Rotational kinematics



Rotational kinematics

Rotational kinematics involves six definitions, here we go again.

● time, t

● time interval, Δt ( tf - ti

● angular position, θ

● angular displacement, Δθ ( θf - θi

● angular velocity, ω ( dθ/dt

● angular acceleration, α ( dω/dt

Note: in going from linear to angular, often we add the adjective angular.

We will measure angles in radians. Recall an angle θ is given by the arc length it subtends divided by the radius of the circle,

θ( s/r.

This definition leads to the following table:

θ(in degrees) s θ(in radians)

360 2πr 2π

180 πr π

90 ½ πr ½ π

60 1/3 πr 1/3 π

45 1/4 πr ¼ π

30 1/6 πr 1/6 π

Definition: Rigid Body

A set of particles for which rij = cij = constant, for every pair of particles in the body, where rij is the distance between any two particles, i and j of the rigid body.

In our work we will concentrate on the rotation of a rigid body about a fixed axis of rotation. In this case, every particle of the body is executing circular motion with its own radius ri( . The following hold for the ith particle of the rigid body,

Δsi = ri( Δθ

vi = ri( ω (velocity)

at = ri( α = dvi/dt (tangential acceleration)

ac = vi2/ri( = ri( ω2 (radial or centripetal acceleration).

CAUTION: ri( is NOT the magnitude of the position vector of the ith particle, but is the radius of its circular motion. See Figure

The motivation for this chapter is based on the fact that certain quantities (ω and α) describe the body as a whole and not just the individual particles of the body.

Special Case of Motion: Constant α

θ(t) = θo + ωot + ½ αt2 and ω(t) = ωo + αt .

Can you guess the rotational “shortcut” equation?

EXAMPLES(in class)

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