A round marble with a given mass m and given radius r is placed inside a smooth

Question

A round marble with a given mass m and given radius r is placed inside a smooth spherical bowl as shown. The radius of curvature for the inside of the bowl is given as R and the marble is released at a position corresponding to a given angle ?0 relative to the the vertical radius line as shown. Assume r << R. Assume that theta0 is small enough so that the marble rolls without slipping. a) Draw a Free Body Diagram indicating the forces on the marble at the point it is released. b) Draw an Extended Free Body Diagram indicating the forces on the marble at the point it is released. c) Calculate the magnitude and direction of the torque about the center of the marble that is applied to the marble at the point it is released. Give you answer in terms of the given parameters. Explain your work. d) Calculate the magnitude of the translational acceleration of the marble at the point it is released? Give you answer in terms of the given parameters. Explain your work. e) Use Conservation of Energy to determine the linear speed of the marble when it is at the bottom of the bowl. Don

Explanation / Answer

force at marble will be mgcos(theta) perpendicluar to point of contact with bowl, mgsin(theta) acting tangential direction from center point of marble, mv^2/r along the center of bowl, so normal force will be: N - mgcos(theta) = mv^2/r at the center theta will be zero so N = mv^2/r + mg, mgsin(theta) = ma, so a = gsin(theta), and at center of bowl theta zero so a = g, total energy = .5mv^2 + 2/3 *m*r^2 * a/r for marble: so from conservation of energy: mg*(r - rcos(theta)) = .5mv^2 + 2/3 *m*r^2 * a/r from here get v linear velocity at center use a=g.

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