Rotational Motion Formula Summary

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
Ixx=Icc+ Md2

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

Perpendicular Axis Therom
Ixx+Iyy=Izz

It is valid for plane laminas only

Torque

τ=rXF

Also τ=Iα

Kinetic Energy is pure Rotating body
KE=(1/2)Iω2

Rotational Work Done

-If a force is acting on a rotating object for a tangential displacement of s = rθ (with θ being the angular displacement and r being the radius) and during which the force keeps a tangential direction and a constant magnitude of F, and with aconstant perpendicular distance r (the lever arm) to the axis of rotation, then the work done by the force is:

W=τθ

• W is positive if the torque τ and θ are of the same direction,
otherwise, it can be negative.

Power

P =dW/dt=τω


Angular Momentum

L=rXp
=rX(mv)
=m(rXv)

For a rigid body rotating about a fixed axis
L=Iω

dL/dt=τ

if τ=0 and L is constant

For rigid body having both translation motion and rotational motion

L=L1+L2

L1 is the angular momnetum of Center mass about an stationary axis
L2 is the angular momentum of the rigid body about Center of mass

Law of Conservation On Angular Momentum

If the external torque is zero on the system then Angular momentum remains contants

dL/dtext

if τext=0
then dL/dt=0


Equilibrium of a rigid body

Fnet=0 and τext=0



Angular Impulse:

∫τdt term is called angular impluse..It is basically the change in angular momentum


Pure rolling motion of sphere/cylinder/disc

-Relative velocity of the point of contact between the body and platform is zero
-Friction is responsible for pure rolling motion
-Friction is non disipative in nature

E = (1/2)mvcm2+(1/2)Iω2+mgh

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