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Conservation of energy questions: skier and a particle in a bowl

39) A particle of mass m=30 g slides inside a bowl whose cross section has circular arcs at each side and a flat horizontal central portion between points a and b of length 20 cm. The curved side oft he bowl are frictionless, and for the flat bottom the coefficient of kinetic friction Mk-0.21. The particle is released from rest at the rim, ,which is 10 cm above the flat part of the bowl. (a) What is the speed of the particle at a and b? (b) Where does the particle final come to rest?

61) A skier skis from rest from a vertical height h1=18 m over two successively lower hills of vertical heights h2=15 m and h3=7 m. The summit of the third hill fits a circle of radius h3 centered at height 0 m. Friction with the snow and air resistance are negligible. (a) Find her speeds at x1, x2, and x3.( x1 is top of first hill, x2 is top of 2nd hill, and x3 is top of 3rd hill.) (b) Does the skier leave the surface at x3? If not, what should h1 be so that she just leaves the surface at x3?

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Two questions explained in the context of conservation of energy. The first is about a particle moving from the rim of a bowl until it stops at rest. The second is about a skier skiing over multiple hills.

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