an object oscillates back and forth at the bottom of africtionless hemispherical

Question

an object oscillates back and forth at the bottom of africtionless hemispherical bowl, the radius of bowl is R adn theangle is small enough that the object oscillates in simpleharmonic motion.   Derive an expression for the angularfrequency w of the motion, express answer in terms of Rand g, the accel due to gravity. sounds easy, but I can't solve it an object oscillates back and forth at the bottom of africtionless hemispherical bowl, the radius of bowl is R adn theangle is small enough that the object oscillates in simpleharmonic motion.   Derive an expression for the angularfrequency w of the motion, express answer in terms of Rand g, the accel due to gravity. sounds easy, but I can't solve it

Explanation / Answer

The idea is that the object rolls back and forth a tiny bit sothat the force acting tangent to the circle is: .            F = - mgsin       where is the angle shown below. . (hold on... I'm having trouble adding the picture...) . cramster wont let me add a diagram... keeps giving me anerror. So here are the steps... .     We can alsowrite     x = Rsin       and the motion ofthe object is approximately horizontal, so that the acceleration isin the x direction. Then... .      ma = - mgsin              a = - g x /R           a +   (g/R) x = 0 . You might recognize this from the spring and mass oscillator.In that case, you had    a + (k/m) x = 0   and you had... .             = k/m . In this case, with the object in the hemisphere, you haveexactly the same equation but with g/R in place ofk/m. . So for yourproblem,    = g/R

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