1

Equilibrium

Objectives:

• Learn how to draw a free body diagram

• Define and learn how to solve problems in static equilibrium

• Define the 3 classes of levers and what each is best suited for

• Introduce stabilizing and dislocating forces and how to compute them

Free Body Diagram• Is a means of representing all of the external forces

and torques acting on a system

• It is the most important step in solving a problem in kinetics

Fquads

System

Wleg+foot

Tflexor

Fcontact

Constructing a Free Body Diagram

1. Identify the system

2. Draw a simple picture (diagram) of the system

3. Identify each of the external forces and torques (i.e. forces or torques acting across the boundary of the system).

4. Identify (or assume) the point of application and direction of each force and draw into the diagram

5. Identify (or assume) the axis of rotation and direction for each torque and draw into the diagram

6. Add a reference frame to the diagram

Identifying the System

• Depends on the problem being solved

• The forces or torques of interest must act across the boundary of the system (i.e. they must be external forces or torques)

• The system should behave as a rigid body (unless you are only interested in the movement of the system’s center of mass)

• Typical systems in biomechanics:

– the whole body– a single body segment

– a rigid group of body segments

2

Identifying External Forces/Torques

• External forces include:– Weight (i.e. force due to gravity)– Contact forces applied from outside the body– Contact forces applied from within the body

• Two methods of representing contact forces from within the body:

– The joint contact force and the force produced by each anatomical structure across the joint (i.e. muscles, ligaments, etc.)

– A resultant joint force and a resultant joint torque

Point of Application & Direction• Usually known for contact forces• Weight acts downward from the center of mass• Resultant joint force is applied at the joint center• Resultant joint torque acts about joint center; can

assume its direction• If direction of a force unknown, assume a positive x-

component and a positive y-component• If point of application of a force unknown, include

point of application as a variable

Fx

Fydirection unknown:

Fd

location unknown:

Example Problem #1We want to compute the ankle torque during the stance

phase of running. Construct an appropriate free body diagram.

Static Equilibrium• A system is at rest and will remain at rest

• No translation or rotation is occurring or will occur• Conditions for static equilibrium

(from Newton’s 1 st Law):

Σ T = 0

Σ Fx = 0

Σ Fy = 0

– Net external force in x direction equals zero

– Net external force in y direction equals zero

– Net torque produced by all external forces and all external torques equals zero

• Can use any point as the axis of rotation

• Can solve for at most 3 unknown quantities

3

Example Problem #2A 60 kg gymnast is standing in the position shown.

Find the ground reaction force acting on her foot.

How far forward can she move her center of mass and remain standing?

20 cm

15 cm

10 cm

body center of mass

80 cm

Example Problem #3During an isometric (static) knee extension, a

therapist measures a force of 100 N using a hand dynamometer in the position shown below

Find the resultant knee joint force and torque.

Does the dynamometer position affect the measured force?

Fdyn = 100 N

m = 4.5 kg

60°

30 cm

KNEE

24 cm

Levers• Most skeletal muscles act using the principle of

leverage

• A lever system consists of:– An axis of rotation (or fulcrum)– A resistance force or load– An effort force (the applied force that is used to

move the load)

• There are 3 classes of leverFload

Feffort

axis of rotation

1st Class Lever• Effort force and load force are applied on

opposite sides of the axis of rotation

• Effort force and load force act in same direction

• For equilibrium: d⊥load Fload = d⊥effort Feffort

Fload Feffortd⊥load d⊥effort

axisFeffortFload

d⊥effortFload =d⊥load

Feffortor:

4

Mechanical Advantage

Feffort

Floadaxis

d⊥effortMechanical Advantage =d⊥load

• When Mechanical Advantage > 1:– Feffort needed is less than Fload

– Point at which Fload applied moves slower and shorter distance than point at which Feffort applied

– Good for strength, poor for moving load quickly or through large range of motion

Mechanical Advantage

Feffort

Fload

axis

• When Mechanical Advantage < 1:– Feffort needed is greater than Fload

– Point at which Fload applied moves faster and greater distance than point at which Feffort applied

– Good for moving load quickly or through large range of motion; poor for strength

• A 1st class lever can have a mechanical advantage greater than, equal to, or less than 1.

2nd Class Lever• Effort force and load force are applied on same

side of the axis of rotation• Effort force applied farther from axis than the load

force (i.e. d⊥effort > d⊥load)• Effort and load force act in opposite directions• Good for strength; poor for moving load quickly or

through large range of motion

Fload

Feffort

d⊥load

d⊥effort

axis

Feffort

Fload

3rd Class Lever• Effort force and load force are applied on same

side of the axis of rotation• Effort force applied closer to axis than the load

force (i.e. d⊥effort < d⊥load)• Effort and load force act in opposite directions• Good for moving load quickly or through large

range of motion; poor for strength

Fload

Feffort

d⊥loadd⊥effort

axis

Feffort

Fload

5

Stabilization vs. Dislocation• Forces do not produce torque only; they also

produce stabilizing or dislocating forces at a joint.

• Can decompose a force into components parallel to (Fll) and perpendicular to (F⊥) the joint surface

• F⊥ points towards joint → stabilization

• F⊥ points away from joint → dislocation

F

Fll

F⊥F

Fll

F⊥

Dislocation:Stabilization:

Example Problem #4A person is holding their upper limb in the abducted

position shown. Find the deltoid muscle force and the force perpendicular to the joint surface.

Is the deltoid force stabilizing or dislocating?What class of lever is this?

Fdeltoid

W = 35 N

30 cm

15 cmShoulder

30°

Join

t Sur

face