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Question
In U. C. M (Uniform Circular Motion), prove the relation `vec v = vec w xx vec r`, where symbols have their usual meanings.
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Solution
Analytical method :
Consider a particle revolving in the anticlockwise sense along the circumference of a circle of radius r with centre O as shown.
Let
`vec omega`= angular velocity of the particle
`vec v`= linear velocity of the particle
`vec r`= radius of the particle
In the vector form, the linear dispalcement is
`vec (delta s) = vec (delta theta) times vec r`
Dividing both sides by `delta t` we get
`vec (delta s)/(delta t) = vec (delta theta)/(delta t) times vec r`
`lim_(delta t -> 0) vec (delta s)/(delta t) = lim_(delta t -> 0) vec (delta theta)/(delta t) times vec r`
`therefore vec(dS)/dt = vec(d theta)/(delta t) times vec r`
but
`vec(dS)/dt = vec v` = Linear velocity
`vec(d theta)/(delta t) = vec omega` = angular velocity
`therefore vec v = vec omega times vec r`
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