Equilibrium

1. Equilibrium A state in which all forces or torques acting on a body or system are balanced A body in equilibrium is either motionless or moving at a constant velocity

2. Static Equilibrium • The sum of forces on all 3 axes = 0 (SFX=SFY=SFZ=0) • The combination of the weight and the reaction forces of an object at rest is an example of static equilibrium

3. When a person is at rest (for example, leaning against a wall), for every force produced by their weight against a surface (the floor or wall) there is an equal force (friction or a reaction force) that is equal but opposite.

4. Static Equilibrium • The sum of all torques = 0 (ST=0) • For a static (or isometric) muscle contraction: The torque produced by the muscle is equal in magnitude to the torque produced by the resistance (TM=TR)

5. FM TM TR Resistance In a state of static equilibrium, the torque produced by muscle force acting on a joint equals the torque produced by resistance acting at the same joint. (TM = TR)

6. Dynamic Equilibrium • All forces on a body or system in motion result in equal and oppositely directed inertial forces. Inertial Force – The result of some acceleration acting on a body’s mass • Weight is an inertial force • In a moving object, the inertial forces equal the object’s mass times the acceleration in each direction (maX, maY, maZ) and the moment of inertia of the object times its angular acceleration (Ia)

7. Equations for Dynamic Equilibrium • The sum of the forces in a given direction is equal to the inertial force in that direction (SFX= maX, SFY= maY,SFZ= maZ OR SFX-maX=0, SFY-maY=0, SFZ-maZ=0) • The sum of the torques around the center of gravity (c-g) of a body is equal to the angular inertial force (STc-g= Ia OR STc-g-Ia = 0)

8. Air Drag and Dynamic Equilibrium • Air resistance exerts fluid drag on a moving object • This drag is a force (essentially a dynamic reaction force) that is dependent on: • The object’s weight • The object’s cross sectional area • The texture or roughness of the surface exposed to the air

9. Air Drag and Dynamic Equilibrium (continued) • Air drag reduces the acceleration applied to an object. • Since air drag is a force, you could say that it produces negative acceleration on a moving object. That is, it tends to slow it down. • For a falling object, air drag changes until the terminal velocity of the object is reached.

10. Terminal Velocity • The state in which drag force (Fd) equals the weight of an object • At terminal velocity, acceleration (not acceleration due to gravity) = 0 • Varies greatly from object to object (A feather appears to float downward because it has a very low terminal velocity.)

11. A skydiver weighing 580 N is in freefall. At one point the skydiver’s acceleration is – 8.8 m/s2. (It is not -9.81 m/s2 because of the effect of drag force (Fd).) Calculate the drag force acting on the skydiver.

12. The forces acting on the skydiver are: weight = -580 N (It’s negative since the weight acts downward) and Fd • The inertial force on the skydiver is: maY • By the equation for dynamic equilibrium: SFY-maY= 0 • m = weight/ag = 580 N/9.81 m/s2 = 59.12 kg

13. SFY=-580 N + Fd maY = (59.12 kg)(-8.8 m/s2) By the equation for dynamic equilibrium: SFY-maY= 0 -580 N + Fd – (59.12 kg)(-8.8 m/s2)=0 Fd = 580 N + (59.12 kg)(-8.8 m/s2) = 59.7 N

14. Stability • Resistance to a change in equilibrium or resistance to acceleration • Balance – The ability to control equilibrium

15. Stability Requirements • Vary greatly from one activity to the next • Activity requirements range from a high degree of static equilibrium (football linemen) to combinations of static equilibrium and controlled instability (wrestlers, gymnasts and dancers)

16. Wrestling requires the competitors to frequently transition from a very stable position to one of controlled instability.

17. Stability Requirements • Some skills require reaching a state as close as possible to instability while remaining stable (start positions in swimming and sprinting)

18. For the staring position of sprint and swim races, the athletes want to be as close to an unstable position as possible.

19. Stability Requirements • Stability of one kind may be required during aggressive movements as in gripping bats, racquets, and golf clubs (these require a stable grip)

20. When swinging bats, rackets, etc. the grip produces a force that counteracts the force trying make the implement slip from the hands.

21. Stability Requirements • Some activities require a constant state of instability (walking, running, etc.)

22. In order to move forward, a runner reaches a state of controlled instability in which the center of gravity is moved continuously past the base of support.

23. Factors Affecting Stability • Resistance to Movement • An object’s mass (increased mass yields increased stability) • Friction (greater friction yields increased stability) • Fluid Drag (can cause an object to resist being moved in a fluid medium)

24. Factors Affecting Stability Balance (Resistance to Falling) Height of c-g (lower c-g yields greater stability) Position of c-g relative to the edge of the base of support (greater horizontal distance yields greater stability) Base of support (wider base yields greater stability)

25. Height of C-G Height of C-G The object on the left is more stable than the object on the right because it has a lower c-g.

26. Assuming that two objects represented by the triangles below have the same weight, the object on the right is more stable than the object on the left. If a force (F) is applied to each object at the same height (h) above each object’s base, the object on the right is harder to tip over because its c-g is farther from the edge. F F h h d d

27. If the two triangles below represent objects of equal density and equal height, the triangle on the right is the more stable of the two because of its wider base.

28. h d d d • Making the base of an object wider can have two effects related to increased stability: • The original height (h) of the c-g is lower. • The distance (d) from the edge of the object to the c-g increases.