A body tied to a thread is made to revolvein a horizontal plane with a definite velocity and the thread does notsnap. But when the body is made to revolve in a vertical plane, the thread snaps. How is it possible?

A body tied to a thread is made to revolvein a horizontal plane with a

1.	A body tied to a thread is made to revolvein a horizontal plane with a definite velocity and the thread does notsnap. But when the body is made to revolve in a vertical plane, the thread snaps. How is it possible?
Answer» Solution :Let the lengthof the thread be l , mass FO the body m, velocity of revolution v and TENSION in the thread T when the body is rotated in the horizontal plane [Fig.1.53]. Required centripetal force for revolution, `(mv^2)/(l sin theta) = Tsin theta.........(1) ` and mg =`T cos theta` .........(2) From equations (1) and (2) we get, `sin^2 theta +cos^2theta=(mv^2)/(lT)+((mg)/T)^2` `or, T^2=(mv^2)/(l)T+m^2g^2` `or, T^2-(mv^2)/lT+m^2g^2=0` `THEREFORE T=((mv^2)/l pmsqrt(((mv^2)/l)^2)+4m^2g^2)/2` `because Tgt0 and (mv^2)/l lt sqrt(((mv^2)/l)^2+4m^2g^2)` `T=1/2 [(mv^2)/l+sqrt(((mv^2)/l)^2+4m^2g^2)]` `or, T=1/2 [(mv^2)/l+sqrt(((mv^2)/l+2mg)^2-2(mv^2)/l2mg)]........(3)` Now, `(mv^2)/l+2mggt sqrt(((mv^2)/l+2mg)^2-2(mv^2)/l2mg)` `therefore` From equation (3) ,it can be ineferred that `Tlt1/2[(mv^2)/l+((mv^2)/l+2mg)]` `or, T lt ((mv^2)/l+mg)........(4)` Again, when TH body is revolved in thevertical plane , tension in the threadat the lowest pointof the circular path, `T^.=(mv^2)/l+mg`.......(5) `therefore` From equation (4) and (5) we get, `Tlt T^.` So, tension in the thread at the lowest point in the second case is greater than that in the first case, and HENCE, in the first casethough the thread does not snap, in the second case the thread may snap.