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Angular quantities Rolling without slipping Rotational kinetic energy Moment of inertia Energy problems Lecture 18: Rotational kinematics and energetics

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Page 1: Angular quantities Rolling without slipping Rotational kinetic energy Moment of inertia Energy problems Lecture 18: Rotational kinematics and energetics

• 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

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

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

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

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

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

Rolling without slipping

Translation of CMRotation about CM

If no slipping

𝑣𝐶𝑀=𝜔𝑅

Rolling w/o slipping

Page 8: Angular quantities Rolling without slipping Rotational kinetic energy Moment of inertia Energy problems Lecture 18: Rotational kinematics and energetics

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

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

Properties of the moment of inertia

1. Moment of inertia depends upon the axis of rotation. Different for different axes of same object:

Page 11: Angular quantities Rolling without slipping Rotational kinetic energy Moment of inertia Energy problems Lecture 18: Rotational kinematics and energetics

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.

Page 12: Angular quantities Rolling without slipping Rotational kinetic energy Moment of inertia Energy problems Lecture 18: Rotational kinematics and energetics

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

Page 13: Angular quantities Rolling without slipping Rotational kinetic energy Moment of inertia Energy problems Lecture 18: Rotational kinematics and energetics

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

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

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 ?