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.

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