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

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

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

6. Solving Dynamic Problems
The type of unknowns and given information could point to the best method to use.
Acceleration suggests 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.
Ken Youssefi
Mechanical Engineering

7. Mechanical Engineering
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
Ken Youssefi
Mechanical Engineering

8. Mechanical Engineering
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?
Ken Youssefi
Mechanical Engineering

9. Mechanical Engineering
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).
Ken Youssefi
Mechanical Engineering

10. Mechanical Engineering
Example 3 A B C D
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 AB and BD .
Relative velocity equation for points A and C
Ken Youssefi
Mechanical Engineering

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

12. Mechanical Engineering
Example 3
Setting the i and j component equal:
Ken Youssefi
Mechanical Engineering

13. Mechanical Engineering
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.
Ken Youssefi
Mechanical Engineering

14. Mechanical Engineering
Example 4 1 2 A B C
Ken Youssefi
Mechanical Engineering

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

16. Mechanical Engineering
Example 5
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Mechanical Engineering

17. Mechanical Engineering
Example 5
i components
j components
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Mechanical Engineering

18. Mechanical Engineering
Example 6
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Mechanical Engineering

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

20. Mechanical Engineering
Example 6
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Mechanical Engineering

21. Mechanical Engineering
Example 7
Force diagram
Motion diagram
Ken Youssefi
Mechanical Engineering

22. Mechanical Engineering
Example 7 G
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Mechanical Engineering

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

24. Mechanical Engineering
Example 8
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Mechanical Engineering

25. Mechanical Engineering
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)
Ken Youssefi
Mechanical Engineering

26. Mechanical Engineering
Example 9
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Mechanical Engineering

27. Mechanical Engineering
Example 10
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28. Mechanical Engineering
Example 10
Ken Youssefi
Mechanical Engineering