Relating the linear and angular variables

We often need to relate the linear variables s, v, and a for a particular point in a rotating body to the angular variables θ, ω, and α for that body. The two sets of variables are related by r, the perpendicular distance of the point from the rotation axis. This perpendicular distance is the distance between the point and the rotation axis, measured along a perpendicular to the axis. It is also the radius r of the circle traveled by the point around the axis of rotation. [Derive the relationship between linear motion variables and angular motion variables.]

A point in a rigid rotating body, at a perpendicular distance r from the rotation axis, moves in a circle with radius r. If the body rotates through an angle θ, the point moves along an arc with length s given by:
s = θr (where θ is in radians)

The linear velocity v of the point is tangent to the circle; the point’s linear speed v is given by:
v = ωr, where ω is the angular speed (in radians per second) of the body, and thus also the point.

The linear acceleration a of the point has both tangential and radial components.

The tangential component of the linear acceleration is:
at = αr, where α is the magnitude of the angular acceleration (in radians per second-squared) of the body.

The radial component of the linear acceleration is:
ar=v2/r = ω2r

If the point moves in a uniform circular motion, the time period T of the motion for the point and the body is:
T = 2π/ω = 2πr/v

Relating the linear and angular variables
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