Force Composition and Force Resolution Flashcards

Force Composition and Force Resolution

Force Composition

• Definition: Vector addition of multiple linear forces to produce a single, representative resultant force.
• Illustration: Demonstrates the combined effect of multiple internal and/or external forces acting simultaneously.
• Applications:
  • Determining normal or typical motion.
  • Identifying poor movement patterns or poor motor control.
• Linear Forces Scenarios:
  • Co-linear Forces: Forces acting in a straight line (e.g., tug-of-war), resulting in linear or translatory motion.
  • Non-co-linear Forces: Two or more vectors acting on the same object at angles to each other.

Methods for Representing/Adding Non-Linear Vectors:

1. Parallelogram Method:

   • Complete the parallelogram using the two vectors, maintaining their magnitudes.
   • The resultant force is drawn from the origin to the opposite end of the parallelogram.
   • $a + b = R$
   • Usefulness: Effective for two vectors but can become complex with more than two.

2. Polygon Method:

   • Start with any vector (addition is cumulative: $a + b = b + a$).
   • Place the tail of the second vector at the tip of the first, maintaining magnitude and direction.
   • Continue until all vectors are used.
   • The resultant is drawn from the tail of the first vector to the tip of the last.
   • $a + b + c = R$

Common Mistakes in Force Composition:

1. Magnitude not maintained.
2. Direction not maintained.
3. Tail-to-tip not followed.
4. Resultant not correctly drawn (tail-to-tip or to the end of the completed parallelogram).

Functional Movement

• Can be used to represent forces generated by muscles.
  • Example: Determining the combined action of rotator cuff muscles.

Dysfunctional Movement

• Example: Analyzing muscle forces with a torn supraspinatus.
• Force composition can be applied to both internal and external forces.

Applications/Discussion

1. Muscle Actions and Testing:

   • Example: Analyzing the actions of the anterior and posterior deltoid muscles during shoulder abduction.
     • Anterior deltoid flexes the shoulder, while the posterior deltoid extends it.
     • Force composition explains why the manual muscle test (MMT) for the full deltoid is shoulder abduction.
   • Example: Understanding how the pectoralis major adducts using force composition with its clavicular and sternal heads.

2. Opposing Forces:

   • Analyzing the resulting action when multiple forces (e.g., G1, W1 pulling down and HF pulling up and to the left) act on an object.

Force Resolution

• Definition: The opposite of force composition; breaking down a resultant force into two component forces.

• The two component forces are at right angles to each other, with the resultant in between.

• All three parts (two components and resultant) share the same point of application.

• One component acts parallel to the segment (or lever) - $F_x$ - often providing joint compression/stabilization or distraction.

• The other component acts perpendicular to the segment (or lever) - $F_y$ - typically providing the majority of the rotary movement at the joint.

• Vector directions depend on the direction of the resultant.

• Connecting the tips of the vectors forms a rectangle.

• Purpose: Used to examine how the resultant force(s) act on the segment/system.

Component Naming Conventions:

• Vary depending on the area of study or application.
• Parallel Component (Examples):
  • Parallel component
  • Tangential component
  • Radial component
  • $F_x$
  • Unit vector designation
  • Other vector designations depending on the free body diagram (FBD).
• Perpendicular Component (Examples):
  • Perpendicular component
  • Normal component
  • Transverse component
  • $F_y$
  • Unit vector designation
  • Other vector designations depending on the FBD.
• For this class:
  • Tangential component: parallel to the lever or segment, primarily for compression or distraction.
  • Normal component: perpendicular to the lever or segment, primarily for rotary movement.

Internal Forces

• Common example: Biceps force resolved into normal and tangential components.
• The normal force ($F_y$) contributes to joint angular motion.
• The tangential force ($F_x$) can create compression (joint stabilization) or distraction.

External Forces

• Common example: Gravity.
• Normal force acts on the segment to produce motion opposite to muscle forces.
• Tangential force may be compressive or distractive.

Tangential Component and Moment Arm

• Basic examples show the tangential component going through the joint axis, resulting in:
  • Effect on the joint from linear force only.
  • No moment arm for the tangential component.
  • No moment from the tangential component.
• Typically, tangential components will not go directly through the joint:
  • Will have a small moment arm, allowing for some rotation at the joint.
  • As well as compressive/distractive forces.
  • $Q_{Lf}$ and its components are examples.

Applications/Discussion

1. Internal and External Forces – What is Really Happening?
   • Forces influence joints and tissues in multiple ways, not just movement.
   • Examples: examining actions/functions/influences of $Q<em>{Lf}$, $F</em>{xQLf}$, $F<em>{yQLf}$, $G</em>{Lf}$, and $F_{pLf}$.
2. Effects on the Bar:
   • Determine the effects on a bar from given forces.
3. Making it ‘Harder’ or ‘Easier’:
   • Example: Determining hand placement to resist or test knee extension more effectively.
4. Practice:
   • Resolve resultant vectors (HF, G1, W1) into their components.

Trig Review

• In a standard coordinate system:
  • Cosine (cos) is related to the adjacent side of a right triangle (typically the x-component).
  • Sine (sin) is related to the opposite side of a right triangle (typically the y-component).
  • Tangent (tan) is the association between the opposite and adjacent sides.

Right Triangle Relationships (Angle and One Side Known):

• $cos \theta = \frac{F<em>x}{QL</em>f}$ (adjacent/hypotenuse)
• $sin \theta = \frac{F<em>y}{QL</em>f}$ (opposite/hypotenuse)
• $tan \theta = \frac{F<em>y}{F</em>x}$ (opposite/adjacent)

Right Triangle Relationships (Two Sides Known):

• $\theta = cos^{-1}(\frac{F<em>x}{QL</em>f})$ [adjacent/hypotenuse]
• $\theta = sin^{-1}(\frac{F<em>y}{QL</em>f})$ [opposite/hypotenuse]
• $\theta = tan^{-1}(\frac{F<em>y}{F</em>x})$ [opposite/adjacent]

Pythagorean Theorem

• $F<em>x^2 + F</em>y^2 = QL_f^2$
• Note: Equations will be provided for exams and quizzes; memorization is not required.

Example

• Force F is resolved into two components: $F<em>x$ & $F</em>y$
• Given: $F = 63$ and $\theta = 25°$
• Goal: Find the magnitudes of $F<em>x$ and $F</em>y$
• Rearrange the drawing to form a right triangle.
  • F is the hypotenuse, $F<em>x$ is the adjacent side, and $F</em>y$ is the opposite side.
• $cos \theta = \frac{F<em>x}{F}$ (adjacent/hypotenuse) --> $cos(25) = \frac{F</em>x}{63}$ --> $F_x = 63 * cos(25) = 57.1$
• $sin \theta = \frac{F<em>y}{F}$ (opposite/hypotenuse) --> $sin(25) = \frac{F</em>y}{63}$ --> $F_y = 63 * sin(25) = 26.6$
• Verification using Pythagorean Theorem: $57.1^2 + 26.6^2 = F^2$ --> $F = \sqrt{57.1^2 + 26.6^2} = 63$
• Finding the angle $\theta$ when F, $F<em>x$, and $F</em>y$ are known: $\theta = tan^{-1}(\frac{F<em>y}{F</em>x}) = tan^{-1}(\frac{26.6}{57.1}) = 24.98°$

Glossary of Terms