Question

How to find minimum speed required for a bob to compelte circle?

Answer 1

See the diagram.

For a bob at the top position in the circle, the forces are centripetal force, weight and tension.

         T = m v² / r - m g   --- (1)

String remains tight if  T > 0.  Hence,  v² >= r g     --- (2)

Let velocity given to the bob at its lowest position in the circle = u.

    1/2 m u² = 1/2  m v² + m g (2 r)

=> u² = v² + 4 g r

=> u >= √(5 rg)        --- (3)    using (2)

Now for the Top bob tied to string of radius L2:

Velocity given at it lowest position =  v2 >= √(5 L2 g)    -- (4)

The collision of two bobs of the same mass m is elastic, and the bob tied to string of length L2, was at rest before collision, we apply conservation of linear momentum and energy. We get

     v1 = v2 >= √(5 L2 g)    --- (5)

Velocity of bob tied to string of length L1 is same as that of the other bob after collision. Then after collision, the first bob's velocity is 0.

Let the velocity of first bob at its lowest position = v0.  Apply conservation of energy:

           1/2 m v0² = 1/2 m v1² + m g (2 L1)

            vo² = v1² + 4 g L1     

                  = 5 g L2 + 4 g L1   --- (6)

    As the first bob was given just enough velocity v0 at its lower position to make a full circle,

             v0² >= 5 L1 g     using (3)

                     = 5 g L2 + 4 g L1           from (6)

Solve these two to get   ans.

Hence,  L1/ L2 = 5