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Fundamentals of Stress and Vibration 1. Mathematics for Structural mechanics
[A Practical guide for aspiring Designers / Analysts]
Let us compute the rate of change of the tangent unit vector T .

From [Fig 1.49] the velocity vector V at time t = t is inclined at an angle 90 − θ with respect
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to the negative ‘x-axis’.

[Fig 1.50: Uniform circular motion] [Fig 1.51: Velocity vector in the cartesian space]

From [Fig 1.51] the velocity vector vT is given by: vT = vsinθ −i + vcosθ j

By where, we get: T = −sinθ i + cosθ j

The angle swept by the radius vector from (t = 0) to (t = t ), that is angle ‘θ’ is given by ‘ωt ’. This
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is because, the radius vector ‘ r ’ is rotating at a constant rate of ‘ω’ radians per second. Therefore,

the tangent vector T is given by: T = sinωt −i + cosωt j

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Differentiating the tangent vector T with respect to time, we get:

dT dT
= ω cosωt −i − ω sinωt j or = −ω cosωt i + sinωt j - - - - (1.40)
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dt dt
Now, let us compute the radius unit vector r at the same instant of time (t = t )
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QP No. SSC/Q4401, Version 1.0, NSQF Level 7, Compliant with Aero and Auto Industries, Page 45