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Show that in angular form, centripetal acceleration is:

ANSWER

Consider a body moving in a circular path. Let at any instant of time t1 it is at a point A on the circumference of the circle. Suppose it moves with a velocity  and at time t2, it is positioned at point B on the circle. The motion is an accelerated one due to its changing direction and let’s further suppose its velocity to be  at B. Let the change in time is  Δt = t2-t1 and change in velocity is  . If Δt is very small, point B and point A will be very close to one another and we can say,

S = vΔt

Similarly, we know that ch5para6

(See question 1, centripetal acceleration).

So, substituting for the value of ‘S’, in the above equation, we got ch5para7 Since , therefore . Considering in vector form, centripetal acceleration3                                                                                                              …       (1) The negative sign is used because the centripetal acceleration is directed towards the center of the circle whereas radius of the circle is directed outward. Therefore, both vaca  and vec r  are opposite in direction to one another. In addition, we know that,

v = rω       …      (2)

So to substitute for v in equation (1) ch5para8 This proves the result. 

4 Comments

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