Linear and Angular motion – relationship equations

Last updated on May 24th, 2023 at 12:12 pm

In this post, we will list down selected equations that can be described as the relationship 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.]

Equations that show the relation between Linear motion 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 a 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 linear acceleration is:
a_t = 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:
a_r=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