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A particle is at rest at the apex A of a smooth fixed hemisphere whose base is horizontal. The hemisphere has centre O and radius a. The particle is then displaced very slightly from rest and moves on the surface of the hemisphere. At the point P on the surface where angle AOP = ¦Á the particle has speed v. Find an expression for v in terms of a, g and ¦Á.

  • Physics -

    Use conservation of energy. At angle A (measured from vertical), the particles elevation has decreased by a (1 - cosA).

    m g (1-cosA) = (1/2) m V^2

    V = sqrt[2*(1-cosA)/g]

    This assumes that the particle remains in contact with the hemisphere. At some point it may (or may not) be fast enough to leave the surface. That is a separate problem.

  • Physics -

    According to the analysis here, the particle does leave the sphere 1/3 of the way down:

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