1. Equilibrium Chapter 9

2. L Center of gravity To calculate the torque due to the gravity force acting on a solid object, place the gravity force vector at the object's center of gravity. The center of gravity is located at the center of uniform, symmetrically shaped objects.

3. Center of gravity for non-uniform systems • Determine the distance from a reference point to the center of gravity of each object. • Calculate (weight) x (distance) for each object • Add all (weight) x (distance) values • Divide by the total weight • Result is the distance from the reference point to the center of gravity Center-of-gravity location is the balance point for the system.

4. Center of gravity for an arm Arm has three parts Upper arm 17 N Lower arm 11 N Hand 4.2 N Find the center of gravity relative to the shoulder joint.

5. Improperly loaded Federal Express airplane Loading too much weight too far back in this plane shifted the center of gravity to a location behind the rear wheels. The plane rotated until the tail of the plane rested on the ground. Sometimes passengers are asked to sit in specific seats so that the center of gravity of the passengers + baggage + plane will be at the correct location for safe operation the airplane.

6. Safe operation region Center of gravity limits for a Cessna 182 airplane

7. Center of gravity of an irregular shape Suspend an object from a string. The center of gravity is directly below the string.

8. Equilibrium requirements Forces are balanced Sum of the externally forces is zero Torques are balanced Sum of the externally torques is zero When an object is in equilibrium, it has ... Constant translational velocity (Zero translational acceleration) Constant angular velocity (Zero angular acceleration) Equilibrium (balanced)

9. Strategy for successful problem solving • Select the object to analyze. • Draw a force diagram. • Gravity force acts at the center of gravity. • Select a convenient rotation axis. • Determine the lever arm for each force. • Calculate the torque for each force. • The torques are balanced. • The sum of the torques is equal to zero. • Choose a convenient set of x andy axes. • The forces are balanced. • The sum of the force x components is equal to zero. • The sum of the force y components is equal to zero. • Solve for the unknown quantities.

10. Example 3 A Diving Board A 530 N woman is standing at the end of a 3.9 m long diving board. The board is supported by a fulcrum 1.4 m from the left end. Find the forces that the bolt and the fulcrum exert on the board. Ignore the weight of the board.

11. 530 N ? ☺ ? Use balanced torques to find one of the unknown forces. Strategy: Choose the axis location so that one force has a zero lever arm distance.That force will have a zero torque.

12. 1476 N 530 N Use balanced forces to find the other unknown force. ?

13. Example 5 Bodybuilding center of gravity A person is holding a dumbbell weight. The deltoid muscle can supply up to 1840 N of force. The arm weighs 31 N. What is the heaviest dumbbell weight that can be held? How much force does the shoulder joint exert on the arm?

14. 1840 N ? ? ☺ 13° 0.15 m It is usually easiest to start with the torque calculations. Calculate torques around an axis at the shoulder joint.

15. 167° 1840 N 31 N 86.1 N Next find the horizontal and vertical components of the shoulder force

16. 0 + 0 - 1792.8 N + Sx = 0Sx = 1792.8 N -31 N - 86.1 N + 413.9 N + Sy = 0 296.8 N + Sy = 0 Sy = -296.8 N

17. y x S Sx = 1792.8 N Sy = -296.8 N Shoulder joint force is

18. The End