A particle falls from a height h of a ramp, and then goes around a circle, how much should the radius of that circle be worth as a function of the potential energy acquired?
I did a calculation for nothing:
By conservation of energy we have the potential energy of height equal to the centripetal Force thus $$mgh=\frac{mv^2}{R}\implies R=\frac{v^2}{gh}$$To find $v^2$ we again use conservation of energy but this time we conserve the kinetic energy $$mgh=\frac{1}{2}mv^2\implies v^2=2gh$$Thus $\boxed{R=\frac{2gh}{gh}=2}$
Wait oops, my approach is wrong. You can’t conserve energy and force (lmao I’m dumb). All you can tell from this problem is the velocity the ball will have once it goes around the circle. The radius can be anything, you can only restrict the radius if you have a period or something
Can you help me?
The particle is able to go around a circle as long as the highest point of the circle is at the same height as $h$, because at which point it has the potential energy $mgh$ and zero velocity, same as those at its initial position of the height $h$.
Thus , the radius of the circle is $\frac h2$.