Parallel axis theorem if MOI about an axis passing through COM of a body is known then MOI of the body about as axis passing parallel to this axis at a distance x from it is given by
 
I = I com + mx2

Where I com is MOI about an axis passing through their COM.

Perpendicular axis theorem this theorem is applicable only to the plane bodies. If X- and Y- be the axes chosen in the plane of the body and Z- axis be perpendicular to this plane, three being mutually perpendicular then

I Z = IX + Iy

Where Ix and Iy are MOI about X – and Y_ axes respectively thus.

Angular velocity (at any instant t is) ω = dθ/dt

Angular acceleration a = dω/dt

Linear velocity v = rω

Tangential acceleration a = ra

ω = ω₀ = αt

θ = ω₀t + 1/2 αt

ω₂ = ω₀2 + 2αθ

Angular momentum (L) L = r x p where P is linear momentum

|L| = p X (perpendicular distance)

|L| = Iω|

If external torque is zero then angular momentum is conserved. Angular momentum is the moment of the linear momentum.

Angular impulse j = ∫_t1^t2τ dt = L₂ – L₁

Rotational kinetic energy = 1/2 Iω₂

If a body only rotates it possesses only rotational kinetic energy. However if a body rolls on a surface then it possesses both linear kinetic energy and rotational kinetic energy.

Total KE = 1/2 mv² + 1/2 Iω

Work done W = ∫τ. d θ

Rotational power p rot. τ.ω

MOI of a parallelepiped I = m (I²+ b²/12

MOI of an elliptical disc I = m/4 (a² + b²)

MOI of a cone (right circular cone)I = 3/10 mr²

MOI of a triangular prism or equilateral triangle = ma²/6

MOI about the base of a triangular lamina is mb²/6

About hypotenuse is mp² / 6 and about perpendicular is

Ih = mb²p²/6(p² + b²)

Acceleration of a body rolling down an inclined plane

A = g sin θ/k² = g sin θ/I²

1 + k²/r²   1 + I²/mr²

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