A solid semi-cylinder of radius r rests under the action of gravity on a hard fixed...

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A solid semi-cylinder of radius r rests under the action of gravity on a hard fixed cylindrical surface of radius R as shown. If rolled slightly to the left or the right and released, the semi-cylinder will rock from side to side (assume no slipping). (a) From first principles, show that the location of the centroid of mass of the solid semi- cylinder lies on the axis of symmetry, from the flat surface. 4r [7%] 3.7 (b) Calculate the moment of inertia of the semi-cylinder about an axis through the mass centre and perpendicular to the page. (You may start with the knowledge that the moment of inertia of a solid cylinder, about its axis, istmr, making use of the parallel axis theorem. Be careful with the meaning of m!) [7%] (c) Using a full A4 sheet, draw accurately and to scale a diagram of the system in its equilibrium position and, superposed on it, when deflected by an angle. Label all points. [4%] (d) Write an expression for the kinetic energy of the semi-cylinder at some instant during the small amplitude oscillation. [15%] (e) At the same instant as used in part (d), write an expression for the gravitational potential energy of the semi-cylinder. [12%] (1) Assuming no damping, apply the principle of conservation of energy to the system to derive the governing equation of motion. [Energy]=0 [9%] (g) Assuming small oscillation amplitude, linearise the differential equation obtained in part (f). Then determine the natural frequency of oscillation of the rocking motion. [15%] 3.7 (h) In light of your answer to part (g), what is the significance of the ratio -? [6%] R Fixed A solid semi-cylinder of radius r rests under the action of gravity on a hard fixed cylindrical surface of radius R as shown. If rolled slightly to the left or the right and released, the semi-cylinder will rock from side to side (assume no slipping). (a) From first principles, show that the location of the centroid of mass of the solid semi- cylinder lies on the axis of symmetry, from the flat surface. 4r [7%] 3.7 (b) Calculate the moment of inertia of the semi-cylinder about an axis through the mass centre and perpendicular to the page. (You may start with the knowledge that the moment of inertia of a solid cylinder, about its axis, istmr, making use of the parallel axis theorem. Be careful with the meaning of m!) [7%] (c) Using a full A4 sheet, draw accurately and to scale a diagram of the system in its equilibrium position and, superposed on it, when deflected by an angle. Label all points. [4%] (d) Write an expression for the kinetic energy of the semi-cylinder at some instant during the small amplitude oscillation. [15%] (e) At the same instant as used in part (d), write an expression for the gravitational potential energy of the semi-cylinder. [12%] (1) Assuming no damping, apply the principle of conservation of energy to the system to derive the governing equation of motion. [Energy]=0 [9%] (g) Assuming small oscillation amplitude, linearise the differential equation obtained in part (f). Then determine the natural frequency of oscillation of the rocking motion. [15%] 3.7 (h) In light of your answer to part (g), what is the significance of the ratio -? [6%] R Fixed

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