Kinetics and Levers Notes

Levers

• Principle of a rigid bar, rod, or segment rotating about its pivot point or fulcrum when acted upon by forces that produce rotation.
• Two or more forces applied in such a way that opposite moments are created.
• Requires at least two linear forces each acting a distance from the pivot point or axis.
• For the graphics presented, the variables are:
  • $EF$ and $R$ for forces
  • $EA$ and $RA$ for distances
  • $EA$ is the moment arm for the force $EF$
  • $RA$ is the moment arm for the force $R$
  • Naming can vary just as component names vary; $EA$ and $RA$ also can be called moment arms or lever arms as well

First Class Lever

• Two forces (one internal ($EA$) – and one external ($R$)) are applied on opposite sides, at some distance from the axis.
• The forces will attempt to cause rotation in opposite directions.

Second Class Lever

• The external resistive force, $RF$, is in between $EF$ (internal force) and the axis.
• Again, both forces act at a distance from the axis. (Strength lever)

Third Class Lever

• The muscle force ($EF$) is between the external resistance force ($RF$) and axis (fast lever).

Mechanical Advantage

• Mechanical advantage: “measure of the mechanical efficiency of the lever system” (Levangie & Norkin, p. 48)
• Effectiveness of $F$ (representing internal force – usually muscle) in comparison to $R$ (representing the external force, resistance)
• $MAd = \frac{FA}{RA}$
  • $FA$ (force arm = moment arm for the internal force or muscle moment arm)
  • $RA$ (resistive arm = moment arm for external force or resistance moment arm)
• If MAd > 1.0, the force has the advantage.
  • A small amount of force can overcome a larger amount of resistance
• If MAd < 1.0, a small amount of resistance can overcome a larger amount of force – or – a larger amount of force is needed to overcome a smaller resistance

Lever Class Advantages

• First Class:
• Second Class:
• Third Class:

Force Couples

• Forces acting together to move an object around a pivot point

Applications/Discussion

1. Lever Recognition: Label which class level is illustrated below in which:
   • $Fms$, Triceps brachii, and $EF$ are the internal forces
   • $A$, External force, and $R$ are the external forces
2. Exercises – SLR
   • Same drawing as before but now it represents the lower extremity with the joint axis at the left side of the bar:
     • $HF$ = hip flexors
     • $G1$ is gravity pulling on the straight leg
     • $W1$ is an ankle weight added to the leg

• Which class level is represented?
• Qualitatively, how much force would be needed to lift the leg in comparison to the external forces?
• Why?

Quantifying Basic Kinetics

• Also use for quantification of forces – doing the math
• Several ways of calculating forces, moments, other kinetic variables
• Calculations for 2D and 3D linear forces, moments, powers, etc. are more involved than what is presented here.
• You may see terms such as “forward solution”, “inverse dynamics approach”, cross-products, matrices and other terms in articles in which kinetics for a given research question were calculated.
• This course and courses in the fall will focus on basic methods. Full methods are beyond scope of the courses and basic calculations sufficient for application concepts.
• Basic methods predominantly utilize:
  • Trigonometry
  • Pythagorean Theorem
  • Geometry
  • Algebra
  • Basic mathematics
• Quantification techniques/rules are not the same thing as the rules for vector composition and vector resolution. Know the differences!

Static Equilibrium

• Forces occur but in a balanced state such that no movement occurs
• Linear
  • $\Sigma Fx = 0$
  • $\Sigma Fy = 0$
  • $a = 0$
• Angular
  • $\Sigma M = 0$
  • $\alpha = 0$
  • $\Sigma M = \Sigma(F * \perp d)$
    • where $M$ is the sum of the moments generated, $F$ are the applied linear forces, and $d$ are the $\perp$ distances between the axis of rotation and the line of action of the applied forces
• Two methods for calculation
  1. Sum of moments created by muscle (internal forces) is set equal to sum of moments created by resistance (external forces)
     • $\Sigma (FMuscle * MAMuscle) = \Sigma (RExternal * RAExternal)$
     • For static equilibrium scenarios, this method is generally easier. It usually takes the directions into account (but not always)
  2. Sum of all forces acting on system at $\perp$ distances to the axis set = 0
     • $\Sigma M = \Sigma(F * \perp d) = 0$
• For example:
  • $QLf * MAQLf = GWbLf * MAGWbLf$
  • Assuming:
    • $GWbLf = 88 N$ (about 20 lbs)
    • $MAGWbLf = 20 cm$
    • $MAQLf = 5 cm$
    • $QLf = ???$
    • $QLf * MAQLf = GWbLf * MAGWbLf$
    • $QLf * 5 cm = 42.4N * 20 cm$
    • $QLf = (88N * 20 cm)/5 cm$
    • $QLf = 352 N$ (79 lbs)

Dynamics

• The system is not in equilibrium so motion is produced
• Can still use $\Sigma M$, but not $\Sigma M = 0$
• $G = 88 N$
• $RAG = 10 cm = .10 m$
• $MA HS = 4 cm =.04 m$
• $HS = 150 N$
• $\Sigma M = (HS * MAHS) + (G * RAG)$
• $\Sigma M = (150 N* .04 m) + (-88 N * .10 m)$
  • [Now must account for direction of gravity]
• $\Sigma M = - 2.8 N.m$
• Is the lower leg flexing or extending?

Applications/Discussion – Forces and Levers

1. To understand which forces can be manipulated and why you would do so. This is commonly used for exercise initiation and progression utilizing the idea of levers.
   • For example: Straight plane shoulder abduction, holding the arm at 90 with the elbow straight vs. elbow bent:
     • $D$ = Deltoid muscle
     • $GAFh$ = weight of the arm as gravity is pulling on it
     • $MAD$ = moment arm of $D$
     • $MAGAFh$ = moment arm of the weight of the arm

• Additional information:
  • Forces will be in Newtons (N); to convert to Newtons, multiple lbs by 4.4482
    • For example: A person weighing 168 lbs = 747.3 N
  • If:
    • $GAFh = 4.9 \%$ of the body weight
    • $MAD = 5 cm$
    • $MAGAFh = 25 cm$
  • Solve for $D$:
    • With the elbow bent →only one variable has changed: MAGAFh = 12 cm
    • So what is $D$ now?

1. To understand joint reaction forces or the sum of all of the forces acting on the joint.
   • First for the arm extended
   • Now for the elbow bent.
   • Using the same examples as in #1 above but with the additional information that the angle of inclination (or angle of application) is roughly 25°:
     • Identify the joint reaction forces produced by the deltoid and the weight of the arm in the x- direction ($Jx$).
     • What are the joint reaction forces acting in the y-direction ($Jy$)?
     • Calculate the component joint reaction forces ($Jx$ and $Jy$) and sum them together.
     • $\Sigma Jx =$
     • $\Sigma Jy =$
     • What is the resultant joint reaction force ($J$)?
     • What is the orientation of the joint reaction force, $J$?
     • $\Sigma Jx =$
     • $\Sigma Jy = 0$
     • What is the resultant joint reaction force ($J$)?
     • What is the orientation of the joint reaction force, $J$?

Global Understanding of Forces

• To have a global understanding of the forces that can be exerted by and on the body:
  • Example: The influence of sitting posture on T5 (5th thoracic vertebra - ish) as demonstrated by the graphics below. The horizontal arrow is pointing at the approximate location. The vertical arrow represents gravity (W) pulling on the head.
  • We will be examining the sagittal plane movements only. The vertebra shown isn’t T5 but it will be used to illustrate the forces.
    • A. What kind of external moment (osteokinematic direction) is being created by the weight of the head and neck on the region of interest?
    • B. What kind of moment needs to be created by the body to counter the weight of the head and neck?
    • C. The $E$ represents the muscle and ligamentous force being exerted on T5. The drawings to the right of the graphic just shows the forces and axis at T5. Using the tip of the triangle as a reference, draw the moment arms for the muscle force $E$ and the weight of the head $W$ for both conditions.
    • D. Variables:
      • For the relaxed sitting condition (R):
        • Moment arm for extensor muscle group and ligaments = $MAER = 2 cm$
        • Moment arm for the weight of the head and vertebra superior to T5 = $MAWR = 5 cm$
        • Weight of head and superior vertebra = $WR = 11\%$ body weight (BW)
        • Force exerted by the muscles and ligaments = $ER$ = unknown
        • Components of Weight = $WRx$ and $WRy$ = unknown, but able to be calculated
        • Components of Muscle+Ligg = $ERx$ and $ERy$ = unknown, but able to be calculated
        • Angle between $WR$ and $WRy = \Theta1 = 15°$
        • Angle between $ER$ and $ERy = \beta = 5°$
      • For the slumped sitting condition (S):
        • Moment arm for extensor muscle group and ligaments = $MAES = 2 cm$
        • Moment arm for the weight of the head and vertebra superior to T5 = $MAWS = 9.5 cm$
        • Weight of head and superior vertebra = $WS = 11\%$ body weight (BW)
        • Force exerted by the muscles and ligaments = $ES$ = unknown
        • Components of Weight = $WSx$ and $WSy$ = unknown, but able to be calculated
        • Components of Muscle+Ligg = $ESx$ and $ESy$ = unknown, but able to be calculated
        • Angle between $WS$ and $WSy = \Theta2 = 30°$
        • Angle between $ES$ and $ESy = \beta = 5°$
  • Additional information:
    • Forces will be in Newtons (N); to convert to Newtons, multiple lbs by 4.4482
      • For example: A person weighing 168 lbs = 747.3 N
    • E. Problems:
      • With $BW$ = to 168 lbs (747.3 N), solve for the unknowns for the relaxed sitting condition.
      • With $BW$ = to 168 lbs (747.3 N), solve for the unknowns for the slumped sitting condition.
    • F. How do you solve for the unknowns?
      • Start with what you know
      • Put what you know if some form of initial equation
      • When get to the full equation, there will only be one unknown – solve via math/trig

Newton’s Laws of Motion

• Anthropometrics/Anthropometry
  • Anthropometry is loosely defined as measurements of physical structures that help to either directly measure or be able to calculate body segment parameters associated with movement.
  • Some anthropometrics include:
    • Mass ($m$)
      • Amount of matter of which a body is composed
    • Location of center of mass
      • Point at which a body’s mass is concentrated in equilibrium (evenly distributed)
      • Point where acceleration of gravity acts on the body (whole body and segment)
    • Height
    • Length
      • May be length of a segment or system
      • May also include breadth (side-to-side) and depth (front-to-back) measurements
    • Weight
    • Volume (circumferential measurements)

Properties of the Body

• Anthropometrics
  • Inertia
    • Property of object that resists initiation or change of motion
    • Inertia is proportionate to the mass of an object
  • Moment of inertia (or mass moment of inertia) typically represented by the letter ‘I’
    • [From Neumann:] Rotational counterpart of mass ($m$)
    • Resistance to change in angular velocity; resistance to rotational forces
    • Dependent upon both mass and distribution of mass in relation to axis of rotation
    • $I = m \times \rho^2$
  • Radius of gyration ($\rho$)
    • The distance between the center of mass and the axis of rotation
  • Methods to Calculate Some of the Anthropometrics
    • Hanavan’s model
    • Dempster’s model
    • Clauser, et al.
  • Precautions

First Law of Motion

• Bodies at rest tend to stay at rest; bodies in motion tend to stay in motion unless either is acted upon by an external force that changes the motion of the body (unbalanced force)
• Applies for both linear and angular velocity

Second Law

• The acceleration for a body of constant mass is proportional of the resultant forces causing it and the change takes place in the direction in which the force acts.
• The acceleration is inversely proportional to the mass of the body
• $a = \frac{\Delta v}{\Delta t}$
• $F = m (\frac{\Delta v}{\Delta t})$
• Linear: $\sum F = ma$
  • $a \propto \frac{F}{m}$
• Angular: $\sum M = I\alpha$
  • $\alpha \propto \frac{M}{I}$

Force-Acceleration Relationship

• With everything else equal, part with lesser mass will move when acted upon.

Third Law

• To every action, there is an equal and opposite reaction
• Consequences of effects one body has on another is dictated by second law: mass dependent
• Can be both internal and external forces

Newton’s Laws of Motion – What Can be Derived from Them?

• [From Neumann:] Impulse-Momentum Relationship
  • Describes additional relationships derived from the second law
  • $F = ma$
  • $a = \frac{\Delta v}{\Delta t}$
  • $F = m (\frac{\Delta v}{\Delta t})$
  • $F \times \Delta t = m \times \Delta v$
    • Right side of equation is the change in momentum
  • Momentum –represented mathematically by letter “p”, where p = mass x linear velocity
    • $p = m \times v$
    • Left side of equation is impulse
  • Impulse – the combination of force and time that is what is required to change the momentum of a body (Ft in the equation below)
    • $p = Ft$
• Angular Equivalents:
  • $M = I \frac{\Delta \omega}{ \Delta t}$
  • $M \times \Delta t = I \times \Delta \omega$

Work-Energy Relationship

• Examines mechanical work in energy expenditure
• Forces or moments applied over a distance in the direction of the force applied
• Distance (in this instance) is NOT a moment arm
• No mechanical work is done if there is no movement
  • Linear: $W = F \times d$
  • Angular: $W = M \times \theta$

Power

• Work done over time
  • $P = W/t$
    or
  • $P = F \times v$ (linear)
  • $P = M \times \omega$ (angular)
• Power as related to angular motion tends to be related to muscle performance:
  • Positive power – rate of work done by muscles during concentric contractions
    • Internal muscle forces over external force
    • Considered as propulsion or acceleration
  • Negative power – rate of work accomplished by muscles undergoing eccentric contractions
    • External load over internal muscle force
    • Considered as absorption or deceleration

Applications/Discussion: Laws of Motion

1. Revisiting Convex-Concave Principle with testing passive accessory motion
   • Promoting ankle DF
2. Injury
   • Important to know about and consider inertia during initiation, continuation and cessation of movements
   • Think about the phrase, “unless acted upon by an external force” as it relates to MVAs, diving, sit-to-stand, and other actions.
   • Now add: $Ft = m \times v$ to #2
   • Impulse-Momentum
3. Functional Movement
   • Some of the items that can be considered for each law include:
     • Inertia – Transfers
     • Momentum – Transfers
     • Equal and opposite reaction – Gait
4. Dysfunctional Movement
   • As with functional movement, now consider the 3 laws as related to dysfunctional movement from multiple perspectives: neurological, musculoskeletal, pediatric, etc.
     • First law: stop/start in transfers/gait
     • Second law: lighter moves on heavier – or – how the deltoid can act as a scapular downward rotator
     • All three laws: Falls

Glossary of Terms