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A particle moves so that its position v...

A particle moves so that its position vertor is given by `overset rarr(r ) = cos omegat hatx + sin omegat haty`, where `omega` is a constant. Which of the following is true?

A

velocity and acceleration both are perpendicular to `overset rarr(r )`

B

velocity and acceleration both are parallel to `overset rarr(r )`

C

velocity is perpendicular to `overset rarr(r )` and acceleeration is directed towards the origin

D

velocity is perpendicular to `overset rarr(r )` and acceleeration is directed away from the origin

Text Solution

Verified by Experts

The correct Answer is:
C
Here, `overset rarr(r ) = cos omega t hatx + sin omega t haty` ..(i)
velocity, `overset rarr (upsilon) = (overset rarr(dr))/(dt) = - omega sin omegat hatx + omega cos omegat haty`..(ii)
Acceleration, `overset rarr(a) = - omega^(2) cos omegat hatx - omega^(2) sin omega t haty`
`= - omega^(2) overset rarr(r )` ..(iii)
From (iii), we note that acceleration is directed towards the origin because `overset rarr(a)` is in opposite direction to `overset rarr(r )`.
Here, `overset rarr(upsilon). overset rarr(r ) = (- omega sin omega t hatx + omega cos omegat haty)`.
`(cos omegat hatx + sin omegat haty)`
`= - omega sin omegat cos omegat + omega sin omegat cos omegat = 0`
or `upsilon r cos theta = 0 or cos theta = 0 or theta = 90^(@)`
i.e. `overset rarr(upsilon) and overset rarr(r )` are perpendicular to each other.
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