Particle on top of a half circle

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SUMMARY

The discussion focuses on determining the minimum initial velocity \( v_0 \) required for a point-particle to clear a half-circle of radius \( R \) without rolling down its sides. The participant suggests that the solution should be expressed as a function of \( R \) and gravitational acceleration \( g \). They correctly identify that the particle's motion will follow a parabolic trajectory rather than a circular one, which complicates the calculations for \( v_0 \). The consensus is that further analysis of projectile motion principles is necessary to derive the required initial velocity.

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Homework Statement



A point-particle sits at rest at the top of a half-circle with radius R. Find the minimum initial velocity v0 the particle has to have in order to clear the half circle without rolling down on one of its sides.

Homework Equations



None are given.

The Attempt at a Solution



Since the radius is given as a variable, I'm guessing it should be possible to write the solution as a function of R and g. The only thing I've managed to write down is:

v0x = ?
v0y = 0
ax=0
ay=g

Any help on how to proceed? I have no idea how to make sure the particle clears the circle... I thought about solving for the initial velocity needed to make sure the particle travels farther than R before hitting the ground, but the path will be a parabola and not a circular path, so again I'm not sure. Thanks!
 
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... I thought about solving for the initial velocity needed to make sure the particle travels farther than R before hitting the ground...

I think you are on the right track.
 

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