Centripetal Acceleration Formula

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Centripetal acceleration is experienced by a body moving around a circular path. It is always directed towards the center of the circle. Equation of centripetal acceleration (ac) is given as ac = v2/ r
Formula for calculating centripetal acceleration can be derived as follows. 

Fig. 1

X and Y are the two points in the circular path where the velocities are ‘v1’ and ‘v2’ respectively. The radius

as shown above is ‘r’. ‘dr’ is the change in positions. Consider here the motion is uniform i.e. velocities are

constant. From the definition of acceleration we can write a = (v2 – v1)/ (t2 – t1) ---------- (1)

It appears that the above expression will result zero. But, due to different direction the expression 1 will result

some value. We can write equation 1 as

a = dv/dt ------- (2)

Representing the velocity vectors by vector diagram we get. 



Example1: A stone is rotating with a velocity of 2 rad/s around a radius of 1m. What is the centripetal acceleration of the stone?

Solution: Angular velocity of the stone is given 2 rev/s. Linear velocity ‘v’ is expressed in terms of angular velocity as 


Where

ω is the angular velocity and r is the radius of the circle. Centripetal acceleration in terms of angular velocity is written as

ac = ω2r

Placing the values in the above equation

ac = 22/1

ac = 4 m/s2

Example: What is the centripetal acceleration experienced by a car that is taking a turn of diameter 6m with a velocity of 25 m/s?

Solution: Centripetal acceleration is expressed in terms of velocity as

ac = v2/r -------------(1)

v = 25 m/s

Radius of the turn is half of the diameter value given in the question.

r = d/2

r = 6/2

r = 3m

Placing the values in equation 1

ac = 252 /3

ac = 208.33 m/s2


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