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I'm stuck in the following question: One end of a light inextensible string of length $a$ is attacheded to a fixed point $A$, a distance $\frac{13a}{27}$ below a horizontal ceiling. The other end supports a particle which is projected from its lowest point with a horizontal speed $\sqrt{\lambda g a}$. Find the smallest value of $\lambda$ for which the particle will reach the ceiling before the string goes slack.

At the lowest point, I'm assuming the particle has kinetic energy $\frac{1}{2}m\lambda g a$ and potential energy of $0$. I'm not sure what to do next.

amika patel
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  • A related question is https://math.stackexchange.com/questions/2499522/describe-the-movement-of-a-string-pendulum-with-initial-speed-v-0-2-sqrtgl. If you increase $v_0$ in that question, the point at which the circular part of the path ends gets higher. You want this point to be at the height of the ceiling. – David K Apr 07 '19 at 17:22
  • You are almost there. You must convert your kinetic energy to potential energy. What is the potential energy of the particle when it has reached the ceiling? Where did that energy come from? – John Douma Apr 07 '19 at 17:27
  • For potential energy at the ceiling, would the height be 14a/27? – amika patel Apr 09 '19 at 11:21

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