2011 Semifinal Exam Part B 7 Question B2 A particle is constrained to move on the inner surface of a frictionless parabolic bowl whose cross- section has equation z=kr2.The particle begins at a height zo above the bottom of the bowl with a horizontal velocity vo along the surface of the bowl.The acceleration due to gravity is g. a.For a particular value of horizontal velocity vo,which we will name uh,the particle moves in a horizontal circle.What is vn in terms of g,zo,and/or k? b.Suppose that the initial horizontal velocity is now vo >Uh.What is the maximum height reached by the particle,in terms of vo,z0,g and/or k? c.Suppose that the particle now begins at a height zo above the bottom of the bowl with an initial velocity vo =0. i.Assuming that zo is small enough so that the motion can be approximated as simple harmonic,find the period of the motion in terms any or all of the mass of the particle m,g,20,and/or k. ii.Assuming that zo is not small,will the actual period of motion be greater than,less than,or equal to your simple harmonic approximation above?(You need not calculate the new value explicitly,but you should show some work to defend your answer.) Copyright C2011 American Association of Physics Teachers