Chapter 1-5 Review: Center of Mass, Kinematics, and Forces

Center of Mass

• Definition: center of mass (COM) is the balance point of the body where the mass is effectively concentrated.
• Practical meaning: it is the point you could balance the object on; it can lie inside or outside the physical body depending on mass distribution.
• How COM moves with redistributing mass:
  • As you move some of your mass in a certain direction, the COM shifts toward that direction.
  • Example: arms are a small fraction of total body mass; moving some mass up (into the arms) shifts the COM slightly up.
  • If a large mass moves (e.g., upper body forward and down), the COM can move forward and down, potentially outside the body.
• Specific motion example: abducting the right shoulder moves mass to the right/up, so the COM moves correspondingly to the right/up.
• Predicting future COM position:
  • To know where the COM will be a little into the future, you need the current velocity of the COM and how quickly that velocity is changing (its acceleration).
  • In calculus terms: you need the first derivative (velocity) and the second derivative (acceleration) of position to forecast future motion.
• Mathematical note: the COM position can be expressed as
  \mathbf{r}{COM} = \frac{1}{M} \sumi mi \mathbf{r}i
  where M is the total mass and \mathbf{r}_i are the positions of the individual masses.
• Important qualitative takeaway: the COM is the balance point, and its motion follows the combined motion of all constituent masses.

Kinematics: Position, Velocity, and Acceleration

• Key definitions:
  • Position: \mathbf{x}(t)
  • Velocity: \mathbf{v}(t) = \frac{d\mathbf{x}}{dt}
  • Acceleration: \mathbf{a}(t) = \frac{d\mathbf{v}}{dt} = \frac{d^2\mathbf{x}}{dt^2}
• What you can infer from graphs:
  • If the position-time graph is flat, velocity is zero at that instant.
  • The slope of the position-time curve at any point equals the instantaneous velocity.
  • When the slope is negative, velocity is negative (motion in the opposite direction).
  • When the slope is positive, velocity is positive (motion in the forward direction).
  • Zero velocity occurs where the slope is zero; the position value at that instant is the same regardless of the slope before or after.
• Practical interpretation: to predict future motion from a snapshot, you need to know the instantaneous velocity and the rate of change of velocity (acceleration).
• Additional note: in physics, higher-order derivatives beyond acceleration (jerk, snap, etc.) can be used for more refined modeling, but velocity and acceleration are the standard starting point.
• Common on-ground condition: acceleration is not constant during normal movement because of changing ground reaction forces and posture; gravity provides a constant downward acceleration, but contact forces under the feet are time-varying.
• In the air (flight or thrown objects): once the object is airborne, the dominant external force is gravity (ignoring air resistance for simple models), giving a nearly constant vertical acceleration downward.

Forces on a Moving Body

• Gravity:
  • Force due to gravity: \mathbf{F}_g = m \mathbf{g}
  • For Earth, |g| ≈ 9.81 m/s^2 downward.
• Ground contact and normal forces:
  • When in contact with the ground, there is a normal force and often time-varying contact forces that segment through the body, typically aligned roughly along the leg from the foot toward the hip.
• The “blue/green/orange/yellow” forces visualization:
  • Blue: center of mass location (reference point for the body’s motion).
  • Green: contact forces with the ground (normal and frictional components).
  • Orange: lift force (when applicable, e.g., certain postures or motions that generate upward components).
  • Yellow: drag force (air resistance acting opposite the direction of motion).
• Net effect during motion:
  • The actual path and acceleration result from the vector sum of gravity, contact forces, and aerodynamic forces (drag and lift).
• On-ground vs in-air distinction:
  • On the ground, acceleration is not constant due to changing contact forces.
  • In the air, the primary external force (in the simple model) is gravity; aerodynamic forces may be small or neglected depending on speed.

Drag and Friction with Air (Aerodynamics)

• Drag force: acts opposite to the direction of motion and resists movement through air.
  • Formula: Fd = \frac{1}{2} \rho Cd A v^2
  • Where: (\rho) is air density, (C_d) is the drag coefficient, (A) is the frontal area, and (v) is speed.
• Frontal area and drag coefficient:
  • Ducking down or changing posture can reduce the frontal area (A), thereby reducing drag.
  • The drag force depends on the velocity squared and the drag coefficient, which encapsulates shape/flow characteristics.
• Relevance of drag:
  • For most everyday human movement (e.g., walking, casual running), drag is relatively small.
  • At very high speeds (e.g., maximum sprinting, fast sprint starts), drag becomes more significant and can influence performance and optimal motion strategies.

Lift and its Role in Flight and Sports

• Lift force: acts perpendicular to the direction of motion.
  • Formula: Fl = \frac{1}{2} \rho Cl A v^2
  • Where: (C_l) is the lift coefficient; other terms as above.
• Physical meaning: lift arises from the interaction of moving air with the body's surface, creating a vertical (or perpendicular) force depending on orientation and flow.
• Analogies and applications:
  • Planes and birds rely on lift to rise; wings generate lift as air flows over surfaces.
  • In sports, lift plays a role in the trajectory of rolling or thrown objects depending on spin and air interaction.
  • Spin and the Magnus effect (e.g., spiraling a football) can create lift-like effects that extend range or alter trajectory.
• Spin and trajectory:
  • Spin can stabilize an object in flight and generate lift that modifies the path, sometimes increasing range or altering bend/curve depending on axis of rotation and velocity.
• Practical takeaway:
  • Lift and drag compete and are both controlled by speed, surface orientation, area, and airflow characteristics; athletes and engineers exploit these forces in design and technique.

Takeoff, Launch, and Trajectory Considerations

• Takeoff kinematics:
  • The state at takeoff (position, velocity, and acceleration) forms the initial conditions for projectile motion.
  • The takeoff angle is a key factor in range, but athletes optimize a combination of takeoff speed, angle, and other factors (like height and post-takeoff velocity) to maximize performance.
• Projectile behavior and angle optimization:
  • The ideal takeoff angle for maximum horizontal range in vacuum is 45°, but real-world conditions (height, air resistance, spin, and launch speed) shift the optimal angle.
• Ground-to-air transition:
  • The moment of leaving the ground is when the kinematics of the mass break from contact constraints and can be analyzed with projectile equations.
• Example connections:
  • A sprinter or jumper starts with a propulsion impulse; the resulting COM trajectory and takeoff angle influence the subsequent flight and landing.

Gravitational Interactions and Inter-body Forces

• Mutual gravity between masses:
  • Any two masses in proximity exert gravitational attraction on each other; for the human body and others nearby, this is a small but conceptually important interaction.
• Direction of gravitational force within the body:
  • The line of action of gravity is typically toward the center of mass; in a standing posture, this line passes roughly through the feet up to the COM.
• Implications for posture and balance:
  • Effective control of COM relative to the base of support is crucial for balance, stability, and maneuverability.
• Ground exchange and internal forces:
  • Internal forces move mass within the body, but external gravitational force remains external to the system of interest (the body).

Practical Considerations and Modeling Assumptions

• When analyzing human motion:
  • On the ground: acceleration is not constant due to changing contact forces; gravity is constant, but the contact force varies with posture and activity.
  • In the air: if air resistance is neglected, the horizontal acceleration is approximately zero and the vertical acceleration is approximately g downward; real situations include drag and lift modifying these results.
• Simplifications and their limits:
  • Neglecting air resistance yields simpler projectile motion; including drag and lift provides more accurate trajectories at high speeds or with large frontal areas.
  • Modeling a body as a single point mass ignores distribution, rotation, joint constraints, and internal movement but captures the essential COM behavior.
• Practical implications across domains:
  • Biomechanics: understanding COM, balance, and force transmission informs coaching, rehabilitation, and ergonomics.
  • Sports engineering: equipment design (e.g., aerodynamics), ball trajectories, and protective gear considerations rely on drag and lift concepts.
  • Safety and safety training: mastering COM and balance reduces injury risk in dynamic activities.

Connections to Foundational Principles and Real-World Relevance

• Foundational physics:
  • Newton's laws (F = m a) underlie the balance of forces (gravity, ground reaction, drag, lift).
  • Conservation concepts connect to how mass distribution affects COM and motion.
• Real-world relevance:
  • Center of mass management is critical in gymnastics, running, jumping, throwing, and even everyday posture.
  • Aerodynamics informs design in sports equipment (balls, shoes) and in vehicles or aircraft.
  • Understanding derivatives of position helps predict motion, plan movements, and interpret experimental data.
• Ethical/philosophical/practical implications:
  • Modeling choices impose simplifications; recognizing limits prevents overinterpretation of predictions.
  • Data collection and interpretation should consider measurement error, variability across individuals, and safety concerns when applying these concepts to training or design.

Summary of Key Equations and Concepts

• Center of Mass:
  \mathbf{r}{COM} = \frac{1}{M} \sumi mi \mathbf{r}i
• Kinematics:
  • \mathbf{v}(t) = \frac{d\mathbf{x}}{dt}
  • \mathbf{a}(t) = \frac{d\mathbf{v}}{dt} = \frac{d^2 \mathbf{x}}{dt^2}
• Gravity:
  \mathbf{F}_g = m \mathbf{g}
• Drag:
  Fd = \frac{1}{2} \rho Cd A v^2
• Lift:
  Fl = \frac{1}{2} \rho Cl A v^2
• Takeoff and trajectory considerations depend on initial conditions (position, velocity, acceleration) and environmental forces (gravity, drag, lift).