IIT JEE , AIEEE Forum - Physics , Mechanics

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discussing a little theory

Consider the figure ...

In this fig. we will find out what should be the velocity of the bob at the lowest position so that the string becomes slack ( i,e, T = 0 ) at the horizontal diameter..

Now the weight acts perpendicular to the tension at this position and so has no component along the string....

So taking v =velocity at that point and T = tension and l = length of string....

we have T = mv²/l..

Now let u = velocity of the bob at lowest position so that the bob acquires velocity = v at the horizontal diameter....

So ... 1/2.mu² = 1/2mv² + mgl

or v² = u²- 2gl

Using this in equ^n of T.... we have

T = m(u² - 2gl)/l

If string becomes slack.... T = 0

i,e, m(u²- 2gl)/l = 0

or u² - 2gl = 0

or u =

2gl

So we see that if velocity at lowest position should be equal to

2gl such that the string becomes slack at the horizontal diameter...

FIGURE 1

And see HCV-1 page 128 , example 8 where you will find that the velocity at lowest position such that the bob complets a full circle is 5gl .

So we conclude that....

if velocity is less than 2gl, then string becomes slack below the horizontal diameter.

if velocity = 2gl, then string becomes slack at horizontal diameter..

if velocity is greater than 2gl but less than 5gl, then string becomes slack in the region between the horizontal diameter and the topmost position....

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