\u903b\u8f91\u7684\u5b8c\u6574\u6e90\u4ee3\u7801\u8bf7\u53c2\u8003\u9644\u52a0\u7684`main.py`\u3002<\/p>\n
# 7. \u5b9e\u9a8c\u65f6\u6ce8\u610f\u4e8b\u9879<\/p>\n
\u672c\u5b9e\u9a8c\u6d89\u53ca\u65cb\u8f6c\u529b\u5b66\u7cfb\u7edf\u7684\u5fae\u5c0f\u53d8\u5316\uff0c\u56e0\u6b64\u4e3a\u4e86\u6700\u5c0f\u5316\u51e0\u4f55\u8bef\u5dee\u548c\u6469\u64e6\uff0c\u5fc5\u987b\u4e25\u683c\u9075\u5b88\u4ee5\u4e0b\u4e8b\u9879\u8fdb\u884c\u5b9e\u9a8c\u3002<\/p>\n
## 7.1 \u65cb\u8f6c\u88c5\u7f6e\u7684\u6c34\u5e73\u4fdd\u6301 (Leveling)<\/p>\n
\u8c03\u6574\u65cb\u8f6c\u7acb\u67b6\u5e95\u5ea7\u4e0a\u5b89\u88c5\u7684\u6c34\u5e73\u4eea(Leveler)\u7684\u87ba\u9489\uff0c\u4f7f\u6c34\u6ce1\u5904\u4e8e\u6b63\u4e2d\u592e\uff0c\u5b8c\u5168\u5bf9\u9f50\u88c5\u7f6e\u7684\u6c34\u5e73\u3002\u82e5\u6c34\u5e73\u4e0d\u5bf9\u9f50\uff0c\u539f\u76d8\u7684\u65cb\u8f6c\u8f74\u4f1a\u503e\u659c\uff0c\u91cd\u529b\u7684\u5206\u529b\u4f1a\u6cbf\u65cb\u8f6c\u65b9\u5411\u4f5c\u7528\uff0c\u4ea7\u751f\u610f\u5916\u7684\u989d\u5916\u5408\u626d\u77e9(Torque)\uff0c\u5bfc\u81f4\u89d2\u52a0\u901f\u5ea6\u6d4b\u5b9a\u4e2d\u4ea7\u751f\u4e25\u91cd\u7684\u7cfb\u7edf\u8bef\u5dee\u3002<\/p>\n
## 7.2 \u7ef3\u5b50\u7f20\u7ed5\u9632\u6b62\u91cd\u53e0\u53ca\u6c34\u5e73\u5bf9\u9f50 (String Winding & Alignment)<\/p>\n
\u5728\u65cb\u8f6c\u8f74(3\u7ea7\u6ed1\u8f6e)\u4e0a\u7f20\u7ed5\u7ef3\u5b50\u65f6\uff0c\u5e94\u4fdd\u8bc1\u7ef3\u5b50\u4e0d\u91cd\u53e0\u5e76\u6574\u9f50\u5730\u5355\u884c\u7f20\u7ed5\u3002\u82e5\u7ef3\u5b50\u91cd\u53e0\u7f20\u7ed5\uff0c\u65cb\u8f6c\u8f74\u7684\u6709\u6548\u534a\u5f84($r$)\u4f1a\u968f\u7ef3\u5b50\u539a\u5ea6\u800c\u53d8\u5316\u3002\u5b9e\u9a8c\u516c\u5f0f$I = mr^2 (\\frac{g}{r\\alpha} - 1)$\u4e2d$r$\u4ee5\u5e73\u65b9\u7684\u5f62\u5f0f\u6bd4\u4f8b\u76f8\u5173\uff0c\u6240\u4ee5\u5fae\u5c0f\u7684\u534a\u5f84\u53d8\u5316\u4e5f\u4f1a\u5bfc\u81f4\u8f6c\u52a8\u60ef\u91cf\u63a8\u5bfc\u7684\u5927\u8bef\u5dee\u3002\u53e6\u5916\uff0c\u5f53\u7ef3\u5b50\u8de8\u8d8a\u667a\u80fd\u6ed1\u8f6e(Smart Pulley)\u65f6\uff0c\u5e94\u8c03\u6574\u6ed1\u8f6e\u9ad8\u5ea6\uff0c\u4f7f\u7ef3\u5b50\u4e0e\u5730\u9762\u5f62\u6210\u5b8c\u7f8e\u7684\u6c34\u5e73\uff0c\u4ee5\u4fbf\u5f20\u529b\u5b8c\u5168\u4f5c\u7528\u4e3a\u65cb\u8f6c\u8f74\u7684\u5207\u7ebf\u65b9\u5411\u626d\u77e9\u3002<\/p>\n
## 7.3 \u5706\u73af\u6295\u653e\u65f6\u7cbe\u786e\u7684\u4e2d\u5fc3\u4e00\u81f4 (Centering the Mass Ring)<\/p>\n
\u5728\u5b9e\u9a8cB(\u89d2\u52a8\u91cf\u5b88\u6052)\u4e2d\uff0c\u5c06\u5706\u73af\u6295\u653e\u5230\u65cb\u8f6c\u5706\u76d8\u4e0a\u65f6\uff0c\u5e94\u4f7f\u5706\u73af\u7cbe\u786e\u5730\u4e0e\u5706\u76d8\u7684\u4e2d\u5fc3\u5bfc\u69fd\u5bf9\u9f50\u5e76\u843d\u4e0b\u3002\u5982\u679c\u504f\u79bb\u4e2d\u5fc3$d$\u800c\u843d\u4e0b\uff0c\u5219\u6839\u636e\u5e73\u884c\u8f74\u5b9a\u7406($I = I_{cm} + Md^2$)\uff0c\u5706\u73af\u7684\u8f6c\u52a8\u60ef\u91cf\u4f1a\u5f02\u5e38\u589e\u5927\uff0c\u5bfc\u81f4\u4e4b\u540e\u7684\u89d2\u52a8\u91cf\u8ba1\u7b97\u4ea7\u751f\u5927\u8bef\u5dee\u3002<\/p>\n
## 7.4 \u6700\u5c0f\u5316\u5916\u90e8\u626d\u77e9\u7684\u5e72\u6270 (Minimizing External Torque)<\/p>\n
\u6295\u653e\u5706\u73af\u65f6\u8981\u6ce8\u610f\u4e0d\u8981\u7528\u624b\u7ed9\u4e88\u65cb\u8f6c\u65b9\u5411\u7684\u529b(\u521d\u59cb\u89d2\u901f\u5ea6)\uff0c\u4e5f\u4e0d\u8981\u5782\u76f4\u5f3a\u529b\u538b\u4e0b\u3002\u53ea\u5e94\u901a\u8fc7\u91cd\u529b\u4f5c\u7528\u8f7b\u8f7b\u653e\u4e0b(Drop)\uff0c\u9664\u78b0\u649e\u65f6\u4ea7\u751f\u7684\u6469\u64e6\u5916\uff0c\u4e0d\u5141\u8bb8\u5176\u4ed6\u5916\u90e8\u626d\u77e9\u4ecb\u5165\u7cfb\u7edf\uff0c\u624d\u80fd\u89c2\u5bdf\u5230\u7eaf\u7cb9\u7684\u89d2\u52a8\u91cf\u5b88\u6052\u3002<\/p>\n
## 7.5 \u781d\u7801\u4e0b\u843d\u5b89\u5168\u8ddd\u79bb\u786e\u4fdd (Safety of Hanging Mass)<\/p>\n
\u8fdb\u884c\u5b9e\u9a8cA\u65f6\uff0c\u8981\u6ce8\u610f\u4e0b\u843d\u7684\u781d\u7801\u4e0d\u4e0e\u5730\u9762\u6216\u667a\u80fd\u95e8\u88c5\u7f6e\u78b0\u649e\u3002\u781d\u7801\u4e0e\u5730\u9762\u78b0\u649e\u7684\u77ac\u95f4\u5f20\u529b($T$)\u4f1a\u7a81\u7136\u53d8\u4e3a0\u6216\u4ea7\u751f\u53cd\u5f39\uff0c\u5bfc\u81f4\u6570\u636e(\u89d2\u52a0\u901f\u5ea6\u7ebf\u6027\u533a\u95f4)\u88ab\u7834\u574f\uff0c\u56e0\u6b64\u5e94\u53ea\u91c7\u96c6\u5230\u781d\u7801\u5373\u5c06\u63a5\u89e6\u5730\u9762\u524d\u7684\u6570\u636e\uff0c\u5e76\u7528\u624b\u6216\u57ab\u5b50\u5b89\u5168\u5730\u63a5\u4f4f\u781d\u7801\u3002<\/p>\n
# 8. \u53c2\u8003\u6587\u732e<\/p>\n
[1] \u5e86\u7199\u5927\u5b66, \"E1-05 \u89d2\u52a8\u91cf\u5b88\u6052,\" APHY1002-11 \u7269\u7406\u5b66\u53ca\u5b9e\u9a8c1 \u5b9e\u9a8c\u8d44\u6599 (PDF), n.d.
\n[2] \u5e86\u7199\u5927\u5b66, \"E1-05_\u89d2\u52a8\u91cf\u5b88\u6052,\" APHY1002-11 \u7269\u7406\u5b66\u53ca\u5b9e\u9a8c1 \u5b9e\u9a8c\u8d44\u6599 (PDF), n.d.
\n[3] \u5e86\u7199\u5927\u5b66, \"EXP05_\u89d2\u52a8\u91cf\u5b88\u6052,\" APHY1002-11 \u7269\u7406\u5b66\u53ca\u5b9e\u9a8c1 \u5b9e\u9a8c\u6570\u636e\u8868 (CSV), 2026.
\n[4] \u5e86\u7199\u5927\u5b66, \"\u7269\u7406\u5b66 \u5b9e\u9a8c-OT-\u674e\u5efa\u5f6c,\" APHY1002-11 \u7269\u7406\u5b66\u53ca\u5b9e\u9a8c1 \u5b9a\u5411\u8d44\u6599 (PDF), n.d.
\n[5] \u6bdb\u9a8f\u5065, \u987e\u7261, \u300e\u5927\u5b66\u7269\u7406\u5b66(\u7b2c\u4e09\u7248)(\u4e0a\u518c)\u300f (\u5927\u5b66\u7269\u7406\u5b66\u7b2c3\u7248\u4e0a\u518c), \u9ad8\u7b49\u6559\u80b2\u51fa\u7248\u793e (\u9ad8\u7b49\u6559\u80b2\u51fa\u7248\u793e), 2020, ISBN: 9787040548822.<\/p>\n","protected":false},"excerpt":{"rendered":"
1.\u5b9e\u9a8c\u6807\u9898<\/p>\n
\u672c\u6b21\u5b9e\u9a8c\u7684\u4e3b\u9898\u662f**\u8f6c\u52a8\u60ef\u91cf\u6d4b\u5b9a\u4e0e\u89d2\u52a8\u91cf\u5b88\u6052**\u3002<\/p>\n
2. \u5b9e\u9a8c\u76ee\u7684<\/p>\n
\u901a\u8fc7\u6d4b\u91cf\u65cb\u8f6c\u521a\u4f53(Rigid Body)\u7684\u89d2\u52a0\u901f\u5ea6(Angular Acceleratio","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"emotion":"","emotion_color":"","title_style":"","license":"","footnotes":""},"categories":[1],"tags":[],"class_list":["post-331","post","type-post","status-publish","format-standard","hentry","category-article-cn"],"_links":{"self":[{"href":"https:\/\/wuhanqing.cn\/wordpress\/wp-json\/wp\/v2\/posts\/331","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/wuhanqing.cn\/wordpress\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/wuhanqing.cn\/wordpress\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/wuhanqing.cn\/wordpress\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/wuhanqing.cn\/wordpress\/wp-json\/wp\/v2\/comments?post=331"}],"version-history":[{"count":1,"href":"https:\/\/wuhanqing.cn\/wordpress\/wp-json\/wp\/v2\/posts\/331\/revisions"}],"predecessor-version":[{"id":332,"href":"https:\/\/wuhanqing.cn\/wordpress\/wp-json\/wp\/v2\/posts\/331\/revisions\/332"}],"wp:attachment":[{"href":"https:\/\/wuhanqing.cn\/wordpress\/wp-json\/wp\/v2\/media?parent=331"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/wuhanqing.cn\/wordpress\/wp-json\/wp\/v2\/categories?post=331"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/wuhanqing.cn\/wordpress\/wp-json\/wp\/v2\/tags?post=331"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}