**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) = Δ

**ω**/Δt

Instantanous Angular Acceleration

α=d

**ω**/dt

**Kinematics of rotational Motion**

**ω**=

**ω**+

_{0}**α**t

**θ**=

**ω**t+1/2

_{0}**α**t

^{2}

**ω**.

**ω**=

**ω**.

_{0}**ω**+ 2

_{0}**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**=

**ω**X

**r**

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

**a**=

**α**X

**r**

**Moment of Inertia**

Rotational Inertia (Moment of Inertia) about a Fixed Axis

For a group of particles,

I =

**∑**mr

^{2}

For a continuous body,

I =

**∫**r

^{2}dm

For a body of uniform density

I = ρ∫r

^{2}dV

**Parallel Axis Therom**

I

_{xx}=I

_{cc}+ Md

^{2}

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

**Torque**

**τ**=

**r**X

**F**

Also τ=Iα

**Kinetic Energy is pure Rotating body**

KE=(1/2)Iω

^{2}

**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**=

**r**X

**p**

=

**r**X(m

**v**)

=m(

**r**X

**v**)

**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**=

**L**+

_{1}**L**

_{2}**L**is the angular momnetum of Center mass about an stationary axis

_{1}L

_{2}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

d

**L**/dt

**=τ**

_{ext}if

**τ**=0

_{ext}then d

**L**/dt=0

**Equilibrium of a rigid body**

**F**=0 and

_{net}**τ**=0

_{ext}**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)mv

_{cm}

^{2}+(1/2)Iω

^{2}+mgh

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