a small 04kg sphere B is given a downward velocity v0 and sw

a small 0.4-kg sphere B is given a downward velocity v0 and swings freely in the vertical plane, first about O and then about the peg A after the cord comes in contact with the peg. Determine the largest allow- able velocity v0 if the tension in the cord is not to exceed 20N

Solution

For the motion at the moment the cord hits the peg, we can say that the sphere will start going about a circular motion about the peg A. The net tension in the wire at this moment will provide the needed centripetal force and also balance the weight of the sphere

Now we need to make sure that the velocity of the sphere must not be so high that the tension in the wire exceeds 20 N

We can write: T = Mg + MV²/R where R = AC and V is the speed of the sphere at the bottom point.

This V can also be written as: V² = V_o² + 2gH where H will be the height through which the sphere falls.

So, we have: T = Mg + M(V_o² + 2gH)/R

or, 20 = 0.4 x 9.81 + 0.4(V_o² + 2 x 9.81 x 0.6)/ 0.3

or, 6 = 1.1772 + 0.4 Vo² + 4.7088

or, Vo² = 0.285

or Vo = 0.534 m/s

Therefore the required speed of the sphere is 0.534 m/s