Relation between Linear Velocity and Angular Velocity
Suppose a particle of mass m moving in a circular path of radius r with constant speed v. let θ be the angular displacement when particle goes from points P to Q in time t as shown in the figure. If s is the arc of length PQ, then θ = s/r
or, s = r. θ
Differentiating both sides with respect to time, we have
ds/dt = r. dθ/dt
As r is constant, v = ds/dt, and angular velocity, ω = dθ/dt
Then, above equation can be written as
V = r. ω
This is the relation between linear velocity and angular velocity.
Since linear acceleration is given by a = dv/dt
And angular acceleration is given by α = dω/dt
Differentiating both sides w.r.t time in V = r. ω, we get
a = r. α
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