Motion of a Particle on a Sphere

In summary, the conversation discusses a problem involving a particle on top of a sphere and how to determine the angle at which it will leave the sphere when perturbed. The equations of motion were written and an angle of departure of theta = acos(2/3) was found assuming no friction. However, when friction was introduced, the results were not as expected and a numerical integrator was used to try and account for the friction force. It is suggested to double check the validity of the equations and the expected angles for both cases.
  • #1
schutte
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I have just recently been given an interesting problem to solve. It goes like this: Imagine a particle (point mass) on top of a sphere. If you perturb the particle from the very top, at what angle will it leave the sphere? Of course this is in a gravity field. Now, after writing the equations of motion, I came up with an angle of departure of theta = acos(2/3). Assuming the sphere is frictionless. However, when I assume some friction, mu, along the sphere things begin to become problematic. My equation of motion for the particle is listed in the attachment. It seems that this equation is only valid when the partilce is in motion. If you numerically integrate this equation for mu=0, the daparture angle matches exactly that of theta = acos(2/3). If you use a value of say mu=0.1, the motion plot doesn't look correct. The particle should leave at a greater angle with friction than without. I tried using an if-statement in my numerical integrator that says if the friction force is greater than the gravity force then theta_double_dot = 0, else it uses the equation listed in the attachment. This provided a greater angle than the frictionless case. In any sense, I was hoping to get someone's thoughts on this problem.
 

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  • #2
That is an interesting problem! It sounds like you have already done some good work on it, but are having some trouble with the friction component. I think it is a great idea to use an if-statement in your numerical integrator as you suggested. However, I would suggest also checking to make sure the equation of motion you are using is valid for both cases (friction and no-friction). Additionally, I think you should double check that the angles you are getting make sense and match with what you would expect from such a system (e.g. does the particle always leave at a greater angle when friction is present?). I hope this helps!
 

1. What is the motion of a particle on a sphere?

The motion of a particle on a sphere refers to the movement of a single point particle along the surface of a three-dimensional sphere. This motion can be described using principles of spherical geometry and trigonometry.

2. What factors affect the motion of a particle on a sphere?

The motion of a particle on a sphere is affected by the particle's initial position, velocity, and direction of motion, as well as the shape and size of the sphere, and the force acting on the particle.

3. How is the motion of a particle on a sphere calculated?

The motion of a particle on a sphere can be calculated using equations derived from spherical coordinates and kinematics. These equations take into account the particle's position, velocity, and acceleration at any given time.

4. What is the difference between linear and angular motion on a sphere?

Linear motion on a sphere refers to the movement of a particle along a straight line on the surface of the sphere, while angular motion refers to the rotation of the particle around the center of the sphere. Both types of motion can occur simultaneously in the motion of a particle on a sphere.

5. Can the motion of a particle on a sphere be described using Newton's laws of motion?

Yes, the motion of a particle on a sphere can be described using Newton's laws of motion, specifically the laws of motion for objects in circular motion. These laws describe the relationship between the force acting on the particle and its acceleration, and can be applied to the motion of a particle on a sphere.

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