IB Physics Motion in a Vertical Circle

A sheet of harder questions on a mass moving on a string in a vertical circle is given below.

  1. A mass m hangs at rest on the end of a light inextensible string of length L. The mass is given a horizontal velocity U and the maximum vertical height to which the mass rises is L. Find U.
  2. In the previous question find the tension in the string and the magnitude of the acceleration of the mass at its highest point.
  3. A mass m is on the end of a string of length L that has its upper end fixed. The mass is released from rest and swings in a circular arc. Find an expression for the tension in the string when the string makes an angle 𝜃 with the vertical.
  4. A mass hangs at rest on the end of a light inextensible string of length L. The mass is given a horizontal speed U. In its subsequent motion the string becomes slack and the mass hits the point of supprt of the string. Find the initial speed U.