**Motion of Object in a Vertical Circle**

Consider an object of mass m tied to the end of a string and whirled in a vertical circle of radius r as shown in the figure. As it goes from lowest point A to highest point C its velocity goes on decreasing and becomes minimum at C and maximum at A. So, in vertical motion, the speed of the object does not remain constant.

Let v

_{A}, v_{C}are the velocities of the object at A and C and T_{A}, T_{C}be the tension in the string at these points respectively. Let T be the tension on the string and v be the velocity of the object when the object is at point P such thatT - mg cosθ = mv

^{2}/ror, T = (mv

^{2}/r) + mg cosθ …....... (i)At the lowest point A, θ = 0. So, cos θ = 1. Then T

_{A}= (mv_{A}^{2}/ r) + mg .....… (ii)At the highest point C, θ = 180. So, cosθ = -1. Then T

_{C}= (mv_{C}^{2}/ r) – mg .....… (iii)At point B, θ = 90, so, cosθ = 0. Then T

_{B}= mv_{B}^{2}/ r ......… (iv)From (ii), (iii) and (iv) it is clear that the tension on the string is maximum when the object moving with speed v is at lowest point A and is minimum when the object is at highest point C i.e.

T

_{max}= T_{A}= (mv^{2}/ r) + mgT

_{min}= T_{C}= (mv^{2}/ r) – mgIf the tension at C is zero i.e. T = 0 then,

mg = mv

_{C}^{2}/ ror, v

_{C}= √ r. gThis is the maximum velocity as C to complete the circle called critical velocity. If the velocity of the object at the highest point C is less than this value, the string gets slack and the object will not complete the circle.

To calculate the minimum speed at A to complete the loop, we proceed as follows:

Total mechanical energy at A = Total mechanical energy at C

(P.E. + K.E.) at A = (P.E. + K.E.) at C

or, K.E. at A = (P.E. + K.E.) at C – P.E. at A

or, K.E. at A = (P.E. at C – P.E. at A) + K.E. at C

or, ½ mv

_{A}^{2}= mg (2r) + ½ mv_{C}^{2}or, v

_{A}^{2}= 4gr + v_{C}^{2}Since, v

_{C}^{2}= rgv

_{A}^{2}= 4gr + gr.: v

_{A}= √ 5grThis equation gives the magnitude of the velocity at the lowest point with which a body can safely go around the vertical circle of radius r.

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