Particle on top of a half circle

You need to use conservation of energy to determine the minimum initial velocity needed for the particle to clear the half-circle without rolling down on one of its sides. Specifically, you need to equate the initial potential energy of the particle at the top of the half-circle with the kinetic energy it has at the bottom of the half-circle. This will allow you to find the minimum initial velocity v0 needed.In summary, the problem is to find the minimum initial velocity v0 needed for a point-particle at rest at the top of a half-circle with radius R to clear the half-circle without rolling down on one of its sides. Conservation of energy can be used to equate the initial potential energy at the top with the kinetic energy at the bottom
  • #1
gralla55
57
0

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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  • #2
... 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.
 

FAQ: Particle on top of a half circle

What is a "particle on top of a half circle"?

A particle on top of a half circle refers to a theoretical scenario in which a small object or particle is placed on the top of a half circle-shaped surface. This is often used as a simplified model to study the behavior of particles in curved surfaces or potential energy landscapes.

How is the motion of the particle affected by the half circle surface?

The motion of the particle on top of a half circle is affected by the shape of the surface, as well as external forces such as gravity or friction. The curved surface creates a potential energy well that the particle must navigate, leading to specific patterns of motion and equilibrium positions.

What factors influence the stability of the particle on top of a half circle?

The stability of the particle on top of a half circle depends on various factors, including the shape and curvature of the half circle, the mass and size of the particle, and external forces acting on the particle. For example, a larger particle may be more stable on a larger half circle surface due to a larger area of contact.

How does the position of the particle on top of a half circle affect its potential energy?

The position of the particle on top of a half circle directly affects its potential energy. The higher the particle is positioned on the half circle, the greater its potential energy will be. As the particle moves towards the bottom of the half circle, its potential energy decreases, reaching a minimum at the bottom before increasing again as it moves towards the top.

What real-world applications does the concept of a particle on top of a half circle have?

The concept of a particle on top of a half circle has various real-world applications, including understanding the motion of particles on curved surfaces, studying the behavior of particles in potential energy landscapes, and designing structures or machines that involve curved surfaces. It is also used in fields such as physics, engineering, and biology to model and analyze various systems and phenomena.

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