IIT PHYSICS:Rotational Short notes -I

Angular Displacement

-When a rigid body rotates about a fixed axis, the angular displacement is the angle Δθ swept out by a line passing through any point on the body and intersecting the axis of rotation perpendicularly
-Can be positive (counterclockwise) or negative (clockwise).
-Analogous to a component of the displacement vector.
-SI unit: radian (rad). Other
units: degree, revolution.

Angular Velocity

Average angular velocity, is defined by
$ = (angular displacement)/(elapsed time) = Δθ/Δt .

Instantanous Angular Velocity ω=dθ/dt


Some points
-Angular velocity can be positive or negative.
-It is a vector quantity and direction is perpendicular to the plane of rotation
-Angular velcity of a particle is diffrent about diffrent points
-Angular velocity of all the particles of a rigid body is same about a point

Angular Acceleration:

Average angular acceleration, is defined by
= (change in angular velocity)/(elapsed time) = Δω/Δt

Instantanous Angular Acceleration
α=dω/dt

Kinematics of rotational Motion
ω=ω₀ + αt
θ=ω₀t+1/2αt²
ω.ω=ω₀.ω₀ + 2 α.θ;

Also
α=dω/dt=ωdω/dθ


Vector Nature of Angular Variables
-The direction of an angular variable vector is along the axis.
- positive direction defined by the right hand rule.
- Usually we will stay with a fixed axis and thus can work in the scalar form.
-angular displacement cannot be added like vectors
-angular velocity and acceleration are vectors


Relation Between Linear and angular variables

v=ωXr

Where r is vector joining the location of the particle and point about which angular velocity is being computed

a=αXr


Moment of Inertia

Rotational Inertia (Moment of Inertia) about a Fixed Axis

For a group of particles,

I = ∑mr²

For a continuous body,

I = ∫r²dm

For a body of uniform density

I = ρ∫r²dV

Parallel Axis Therom
I_xx=I_cc+ Md²

Where I_cc is the moment of inertia about the center of mass

Perpendicular Axis Therom
I_xx+I_yy=I_zz

It is valid for plane laminas only