Post on 13-Mar-2020
β’ Angular quantities
β’ Rolling without slipping
β’ Rotational kinetic energy
β’ Moment of inertia
β’ Energy problems
Lecture 20:
Rotational kinematics and energetics
Angle measurement in radians
π(in radians) = π /π
s is an arc of a circle of radius π
Complete circle:
π = 2ππ , π=2π
Angular Kinematic Vectors
Position: ΞΈ
Angular displacement:
ΞΞΈ = ΞΈ2 β ΞΈ1
Angular velocity:
Οπ§ =πΞΈ
ππ‘
Angular velocity vector
perpendicular to the plane of
rotation
Angular acceleration
Ξ±π§ =πΟπ§ππ‘
Angular kinematics
For constant angular acceleration:
π = π0 + π0π§π‘ +1
2πΌπ§π‘2
ππ§ = π0π§ + πΌπ§π‘
ππ§2 = π0π§
2 + 2 πΌπ§ π β π0
π₯ = π₯0 + π£0π₯π‘ + Β½ππ₯π‘2
π£π₯ = π£0π₯ + ππ₯π‘
π£π₯2 = π£0π₯
2 + 2ππ₯(π₯ β π₯0)
Compare constant ππ₯:
Relationship between angular and linear motion
ππππ =π£2
π= π2 π
Angular speed π is the same for all points of a rigid
rotating body!
π£ =ππ
ππ‘=π
ππ‘ππ = π
ππ
ππ‘
linear velocity tangent to circular path:
π£ = ππ
π distance of particle
from rotational axis
ππ‘ππ = Ξ±π
Rolling without slipping
Translation of CMRotation about CM
If π£πππ = π£πΆπ: π£πππ‘π‘ππ = 0 no slipping
π£πΆπ = ππ
Rolling w/o slipping
Rotational kinetic energy
πΎπππ‘ππ‘πππ = βΒ½πππ£π2 = βΒ½ππ (Οπππ)
2
= βΒ½ππ (Ο ππ)2 = Β½ [ βππππ
2] π2
πΌ =
π
ππππ2
Moment of inertia
πΎπππ‘ = Β½ πΌπ2
with
Moment of inertia
πΌ =
π
ππππ2
ππ perpendicular distance from axis
Continuous objects:
πΌ =
π
ππππ2βΆ π2ππ
Table p. 291* No calculation of moments of inertia
by integration in this course.
Properties of the moment of inertia
1. Moment of inertia πΌ depends upon the axis of
rotation. Different πΌ 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 ππ from the axis
counts.
Example:
πΌ for cylinder of mass π and radius π : πΌ = Β½ππ 2
same for long cylinders and short disks
Parallel axis theorem
πΌπ = πΌππ||π +ππ2
Example:
πΌπ =1
12ππΏ2 +π
πΏ
2
2
=1
12ππΏ2 +
1
4ππΏ2 =
1
3ππΏ2
Rotation and translation
πΎ = πΎπ‘ππππ + πΎπππ‘ = Β½ππ£πΆπ2 + Β½ πΌπ2
π£πΆπ = ππ with
Hoop-disk-race
Demo: Race of hoop and disk down incline
Example: An object of mass π, radius π and
moment of inertia πΌ is released from rest and is
rolling down incline that makes an angle ΞΈ with the
horizontal. What is the speed when the object has
descended a vertical distance π»?