RIGID BODY ROTATIONS!

As r sweeps out an angle θ we can define ω = dθ/dt. Similarly we can define an angular acceleration α = dω/dt. Both the angular velocity and the angular acceleration are vector quantities. For now we will just indicate the direction by a right-hand rule. Sweep the fingers of your right hand in the sense of rotation, and your thumb will point in the direction of the angular velocity. The angular acceleration vector points in the direction the angular velocity vector is changing. Angular and linear quantities are simply related: v = rω, ar = rω2 and at = rα.

ANGULAR VELOCITY ANIMATION!

For constant angular acceleration α, we have simple relations identical in form to those seen in Chapter 2:
ω(t) = ω(0) + αt.
θ(t) = θ(0) + ω(0)t + (1/2)αt2.
ω2 = ω(0)2 + 2α[θ - θ(0)].


The vector cross product. Most rotational relations can be written in cross-product form, for example v = ω × r.

The torque on a rigid body due to a force acting at a point r away from the pivot is τ = r × F.

If θ is the angle between r and F, the magnitude of the torque is of course just τ = rFsinθ. Note this can be written τ = r⊥F or τ = rF⊥.



I = ∑i miri2 = ∫ r2 dm



ROTATIONAL INERTIA!


All this is leading up to the key result, the 2nd Law for Torques: Σiτi = I α.

Rotational Kinetic Energy Kr = (1/2)Iω2

Because x = s = Rθ, for rolling motion without slipping, we will always have the very simple relation vcm = Rω. Notice a natural instantaneous pivot for the system is the point where the rolling object touches the floor.

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