what is uniform circular motion by using proper figure,obtain equation of acceleration ac=v2/r for uniform circular motion.show that its direction is towards its centre.
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The motion of a body moving with constant speed along a circular path is called Uniform Circular Motion. Here, the speed is constant but the velocity changes. If a particle is moving in a circle, it must have some acceleration acting towards the centre which is making it move around the centre.
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Using a uniform circular motion, obtain the equation acceleration ac = v²/r for uniform circular motion
Consider a particle of mass, ‘m’ moving with a constant speed, ‘v’’ having a uniform angular velocity, ‘ω,’ around a circular path of radius, ‘r’ with center O.
Let the particle be at P in any time,
Since v = r × ω
Differentiating both side with respect to time,
dv/dt = ω dr/dt + r dω/dt
Here, dv/dt = a (resultant acceleration of the particle at P)
dω/dt = α (angular acceleration at P)
Where angular acceleration is the time rate of change in the angular velocity of an object traversing a circular path.
dr/dt = v (linear velocity at P)
Putting values in equations
a = ω x v + r x α
|a(c)| = |v x ω|
a(c) = radial or centripetal acceleration.
Both are perpendicular to each other.
|a(c)| = |v x ω| = v ωSin 90° =vω
Putting v = rω, we get,
Centripetal acceleration, a(c) = ω²r = v²/r
Therfore, the centripetal acceleration is v²/r.