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Engr 240 – Week 12

Explore the dynamics of rigid bodies and their motion in plane, including forces, accelerations, moment of inertia, and equations of motion. Learn about D'Alembert's principle and apply it to scenarios involving rotation and translation.

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Engr 240 – Week 12

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  1. Engr 240 – Week 12 Chapter 16. Kinetics of Rigid Bodies: Forces And Accelerations

  2. and The system of the external forces is equivalent to the system consisting of the vector attached at G and the couple of moment . Equations of Motion for a Rigid Body .

  3. Consider a rigid slab made up of a large number of particles of mass . - position vector of ith particle where: - linear momentum of ith particle But: which has the same direction as Therefore and of magnitude . Centroidal moment of inertia to the slab. Differentiating both sides with respect to time: Angular Momentum In Plane Motion The angular momentum about the centroidal frame Gx’y’ is Hence:

  4. The external forces acting on a rigid body is equivalent to the system consisting of a vector attached to the center of mass G, and a couple of moment . and D’ALEMBERT’S PRINCIPLE For a rigid body in plane motion: and , . The two vector equations: three independent scalar equations:

  5. Translation: Centroidal Rotation: NOTE: Before applying D’Alembert’s principle, it may be necessary to analyze kinematics of motion to reduce the number of unknown kinematic variables.

  6. Example 1. A pulley weighing 12 lb and having a radius of gyration of 8 in is connected to two blocks as shown. Assuming no axle friction, determine the angular acceleration of the pulley and the acceleration of each block.

  7. Example 2. The spool has a mass of 8 kg and a radius of gyration of kG=0.35 m about its center. If cords of negligible mass are wrapped around its inner hub and outer rim as shown, determine its angular acceleration.

  8. Example 3. A slender bar AB weighs 60 lbs and moves in the vertical plane, with its ends constrained to follow the smooth horizontal and vertical guides. If the 30-lb force is applied at A with the bar initially at rest in the position shown for which =30, calculate the resulting angular acceleration of the bar and the forces on the small end rollers at A and B.

  9. Rolling Motion: Rolling Without Sliding

  10. Rolling With Sliding:

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