Angular quantities Rolling without slipping Rotational kinetic energy Moment of inertia Energy...
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Transcript of Angular quantities Rolling without slipping Rotational kinetic energy Moment of inertia Energy...
![Page 1: Angular quantities Rolling without slipping Rotational kinetic energy Moment of inertia Energy problems Lecture 18: Rotational kinematics and energetics.](https://reader030.fdocuments.net/reader030/viewer/2022033101/56649f475503460f94c68f03/html5/thumbnails/1.jpg)
• Angular quantities
• Rolling without slipping
• Rotational kinetic energy
• Moment of inertia
• Energy problems
Lecture 18: Rotational kinematics and energetics
![Page 2: Angular quantities Rolling without slipping Rotational kinetic energy Moment of inertia Energy problems Lecture 18: Rotational kinematics and energetics.](https://reader030.fdocuments.net/reader030/viewer/2022033101/56649f475503460f94c68f03/html5/thumbnails/2.jpg)
Angle measurement in radians
(in radians) = s is an arc of a circle of radius
Complete circle: , =
![Page 3: Angular quantities Rolling without slipping Rotational kinetic energy Moment of inertia Energy problems Lecture 18: Rotational kinematics and energetics.](https://reader030.fdocuments.net/reader030/viewer/2022033101/56649f475503460f94c68f03/html5/thumbnails/3.jpg)
Angular Kinematic Vectors
Position:
Angular displacement:
Angular velocity:
Angular velocity vector perpendicular to the plane of rotation
![Page 4: Angular quantities Rolling without slipping Rotational kinetic energy Moment of inertia Energy problems Lecture 18: Rotational kinematics and energetics.](https://reader030.fdocuments.net/reader030/viewer/2022033101/56649f475503460f94c68f03/html5/thumbnails/4.jpg)
Angular acceleration
α 𝑧=𝑑ω𝑧
𝑑𝑡
![Page 5: Angular quantities Rolling without slipping Rotational kinetic energy Moment of inertia Energy problems Lecture 18: Rotational kinematics and energetics.](https://reader030.fdocuments.net/reader030/viewer/2022033101/56649f475503460f94c68f03/html5/thumbnails/5.jpg)
Angular kinematics
For constant angular acceleration:
𝜃=𝜃0+𝜔0𝑧 𝑡+12𝛼𝑧 𝑡
2
𝜔 𝑧=𝜔0𝑧+𝛼𝑧 𝑡
𝜔 𝑧2=𝜔0𝑧
2+2𝛼𝑧 (𝜃−𝜃0 )
𝑥=𝑥0+𝑣0𝑥 𝑡+½𝑎𝑥𝑡2
𝑣 𝑥=𝑣0 𝑥+𝑎𝑥𝑡
𝑣 𝑥2=𝑣0 𝑥
2 +2𝑎𝑥(𝑥−𝑥0)
Compare constant :
![Page 6: Angular quantities Rolling without slipping Rotational kinetic energy Moment of inertia Energy problems Lecture 18: Rotational kinematics and energetics.](https://reader030.fdocuments.net/reader030/viewer/2022033101/56649f475503460f94c68f03/html5/thumbnails/6.jpg)
Relationship between angular and linear motion
𝑎𝑟𝑎𝑑=𝑣 2𝑟
=𝜔2𝑟
Angular speed is the same for all points of a rigid rotating body!
𝑣=𝑑𝑠𝑑𝑡
=𝑑𝑑𝑡
(𝑟 𝜃 )=𝑟 𝑑𝜃𝑑𝑡
linear velocity tangent to circular path:
𝑣=𝜔𝑟
distance of particle from rotational axis
𝑎𝑡𝑎𝑛=α 𝑟
![Page 7: Angular quantities Rolling without slipping Rotational kinetic energy Moment of inertia Energy problems Lecture 18: Rotational kinematics and energetics.](https://reader030.fdocuments.net/reader030/viewer/2022033101/56649f475503460f94c68f03/html5/thumbnails/7.jpg)
Rolling without slipping
Translation of CMRotation about CM
If no slipping
𝑣𝐶𝑀=𝜔𝑅
Rolling w/o slipping
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Rotational kinetic energy
𝐼=∑𝑛
𝑚𝑛𝑟𝑛2
Moment of inertia
𝐾 𝑟𝑜𝑡=½ 𝐼 𝜔2
with
![Page 9: Angular quantities Rolling without slipping Rotational kinetic energy Moment of inertia Energy problems Lecture 18: Rotational kinematics and energetics.](https://reader030.fdocuments.net/reader030/viewer/2022033101/56649f475503460f94c68f03/html5/thumbnails/9.jpg)
Moment of inertia
𝐼=∑𝑛
𝑚𝑛𝑟𝑛2
perpendicular distance from axis
Continuous objects:
Table p. 291* No calculation of moments of inertia by integration in this course.
![Page 10: Angular quantities Rolling without slipping Rotational kinetic energy Moment of inertia Energy problems Lecture 18: Rotational kinematics and energetics.](https://reader030.fdocuments.net/reader030/viewer/2022033101/56649f475503460f94c68f03/html5/thumbnails/10.jpg)
Properties of the moment of inertia
1. Moment of inertia depends upon the axis of rotation. Different for different axes of same object:
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Properties of the moment of inertia
2. The more mass is farther from the axis of rotation, the greater the moment of inertia. Example:Hoop and solid disk of the same radius R and mass M.
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Properties of the moment of inertia
3. It does not matter where mass is along the rotation axis, only radial distance from the axis counts.
Example:
for cylinder of mass and radius : same for long cylinders and short disks
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Parallel axis theorem
𝐼𝑞= 𝐼𝑐𝑚∨¿𝑞+𝑀𝑑2
Example:
𝐼𝑞=112
𝑀 𝐿2+𝑀 ( 𝐿2 )2
¿112
𝑀𝐿2+14𝑀𝐿2=
13𝑀𝐿2
![Page 14: Angular quantities Rolling without slipping Rotational kinetic energy Moment of inertia Energy problems Lecture 18: Rotational kinematics and energetics.](https://reader030.fdocuments.net/reader030/viewer/2022033101/56649f475503460f94c68f03/html5/thumbnails/14.jpg)
Rotation and translation
𝐾=𝐾 𝑡𝑟𝑎𝑛𝑠+𝐾 𝑟𝑜𝑡=½𝑀𝑣𝐶𝑀2 +½ 𝐼𝜔2
𝑣𝐶𝑀=𝜔𝑅with
![Page 15: Angular quantities Rolling without slipping Rotational kinetic energy Moment of inertia Energy problems Lecture 18: Rotational kinematics and energetics.](https://reader030.fdocuments.net/reader030/viewer/2022033101/56649f475503460f94c68f03/html5/thumbnails/15.jpg)
Hoop-disk-race
Demo: Race of hoop and disk down incline
Example: An object of mass , radius and moment of inertia is released from rest and is rolling down incline that makes an angle θ with the horizontal. What is the speed when the object has descended a vertical distance ?