Launching a particle at the highest point inside a sphere

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SUMMARY

A particle projected horizontally at the highest point inside a fixed smooth sphere with radius 'a' and speed (4ag/5)^0.5 will lose contact with the sphere at an angle theta, where theta is cos^-1(4/5). This behavior mirrors that of a particle launched from the highest point outside the sphere. The particle does not complete a full circle inside the sphere due to the dynamics of the normal force and gravitational acceleration acting on it. If the initial speed allows for contact outside, it will result in immediate loss of contact when launched from within.

PREREQUISITES
  • Understanding of classical mechanics principles, particularly projectile motion.
  • Familiarity with concepts of normal force and gravitational forces.
  • Knowledge of angular motion and circular dynamics.
  • Basic proficiency in trigonometric functions, specifically inverse cosine.
NEXT STEPS
  • Study the dynamics of particles in circular motion using Newton's laws.
  • Explore the implications of normal force in varying gravitational contexts.
  • Learn about the equations of motion for particles in constrained systems.
  • Investigate the effects of initial velocity on particle trajectories in spherical coordinates.
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Physics students, mechanical engineers, and anyone interested in the dynamics of motion within constrained environments, particularly in spherical geometries.

LouysHong
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If a smooth sphere with radius a is fixed on a plane, and a particle is projected horizontally at the highest point outside/on of the sphere with speed (4ag/5)^0.5, I know that the particle will lose contact with the sphere when it makes an angle of theta with the upward vertical, where theta is cos^-1(4/5).

But what if I were to launch the particle with that speed horizontally at the highest point inside the sphere ? It seems that the particle would also lose contact at the same point ? But wouldn't the particle complete a full circle inside the sphere ?
 
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LouysHong said:
It seems that the particle would also lose contact at the same point ?
Why? Which way is the normal force pointing now? What equation do you get?
 
LouysHong said:
It seems that the particle would also lose contact at the same point ?
If the initial speed is such that it stays in contact outside, then it would immediately loose contact inside.
 

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