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Rolling_Racers_-_Moment_of_inertia.gif(480 × 270 像素,檔案大小:1.6 MB,MIME 類型:image/gif、​循環、​126 畫格、​4.2秒)


描述
English: An object's moment of inertia I determines how much it resists rotational motion. In this simulation, four objects are placed on a ramp and left to roll without slipping. Starting from rest, each will experience an angular acceleration based on their moment of inertia.

The objects are, from back to front:

  1. A hollow spherical shell (red)
  2. A solid ball (orange)
  3. A ring (green)
  4. A solid cylinder (blue)

At any moment in time, the forces acting on each object will be its weight, the normal force exerted by the plane on the object and the static friction force. As the weight force and the normal force act on a line through each object's center of mass, they result in no net torque. However, the force due to friction acts perpendicular to the contact point, and therefore it does result in a torque, which causes the object to rotate.

Since there is no slipping, the object's center of mass will travel with speed , where r is its radius, or the distance from a contact point to the axis of rotation, and ω its angular speed. Since static friction does no work, and dissipative forces are being ignored, we have conservation of energy. Therefore:

Solving for , we obtain:

Since the torque is constant we conclude, by Newton's 2nd Law for rotation , that the angular acceleration α is also constant. Therefore:

Where, v0 = 0 and d is the total distance traveled. Therefore, we have:

For a ramp with inclination θ, we have sin θ = h / d. Additionally, for a dimensionless constant k characteristic of the geometry of the object. Finally, we can write the angular acceleration α using the relation :

This final result reveals that, for objects of the same radius, the mass the object are irrelevant and what determines the rate of acceleration is the geometric distribution of their mass, which is represented by the value of k. Additionally, we observe that objects with larger values of k will accelerate more slowly.

This is illustrated in the animation. The values of k for each object are, from back to front: 2/3, 2/5, 1, 1/2. As predicted by the formula found above, the solid ball will have a larger acceleration, reaching the finish line first.
日期
來源 自己的作品
作者 Lucas Vieira
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其他版本 OGG Theora Video: small and [[:File:Rolling Racers - Moment of inertia (HD).ogv|large (HD) and for classroom educational purposes a static image of the finish at File:Rolling Racers - Moment of inertia Photofinish.jpg ]]

POV-Ray source code

Available at the video version's description page.

此圖像經優質圖像指引的評估,被認為是一張優質圖像

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Comparision Of Moment Of Inertia for Different Solids.

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作者姓名字串 繁體中文 (已轉換拼寫):​Lucas Vieira
維基媒體使用者名稱 繁體中文 (已轉換拼寫):​LucasVB

著作權狀態 繁體中文 (已轉換拼寫)

共享資源質素評價 繁體中文 (已轉換拼寫)

維基共享資源品質形象 中文 (已轉換拼寫)

檔案來源 Chinese (Taiwan) (已轉換拼寫)

上傳者的原創作品 繁體中文 (已轉換拼寫)

多媒體型式 繁體中文 (已轉換拼寫)

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日期/時間縮⁠圖尺寸用戶備⁠註
目前2021年6月16日 (三) 21:01於 2021年6月16日 (三) 21:01 版本的縮圖480 × 270(1.6 MB)TomFryersImprove render quality and increase resolution and framerate slightly
2012年12月23日 (日) 03:10於 2012年12月23日 (日) 03:10 版本的縮圖444 × 250(1.49 MB)LucasVB{{Information |Description=... |Source={{own}} |Date=2012-12-23 |Author= Lucas V. Barbosa |Permission={{PD-self}} |other_versions=OGG Theora video }}

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