Rotational dynamic

Equation of angular motion

 The equations of angular motion describe the motion of an object rotating around an axis. 

The equations of linear motion describe the motion of an object in a straight line.

The equations of angular motion are:

1. θ = ω₀t + ½αt² 

 (where θ is the angular displacement, ω₀ is the initial angular velocity, α is the angular acceleration, and t is the time)

2. ω = ω₀ + αt 

 (where ω is the angular velocity)

3. ω² = ω₀² + 2αθ 

 (where θ is the angular displacement)

4. θ = (ω + ω₀)t/2 

 (where ω and ω₀ are the final and initial angular velocities, respectively, and t is the time)

The equations of linear motion are:

1. S= ut + ½at² 

 (where x is the displacement, x₀ is the initial displacement, v is the initial velocity, a is the acceleration, and t is the time)

2. v = u+ at 

 (where v is the velocity)

3. v² = u² + 2as 

(where x is the displacement)

4. S = (u + u₀)t/2 

 (where v and v₀ are the final and initial velocities, respectively, and t is the time)

Relation between torque and moment of inertia 

Let us consider a body of mass rotating in a circle of radius "r" with

angular velocity ω 

We know 

Torque produce τ.

τ=Fr

τ=mar

But again we know,

Angular acceleration α

α=dω/dt

Or, α=d(v/r)/dt

Or, α= (1/r)dv/dt

Or, α=a/r

i.e a=αr............(I)

Then equation (I) become

τ=mar

τ=mαr.r = mr^2α

i.e τ=Iα