Useful Equations in Planar Rigid-Body Dynamics

1. Useful Equations in Planar Rigid-Body Dynamics Kinematics (Ch. 16) Linear Motion Angular Motion r = r{t} Constantlinear Acceleration Constant AngularAcceleration 2D Rigid Body Kinematics, relative velocity and acceleration equations Mechanical Engineering

2. Useful Equations in Planar Rigid-Body Dynamics Equations of Motion (Ch. 17)  F= ma Translation • Rotation about the axis thru the center of gravity  MG= IGα • Rotation about a fixed axis thru point O  MO= IOα  MO=  (MO)k Mass Moment of Inertia Parallel Axis Theorem Radius of Gyration Mechanical Engineering

3. Work and Energy (Ch. 18) Total kinetic energy of a rigid body rotating and translating Principle of Work and Energy Work done by a moment Work done by a force Work done by gravity and spring Due to gravity Due to torsional spring Due to linear spring Conservation of Energy Mechanical Engineering

4. Linear and Angular Momentum (Ch. 19) Principle of Angular Impulse and Momentum Principle of Linear Impulse and Momentum Non-Centroidal Rotation Mechanical Engineering

5. Solving Dynamic Problems • Newton’s second law (force and acceleration method) may be the most thorough method, but can sometimes be more difficult. • If all forces are conservatives (negligible friction forces), the conservation of energy is easier to use than the principle of work and energy. • Conservation of linear and angular impulse and momentum can be used if the external impulsive forces are zero (conservation of linear impulse and momentum, impact) or the moment of the forces are zero (conservation of angular momentum) • Energy method tends to be more intuitive and easier to use. • Momentum method is less intuitive, but sometime necessary. Mechanical Engineering

6. Solving Dynamic Problems The type of unknowns and given information could point to the best method to use. • Accelerationsuggests that the equations of motion should be used, kinematic equations are used when there are no forces or moments involved. • Displacement or velocity (linear or angular) indicates that the work and energy method is easier to use. • Time suggests that the impulse and momentum method is useful • If springs (linear or torsional) exist, the work and energy is a useful method. Mechanical Engineering

7. Example 1 A homogeneous hemisphere of mass M is released from rest in the position shown. The moment of inertia of a hemisphere about its center of mass is (83/320)mR2. What is the angular velocity when the object’s flat surface is horizontal? Conservation of energy Mechanical Engineering

8. Example 2 • The angular velocity of a satellite can be altered by deploying small masses attached to very light cables. The initial angular velocity of the satellite is 1 = 4 rpm, and it is desired to slow it down to 2 = 1 rpm. • Known information: • IA = 500 kg·m2 (satellite) • mB = 2 kg (small weights) • What should be the extension length d to slow the satellite as required? Mechanical Engineering

9. Example 2 The only significant forces and moments in this problem are those between the two bodies, so conservation of momentum applies. The point about the deployed masses being small mean they have insignificant I, and the (r x mv) for the main body is zero because it spins about its center of mass (r = 0). Mechanical Engineering

10. Example 3 The left disk rolls at constant 2 rad/s clockwise. Determine the linear velocities of joints A and B, vA and vB. Also determine the angular velocities ABand BD. B A Relative velocity equation for points A and C D C Mechanical Engineering

11. Example 3 Relative velocity equation for points B and D B A D C Relative velocity equation for points B and A Mechanical Engineering

12. Example 3 Setting the iandj component equal: Mechanical Engineering

13. Example 4 • The system shown is released from rest with the following conditions: • mA=5 kg, mB=10 kg, Ipulley=0.2 kg·m2, R = 0.15 m • No moment is applied at the pivot. What is the velocity of mass B when it has fallen a distance h= 1 m? The only force or moment that exists and does work is gravity, and it is a conservative force, so conservation of energy applies. Mechanical Engineering

14. Example 4 C 2 A B 1 0 Mechanical Engineering

15. Example 5 If bar AB rotates at 10 rad/s, what is the rack velocity vR? Relative velocity equations Mechanical Engineering

16. Example 5 Mechanical Engineering

17. Example 5 i components j components Mechanical Engineering

18. Example 6 Mechanical Engineering

19. Example 6 Force and motion diagrams for the plate. Kinematics G Relative acceleration equation for points G and A A Mechanical Engineering

20. Example 6 Mechanical Engineering

21. Example 7 Force diagram Motion diagram Mechanical Engineering

22. Example 7 G Mechanical Engineering

23. Example 8 vB = 21  AB vC  slider dir. vC/B CB  = (vC/B)/CB Mechanical Engineering

24. Example 8 Mechanical Engineering

25. Example 9 At the instant shown, the disk is rotating with an angular velocity of  and has an angular acceleration of α. Determine the velocity and acceleration of cylinder B at this instant. Neglect the size of the pulley at C Use position coordinate method Determine the length s = AC in terms of the angle θ( Law of Cosines) Mechanical Engineering

26. Example 9 Mechanical Engineering

27. Example 10 Mechanical Engineering

28. Example 10 Mechanical Engineering