For hydrogen like atom, find velocity from potential energy


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> PE of system is given by $$\mathrm{U(r) = \frac{-Ke²}{r^3}}$$ assume bohr model to be valid Find velocity
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For this I used $\mathrm{\frac{-PE}{2} =KE}$ and then used $$\frac{mv^2}{2} =\frac{Ke^2}{2r^3}$$ and $$mvr=\frac{nh}{2\pi}$$


However, I am getting the answer as $$v=\frac{n^3h^3}{16Ke^2\pi^3m^2}$$ but the answer given has a factor of 24 rather than 16. I don't know where I'm going wrong.




The fault in your solution is the part where you assume $\mathrm{\frac{-PE}{2} =KE}$ this is true only if the force follows [inverse square law](https://en.wikipedia.org/wiki/Inverse-square_law).


In this particular case we need to find the force first by using


$$F =\frac{-dU}{dr}$$


Which gives us


$$F =\frac{-3Ke^2}{r^4}$$


Now this force is central and provides centripetal force


$$\frac{mv^2}{r} =\frac{3Ke^2}{r^4}$$ 


This along with 
$$mvr=\frac{nh}{2\pi}$$


gives us
$$v=\frac{n^3h^3}{24Ke^2\pi^3m^2}$$


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The solution of the questions is given in steps in the figure.


[![enter image description here](https://i.stack.imgur.com/fM5fQ.jpg)](https://i.stack.imgur.com/fM5fQ.jpg)


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