Meriam, Kraige & Bolton — Rigid Bodies: Equivalent Systems of Forces: Problem 3_93

⚡ Mecademy AIENG정역학 · ch3  Problem Statement A slender rod of mass is welded to the horizontal edge of a uniform semicylindrical shell of mass . Determine an expression for the angle with the horizontal made by the diameter of the shell through . (Consult Table D/3 in Appendix D to locate the center of gravity of the semicircular section.) Problem 3/93 (a) Expression for the angle ● Calculation Process 1. Establish Coordinate System and Locate Centroids: We establish a body-fixed coordinate system with the origin at the center of the semicylindrical shell's diameter. Let the -axis lie along the diameter directed toward the rod of mass , and the - axis lie along the axis of symmetry directed toward the centroid of the semicircular shell. For the slender rod: , position . For the semicylindrical shell: , position (from Table D/3 for a semicircular arc). 2. Calculate the Center of Mass of the System (): The coordinates of the center of mass are given by: 3. Apply Equilibrium Condition: For a body with a circular profile resting on a horizontal plane, static equilibrium requires the system's center of mass to lie on the vertical line passing through the geometric center and the point of contact. Thus, the m 1 m 2 θ m 1 θ O xm 1 y m=m 1 (x ,y )= 11 (r,0) m=m 2 (x ,y )= 22 (0, ) π 2r G(, )xˉyˉ G =xˉ = m +m 12 m (r)+m (0) 12 m +m 12 m r 1 =yˉ = m +m 12 m (0)+m ( ) 12 π 2r π(m +m ) 12 2m r 2 G O vector is vertical. Let be the angle that makes with the diameter (-axis): 4. Relate to