Biomechanics: Ground Reaction Forces, Center of Mass, and Movement Analysis

Context and class logistics

• Next Tuesday: speaker will be at a wrist and hand specialist; hand numbness is the issue.
• Thursday: exam; accommodations for students; those without accommodations must explain their answer and rationale in class.
• Example problems will be posted for today and more problems before next class to give time to review.
• Problems settings: the instructor expects problems to stay open through the end of the semester; Catherine can adjust settings if needed.
• Knowledge check: average score 91%; this was roughly the target; most points lost were on one main question.
• Purpose of today’s recap: ensure everyone is on the same page before moving forward; foundational material will be used for upcoming topics (vertical jumping, squatting).

Summary of the core framework for motion analysis

• Anatomical position is the basis for all movement analyses; it’s used to define planes, axes, and positive directions.
• Key steps to analyze a motion (systematic approach):
  • 1) Start from anatomical position.
  • 2) Identify the plane of motion.
  • 3) Identify the axis perpendicular to that plane.
  • 4) Establish the positive direction along that axis (using the right-hand rule).
  • 5) Place your hand near the joint of interest with your thumb aligned with the positive axis; curl the fingers along the moving segment to see whether the motion is in the positive direction.
• This approach creates a consistent, step-by-step method similar to solving a geometry proof or angular-acceleration problem in physics.
• Examples discussed:
  • Right hip flexion is in the sagittal plane; axis is mediolateral; the positive direction corresponds to the direction your thumb points when the distal segment (femur) moves in the flexion direction.
  • Right shoulder external rotation is in the transverse plane; axis is longitudinal; interpreting whether the motion is positive requires the same right-hand rule approach.
  • Left shoulder adduction/abduction occurs in the frontal plane; axis is anterior-posterior; the positive direction is defined by the axis orientation via the right-hand rule.
• The main takeaway: master the anatomical position → plane → axis → positive direction, then determine joint motion using the right-hand rule and the relative position of the moving segment.

Planes, axes, and positive direction (foundation)

• Planes of motion:
  • Sagittal plane: flexion/extension; keeps body in a front/back division.
  • Frontal (coronal) plane: abduction/adduction; divides body into front/back.
  • Transverse (horizontal) plane: rotations (internal/external rotation); divides body into top/bottom.
• Axes associated with planes (perpendicular to each plane):
  • Sagittal plane ↔ mediolateral axis (roughly left-to-right axis)
  • Frontal plane ↔ anterior-posterior axis
  • Transverse plane ↔ longitudinal axis
• Positive direction (via right-hand rule):
  • For each axis, orient your right hand so the thumb points in the positive direction; curl the fingers to see the direction of positive rotation for the corresponding plane.
  • In class examples, the positive axis direction is described with reference to anterior/superior as appropriate, and the “positive” rotation is determined by whether the distal segment moves in the same sense as the curled fingers.
• Practical note: always connect anatomical position → plane → axis → positive direction before deciding if a motion is positive or negative.

Center of mass (COM) and body modeling

• COM definition: the single point where the mass of a body can be considered to be concentrated; for a real body, COM location shifts with posture and movement.
• For a symmetrical, simple shape (e.g., a hollow sphere), COM is at the geometrical center; real human bodies are asymmetrical, so COM is not fixed and must be estimated segmentally.
• Segmental approach to COM:
  • Model the body as multiple segments with mass mi and center of mass location ri for each segment.
  • Whole-body COM:
    oxed{\mathbf{r}{COM} = \frac{1}{M} \sumi mi \mathbf{r}i}
    where M = total body mass and mi/outstanding ri are the segment masses and segment COM positions.
• Static vs dynamic COM:
  • In static stance, COM stays in the same spot.
  • When moving, mass redistributes, shifting COM location (e.g., bringing mass anteriorly or superiorly to reach or stabilize a position).
• COM and gravity:
  • Gravity acts at COM; acceleration due to gravity on Earth is
    g \approx 9.81\ \mathrm{m\,s^{-2}}.
  • The gravitational force on the body is
    \boxed{\mathbf{F}_g = m \mathbf{g}}\quad(\text{m is body mass; direction downward})
• Relationship to stability and posture:
  • Lower COM generally increases stability; raising COM can increase reach but reduce stability.
• How COM is used in biomechanics labs:
  • Planes and axes are assigned to each segment; a whole-body plane/axis system allows analysis of joint behavior and whole-body motion.
• Practical note: COM is a central concept in analyzing external loads and movement strategies in daily activities and athletic tasks.

Gravity, ground reaction forces (GRF), and measurement

• Ground Reaction Force (GRF):
  • The force the ground exerts on the body in response to body-ground interaction; it is the equal-and-opposite reaction to the force the body exerts on the ground (Newton’s third law).
  • Focus: vertical component of GRF during stance phases (walking, squatting, etc.).
• Measurement: force plates embedded in floors, gym racks, courts, etc.; record vertical GRF over time during stance phases.
• Relationship to gravity and COM motion (Newton’s second law):
  • In vertical direction (upward positive):
    \boxed{\sum Fy = m ay \Rightarrow F{GRF} - m g = m ay \Rightarrow F{GRF} = m (g + ay)}
  • Interpretation:
  • If COM accelerates downward (a_y < 0), GRF < m g (less than body weight).
  • If COM accelerates upward (a_y > 0), GRF > m g (greater than body weight).
• Observations during gait (stance phase):
  • GRF curve typically fluctuates around body weight; it can exceed body weight during late stance (push-off) when accelerating the body upward or forward.
  • Sometimes the GRF visually appears to exceed 100% body weight due to dynamic loading, not a literal instantaneous weight gain.
• Why GRF matters:
  • GRF is a primary external force acting on the body; understanding it helps explain how we walk, run, jump, squat, stand up, and perform many daily tasks.
• Additional note on gravity and static measures:
  • When standing still, with no acceleration, GRF equals body weight: F_{GRF} = m g.
• Units and interpretations:
  • GRF is measured in Newtons (N) or kilonewtons (kN);
  • Body weight is a force, expressed in newtons; body mass is in kilograms (kg).
• Intuition for practice:
  • To compare people or tasks, often normalize GRF to body weight: use the ratio \frac{F_{GRF}}{m g} or express as multiple of body weight.

Ground reaction forces during a squat: a worked example

• Movement phases (relevant for COM and GRF analysis):
  • Eccentric phase: lowering (COM moves downward, negative displacement, negative velocity, negative acceleration).
  • Isometric phase: at the bottom of the squat, velocity ~0; COM displacement is approximately zero during the hold.
  • Concentric phase: rising (COM moves upward, positive displacement, positive velocity, acceleration).
• Sign conventions used in the example:
  • Positive direction for vertical movement is upward; negative is downward.
  • Negative displacement and velocity indicate downward movement (eccentric phase).
• Worked problem setup (example from the lecture):
  • Mass m = 70 kg.
  • COM velocity at start of the eccentric phase: v_0 = 0 m/s.
  • COM velocity at end of the considered interval: v_1 = -2 m/s.
  • Time interval: Δt = 0.5 s.
• Compute acceleration during the interval:
  a = \frac{\Delta v}{\Delta t} = \frac{v1 - v0}{\Delta t} = \frac{-2 - 0}{0.5} = -4\ \mathrm{m\,s^{-2}}.
• Gravity and GRF calculation for this interval:
  • Gravitational acceleration: $g \approx 9.81\ \mathrm{m\,s^{-2}}$ downward.
  • If we use the vertical GRF formula with upward positive:
    F{GRF} = m (g + ay) = 70 \times (9.81 + (-4)) = 70 \times 5.81 \approx 406.7\ \mathrm{N}.
  • Body weight (static) for comparison:
    W = m g = 70 \times 9.81 \approx 686.7\ \mathrm{N}.
  • Percent of body weight represented by GRF during this interval:
    \frac{F_{GRF}}{W} = \frac{406.7}{686.7} \approx 0.592 \Rightarrow 59.2\%.
• Interpretation of the result:
  • During the eccentric (downward) phase with downward COM acceleration, the vertical GRF is less than body weight (here about 59% of body weight).
  • During the concentric (upward) phase, the COM acceleration would be positive, and GRF would typically be greater than body weight.
• Additional modeling notes:
  • If the motion were continuous through the complete squat, the net impulse over the whole cycle (start versus end states) can be zero if the initial and final states are the same (static posture), even though the COM velocity and acceleration are nonzero during phases.
  • This example treats the movement as a linear, time-segmented problem with constant acceleration within the interval; real human motion may have non-constant acceleration, requiring piecewise or more advanced modeling.
• How this problem connects to broader biomechanics:
  • Demonstrates how to use Newton’s laws to connect internal muscular actions, COM motion, gravity, and external GRF measured by force plates.
  • Provides a framework for comparing individuals or movements by normalizing GRF to body weight and analyzing phases of movement.

Practical implications and real-world relevance

• Why this matters for daily life:
  • Ground reaction forces underpin how we walk, run, stand up, sit down, jump, and perform sports actions; they influence joint loading, injury risk, and training design.
• How COM and GRF interplay affect stability and control:
  • Shifts in COM position and velocity change the required GRF to maintain or modify posture and to accelerate the body in different directions.
  • Training can target improving control during eccentric deceleration and concentric acceleration to optimize performance and reduce injury risk.
• Connecting to broader topics in the course:
  • These concepts build a foundation for later units on vertical jumping, squatting mechanics, and more complex musculoskeletal analyses.
• Ethical, philosophical, and practical implications:
  • Understanding how external forces interact with body mass can inform safe exercise guidelines and rehabilitation planning.
  • Clear, methodical problem-solving approaches reduce misinterpretation of dynamic data and promote evidence-based training and therapy.

Quick reference: key formulas and terms (recap)

• Gravitational force (weight):
  F_g = m g, \quad g \approx 9.81\ \mathrm{m\,s^{-2}}.
• Vertical force balance (GRF analysis):
  \boxed{\sum Fy = m ay \,\Rightarrow\, F{GRF} - Fg = m ay \,\Rightarrow\, F{GRF} = m (g + a_y).}
• Center of mass (whole body):
  \boxed{\mathbf{r}{COM} = \frac{1}{M} \sumi mi \mathbf{r}i.}
• COM motion and phase signs in a squat (example conventions):
  • Eccentric phase: negative displacement, velocity, and likely negative acceleration; COM moves downward.
  • Isometric phase: velocity ~0; COM displacement ~0.
  • Concentric phase: positive displacement and velocity; COM moves upward.
• Ground reaction force interpretation:
  • GRF magnitude relative to body weight indicates whether the body is accelerating upward or downward during that phase.
• Units:
  • Mass: kilograms (kg)
  • Force: newtons (N) or kilonewtons (kN)
  • Acceleration: meters per second squared (m/s^2)

Notes on terminology observed in the lecture

• Ground reaction force (GRF): the vertical component is the primary focus for many foundational analyses.
• Body weight: the static gravitational force acting on the body, equal to $m g$.
• Force plates: tools used to measure GRF during movement; common in labs and training facilities.
• Pin deck: industry term mentioned as the surface where pins are placed in a bowling context; used here as an analogy for COM location considerations in simple shapes.
• Framing of the problem: break complex, whole-body tasks into simpler phases (eccentric, isometric, concentric) to isolate mechanics and sign conventions.

Quick study tips based on today’s content

• Always start with anatomical position, then identify the plane, axis, and positive rotation direction before judging motion as positive or negative.
• When analyzing GRF, keep track of the sign convention for vertical forces (upward as positive in the equations) and relate GRF to COM acceleration via F{GRF} = m(g + ay).
• Use the COM as the focal point for gravity and external loading; remember that COM can shift with posture and movement, affecting stability and loading.
• For problem solving, compute acceleration first from velocity change and time, then compute GRF to compare with body weight, and interpret whether GRF is greater or less than $m g$.
• Practice with real data or simulated data: compare predicted GRF curves to measured force-plate data to reinforce understanding of during-activity loading and phases.