Projectile + Vertical Circle = Double Trouble

A heavy particle is suspended by a string of length $L$ from a fixed point $O$ . The particle is given a horizontal velocity $v$ . The string slacks at some angle and particle proceeds on a projectile path. The particle proceeds such that after becoming a projectile it passes through the point of suspension.

If the value of $v^2$ can be expressed as $v^2 = gL \ \tan(x)$ , for positive acute angle $x$ then find the measure of $x$ in degrees.

Details And Assumptions :

• Take acceleration due to gravity to be constant $10 \text{ m/sec}^2$ .

• The initial motion of the particle (that is before becoming a projectile) is motion along a vertical circle (motion is in vertical plane).

• At the instant of slacking the tension in the string will become zero.

• The string is light and inextensible.

• Neglect air resistance and consider the particle to be a point mass.

This is not original.