Motion in a Plane — Physics STD 11 Science — Question
Maharashtra BoardEnglish MediumSTD 11 SciencePhysicsMotion in a Plane4 Marks
Question
For a particle performing uniform circular motion, derive an expression for angular speed and state its unit.
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Answer
Consider an object of mass m, moving with a uniform speed v, along a circle of radius r. Let T be the time period of revolution of the object, i.e., the time taken by the object to complete one revolution or to travel a distance of 2πr. Thus, T = 2πr/v $\therefore$ Speed,$v=\frac{\text { Distance }}{\text { Time }}=\frac{2 \pi r}{T}$ …….. (1)
During circular motion of a point object, the position vector of the object from centre of the circle is the radius vector r. Its magnitude is radius r and it is directed away from the centre to the particle, i.e., away from the centre of the circle.
As the particle performs UCM, this radius vector describes equal angles in equal intervals of time.
The angular speed gives the angle described by the radius vector.
During one complete revolution, the angle described is 2π and the time taken is period T. Hence, the angular speed ω is given as, ….[From (1)] $\omega=\frac{\text { Angle }}{\text { time }}=\frac{2 \pi}{T}=\frac{\left(\frac{2 \pi r}{T}\right)}{ r }$ …………….. [From (1)] $=\frac{ v }{ r }$
The unit of angular speed is radian/second.
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