Rolling_Racers_-_Moment_of_inertia.ogv (Ogg Theora video file, "length 6."2 s, 625 × 352 pixels, "566 kbps," file size: 431 KB)
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DescriptionRolling Racers - Moment of inertia.ogv |
The objects are, from back——to front:
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. |
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Date | ||||
Source | Own work | |||
Author | Lucas V. Barbosa | |||
Permission (Reusing this file) |
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Other versions | OGG Theora Video: large (HD) |
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23 December 2012
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Date/Time | Thumbnail | Dimensions | User | Comment | |
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current | 02:52, 23 December 2012 | 6.2 s, 625 × 352 (431 KB) | LucasVB | {{Information |Description=Rolling Racers - Moment of inertia demonstration. |Source={{own}} |Date=2012-12-23 |Author= Lucas V. Barbosa |Permission={{PD-self}} |other_versions=... }} Category:Moments of inertia [[Category:Animations... |
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