MATLAB Problem: Half-Pipe In this problem you will analyze the motion of a block on a half-circle depicted in the figure. The dashed block shows the initial position. It is released from rest. The coefficient of kinetic friction is , the mass is m , the circle radius is r , the gravitational constant is g . Let be the angle of rotation from the Cartesian coordinate basis to the polar coordinate basis . 1) What is the relation between and ? 2) Derive the linear transformation from to in terms of . 3) Show the linear transformation derived in 2) is a rotation matrix. 4) Derive expressions for the forces acting on the block in polar coordinates, in terms of m , g , r , , , , N (the normal force in the direction of ). 5) Derive an expression for the acceleration of the block in polar coordinates using (instead of ). 6) From the balance of linear momentum (combining 4) and 5)), derive an expression for in terms of g , r , , , . 7)
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