What is the angle at which a ball falls off a hemisphere?

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SUMMARY

The angle at which a ball falls off a hemisphere can be determined by analyzing the forces acting on the ball as it slides down a frictionless surface. The key equations involved include the gravitational force components, mgsin(theta) and mgcos(theta), and the centripetal force equation, mv^2/r. The ball will detach from the hemisphere when the normal force becomes zero, which can be calculated using energy conservation principles or by applying calculus to find the critical angle theta.

PREREQUISITES
  • Understanding of gravitational force components (mgsin(theta) and mgcos(theta))
  • Familiarity with centripetal force concepts (mv^2/r)
  • Basic knowledge of energy conservation principles
  • Ability to apply calculus to physics problems (optional for alternative solutions)
NEXT STEPS
  • Study the derivation of the centripetal force equation (mv^2/r)
  • Learn about energy conservation in mechanical systems
  • Explore calculus applications in physics for finding critical points
  • Investigate the dynamics of motion on curved surfaces
USEFUL FOR

Students studying physics, particularly those focusing on mechanics and dynamics, as well as educators seeking to explain the principles of motion on curved surfaces.

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Homework Statement


A ball with initial velocity 0 and mass m starts at the top of a hemisphere and begins to slid down the frictionless side. The radius of the hemisphere is r. At what angle does the ball fall off the hemisphere?

Homework Equations


mgsintheta, mgcostheta, fn
mv^2/r

The Attempt at a Solution


I guess that you can split mg into mgsin theta and mg cos theta and that one of them would create centripetal force. Then you can find where the centripetal force is 0 because there would be no normal force when the ball falls off? I'm not sure. Also, my teacher mentioned that there is a way to solve this with and without calculus.
 
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Find the place (or time) at which the normal force becomes zero, if you constrain the ball is constrained to remain on the surface of the hemisphere.
 
you can use energy to find theta, solve v then substitute into your force equation
 

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