2.2 Friction, Centripetal Force & Rotational Equilibrium in Biomechanics

2.2 Friction, Centripetal Force & Rotational Equilibrium in Biomechanics

Friction in Biological Systems

Friction is a non-conservative contact force that opposes relative sliding motion between two contacting surfaces.

Static Friction ($f_s$)

Static friction operates between surfaces that are not in relative motion. It is an adaptive force that matches applied lateral force up to a critical threshold:

$$0 \le f_s \le \mu_s N$$ $$f_{s,\max} = \mu_s N$$

where $\mu_s$ is the dimensionless coefficient of static friction and $N$ is the normal contact force. If the applied tangential force is less than $f_{s,\max}$, the object remains static ($a = 0$), and $f_s$ exactly balances the applied force.

Kinetic Friction ($f_k$)

Once the applied force exceeds $f_{s,\max}$, the surfaces slide past one another. The resisting force immediately transitions to kinetic friction:

$$f_k = \mu_k N$$

where $\mu_k$ is the coefficient of kinetic friction. For virtually all materials, $\mu_k < \mu_s$, meaning it requires more force to initiate sliding than to maintain it. Furthermore, $f_k$ is independent of both the contact surface area and the sliding velocity over typical physiological ranges.

Biological Tribology: Synovial Joints

In healthy human diarthrodial (synovial) joints—such as the knee or hip—articular cartilage is capped by a layer of synovial fluid containing hyaluronic acid and lubricin glycoproteins. This creates an extraordinary fluid-film and boundary lubrication system with $\mu \approx 0.001 - 0.01$. In comparison, dry prosthetic joints or unlubricated cartilage exhibit $\mu \approx 0.1 - 0.3$. In osteoarthritis, enzymatic breakdown of synovial proteoglycans elevates friction, increasing shear stress, frictional heating, and cartilage degradation.

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Centripetal Force & Laboratory Centrifugation

When a body moves along a circular path of radius $r$ at constant speed $v$, its velocity vector continuously alters direction toward the center of curvature. This directional change produces a centripetal acceleration ($a_c$):

$$a_c = \frac{v^2}{r} = \omega^2 r$$

where $\omega = v/r$ is the angular velocity in rad/s.

According to Newton's second law, an acceleration requires a net inward force, termed the centripetal force ($F_c$):

$$F_c = m a_c = \frac{m v^2}{r}$$

Centripetal force is not an independent physical interaction; it is the net radial force provided by physical mechanisms such as tension in a muscle or ligament, friction between tires and a track, or gravitational pull.

Ultracentrifugation and Sedimentation

In biochemistry, differential centrifugation separates subcellular organelles, macromolecules, and proteins based on density, size, and shape. The centrifugal field accelerates particles outward relative to the rotating frame:

$$\text{RCF} (\text{Relative Centrifugal Field}) = \frac{r \omega^2}{g}$$

Particles sediment until the buoyant force and viscous drag balance the centrifugal force. The sedimentation rate is expressed in Svedberg units ($1\text{ S} = 10^{-13}\text{ s}$), where ribosomal subunits (e.g., bacterial 70S formed from 50S and 30S subunits) sediment according to their hydrodynamic radius and mass.

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Torque & Rotational Equilibrium in Biomechanics

Torque ($\tau$), or moment of force, quantifies the tendency of a force to induce rotation about a specific pivot point (fulcrum):

$$\tau = r F \sin\theta = r_\perp F = r F_\perp$$

where:

• $r$ is the distance from the pivot point to the line of action of the force (lever arm distance),
• $F$ is the applied force magnitude,
• $\theta$ is the angle between the position vector $\vec{r}$ and the force vector $\vec{F}$.

Torque is maximized when force is applied perpendicularly to the lever ($\theta = 90^\circ$, $\sin 90^\circ = 1$). By standard convention, counterclockwise (CCW) torques are defined as positive ($+$), and clockwise (CW) torques are negative ($-$).

Conditions for Complete Mechanical Equilibrium

A biological structure or body segment is in static equilibrium if and only if two conditions are satisfied simultaneously:

1. Translational Equilibrium: Net external force is zero: $$\sum \vec{F} = 0 \iff \sum F_x = 0, \quad \sum F_y = 0$$
2. Rotational Equilibrium: Net external torque about any chosen pivot axis is zero: $$\sum \vec{\tau} = 0$$

Choosing a pivot point that coincides with an unknown contact force (such as a joint reaction force) eliminates that force from the torque equation because its lever arm is zero ($r = 0$), allowing direct solution for muscle forces.

Musculoskeletal Lever Systems

The human body employs three classes of levers:

1. Class 1 Lever: Fulcrum located between effort and load (e.g., the atlanto-occipital joint of the neck holding the head erect; triceps extending the elbow).
2. Class 2 Lever: Load located between fulcrum and effort (e.g., standing on tiptoes via the gastrocnemius muscle contracting on the calcaneus; fulcrum is the metatarsophalangeal joints). Mechanical advantage is always $> 1$.
3. Class 3 Lever: Effort located between fulcrum and load (e.g., biceps brachii flexing the forearm). Mechanical advantage is always $< 1$.

In the forearm biceps system, the fulcrum is the humeroulnar joint. The biceps tendon inserts approximately $3\text{–}5\text{ cm}$ anterior to the joint, while the hand holds a load $30\text{–}35\text{ cm}$ away. Because $r_{\text{muscle}} \ll r_{\text{load}}$, the muscle must generate a contractile force roughly 8 to 10 times larger than the weight of the load. This anatomical arrangement sacrifices force for speed and excursion: a tiny shortening of the biceps produces a wide, rapid angular displacement of the hand.

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Worked Numerical Example: Biceps Tension and Joint Compressive Force

Problem: A patient holds a $10.0\text{ kg}$ dumbbell in their hand with the forearm horizontal (angle of $90^\circ$ relative to the upper arm). The forearm has a mass of $2.0\text{ kg}$, with its center of gravity located $0.15\text{ m}$ from the elbow joint. The dumbbell is held at a distance of $0.35\text{ m}$ from the elbow joint. The biceps tendon attaches to the radius at a distance of $0.05\text{ m}$ from the joint and pulls vertically upward ($\theta = 90^\circ$). Assuming $g = 10\text{ m/s}^2$, calculate:

1. The contractile force generated by the biceps brachii ($F_m$).
2. The joint reaction force exerted by the humerus on the ulna at the elbow joint ($F_j$).

Step 1: Identify forces and lever arms

• Fulcrum: Elbow joint ($r = 0$).
• Forearm weight: $W_{\text{arm}} = m_{\text{arm}} g = (2.0\text{ kg})(10\text{ m/s}^2) = 20\text{ N}$, downward at $r = 0.15\text{ m}$ (CW torque).
• Dumbbell weight: $W_{\text{load}} = m_{\text{load}} g = (10.0\text{ kg})(10\text{ m/s}^2) = 100\text{ N}$, downward at $r = 0.35\text{ m}$ (CW torque).
• Muscle force: $F_m$, upward at $r = 0.05\text{ m}$ (CCW torque).
• Joint reaction force: $F_j$ acting at the elbow ($r = 0$).

Step 2: Apply rotational equilibrium ($\sum \tau = 0$) about the elbow joint

$$\sum \tau = + (F_m \cdot r_m) - (W_{\text{arm}} \cdot r_{\text{arm}}) - (W_{\text{load}} \cdot r_{\text{load}}) = 0$$ $$F_m (0.05\text{ m}) - (20\text{ N})(0.15\text{ m}) - (100\text{ N})(0.35\text{ m}) = 0$$ $$F_m (0.05\text{ m}) - 3.0\text{ N}\cdot\text{m} - 35.0\text{ N}\cdot\text{m} = 0$$ $$F_m (0.05\text{ m}) = 38.0\text{ N}\cdot\text{m}$$ $$F_m = \frac{38.0\text{ N}\cdot\text{m}}{0.05\text{ m}} = 760\text{ N}$$

The biceps must pull upward with $760\text{ N}$ of force to support a combined downward gravitational load of only $120\text{ N}$.

Step 3: Apply translational equilibrium ($\sum F_y = 0$) to find joint force $F_j$ Assuming downward joint reaction force on the forearm:

$$\sum F_y = F_m - W_{\text{arm}} - W_{\text{load}} - F_j = 0$$ $$760\text{ N} - 20\text{ N} - 100\text{ N} - F_j = 0$$ $$F_j = 760\text{ N} - 120\text{ N} = 640\text{ N}$$

The humerus pushes downward onto the ulna with a massive compressive force of $640\text{ N}$. This illustrates why joint articular cartilage is subjected to high mechanical stresses during routine motor tasks.