Angular quantities Rolling without slipping Rotational kinetic energy Moment of inertia Energy...

• 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: , =

Angular Kinematic Vectors

Position:

Angular displacement:

Angular velocity:

Angular velocity vector perpendicular to the plane of rotation

Angular acceleration

α 𝑧=𝑑ω𝑧

𝑑𝑡

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 :

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

Rolling without slipping

Translation of CMRotation about CM

If no slipping

𝑣𝐶𝑀=𝜔𝑅

Rolling w/o slipping

Rotational kinetic energy

𝐼=∑𝑛

𝑚𝑛𝑟𝑛2

Moment of inertia

𝐾 𝑟𝑜𝑡=½ 𝐼 𝜔2

with

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.

Properties of the moment of inertia

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

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.

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

Parallel axis theorem

𝐼𝑞= 𝐼𝑐𝑚∨¿𝑞+𝑀𝑑2

Example:

𝐼𝑞=112

𝑀 𝐿2+𝑀 ( 𝐿2 )2

¿112

𝑀𝐿2+14𝑀𝐿2=

13𝑀𝐿2

Rotation and translation

𝐾=𝐾 𝑡𝑟𝑎𝑛𝑠+𝐾 𝑟𝑜𝑡=½𝑀𝑣𝐶𝑀2 +½ 𝐼𝜔2

𝑣𝐶𝑀=𝜔𝑅with

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 ?