# Law of Motion XVII

Vertical Circular Motion
Whenever the plane of circular path of body is vertical its motion is said to be vertical circular motion.

Vertical Circular Motion of a Body Tied to a String
Consider a body of mass m tied to a string and performing vertical circular motion on a circular path of radius r. At the topmost point A of the body weight of the body mg and tension Tboth are acting in the vertically downward direction towards the center of the circular path and they together provide centripetal force.
TA + mg = mvA2 / r

Critical velocity at the top most point
As we go on decreasing the vA, tension Talso goes on decreasing and in the critical condition when vA is minimum tension TA = 0. The minimum value of vin this case is known as critical velocity TA(Critical) at the point A. From above
If the velocity at point A is less than this critical velocity then the string will slag and the body in spite of moving on a circular path will tend to fall under gravity.

Critical velocity at the lower most point
Taking B as reference level of gravitational potential energy and applying energy conservation
This is the minimum possible velocity at the lower most point for vertical circular motion known as critical velocity at point B.
vB(Critical) = √5gr

Tension at lowermost point in critical condition
For lowermost point B net force towards the center is centripetal force. Tension TB acts towards the center of the circular path whereas weight mg acts away from it.
Hence,
TB – mg = mvB2 / r
Putting,
vB = √5rg
TB – mg = m5gr / r
or, TB = 6mg

Hence in critical condition of vertical circular motion of a body tied to a string velocities at topmost and lowermost be √(rg) and √(5rg) respectively and tensions in the strings be 0 and 6mg respectively.

General Condition for Slagging of String in Vertical Circular Motion

For the body performing vertical circular motion tied to a string, slagging of string occurs in the upper half of the vertical circle. If at any instant string makes angle θ with vertical then applying net force towards center is equal to centripetal force, we have

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