A ball of mass 100 grams is hanging from a fixed point by a string 2.5 metres long. It is struck with a bat so that it starts to describe a circle in a vertical plane. When the ball reaches a height of 4 metres the string becomes slack. Find the speed of the ball immediately after it was struck and the tension in the string at that instant.
PS - I know how to find the tension at that instant, but finding the value for the speed is quite a challenge for me. I have used energy conservation principle to equate 0.5 mu^2 to (m x g x 4 + 0.5 x m x v^2) - but how do I find the speed of the ball at this height? If there's any other way anyone suggests I could do this question, please ...
At any instant while the ball is in its circular "orbit" around the fixed point, the force of gravity on the ball can be broken down into two perpendicular components, one component in the direction of travel (tangent to the circle) and one perpendicular to the direction of travel (along the radius of the circle).
When the ball is in the lower half of the circle, the radial component of gravity pulls the ball away from the fixed point. The tension in the string has to be enough so that after canceling the radial component of gravity, there is just enough force to keep the ball traveling along the circle. But in the upper half of the circle, the radial component of gravity pulls the ball toward the fixed point. When the radial component of gravity alone is sufficient to bend the path of the ball along the circular arc, the tension in the string is zero and the string goes slack.
After that point the centripetal force is greater than required and the ball will travel inside the circle, but we're not asked to concern ourselves with that.
You can work out the radial direction at the height of $4$ meters, from which you can determine the force of gravity in that direction. You should then be able to work backward to get velocity at that instant.