Unit 1: Fundamentals of mechanics

Biomechanics applies the laws of classical (Newtonian) mechanics to living tissue, treating bones as rigid levers, joints as pin or roller supports, and muscles as force generators. This unit establishes the mechanical framework first, then applies it to the musculoskeletal system.

• Scope: Statics (bodies at rest or constant velocity) and dynamics (accelerating bodies); this unit is dominated by statics and equilibrium analysis.
• Rigid-body assumption: Bones are modelled as rigid segments so that internal deformation is ignored and only external forces/moments matter.
• SI units: force in newtons (N), moment/torque in newton-metres (N·m), work and energy in joules (J), power in watts (W).
• Vector convention: Forces have magnitude, direction and point of application; moments are computed about a chosen axis with sign (counter-clockwise positive).

II. Newton's Laws

The three axioms governing motion of masses.

A. Statement and conditions

Newton's laws hold in an inertial (non-accelerating) reference frame for point masses or rigid bodies.

• First law (inertia): A body remains at rest or in uniform motion unless acted on by a net external force. Defines the equilibrium condition ΣF = 0.
• Second law: F = ma, where F is net force (N), m is mass (kg), a is acceleration (m/s²). Extended rotational form: M = Iα (M = moment, I = moment of inertia, α = angular acceleration).
• Third law (action–reaction): Forces occur in equal, opposite pairs on different bodies — the basis for joint reaction forces and ground reaction force during standing.

B. Applications and limitations

• Application: Ground reaction force on the foot equals body weight in quiet standing (third law); a 700 N person exerts 700 N down, ground pushes 700 N up.
• Limitation: Fails at speeds near light and at atomic scale; irrelevant to biomechanics but the rigid-body idealisation itself ignores tissue viscoelasticity.

III. Mechanical behavior of bodies in contact

How surfaces transmit force where they touch.

A. Contact forces and friction

Two bodies in contact exchange a normal force (perpendicular) and a friction force (tangential).

• Normal force (N): Reaction perpendicular to the contact surface, e.g. femoral head pressing into acetabulum.
• Friction: f ≤ μN, μ = coefficient of friction. Static μ prevents slipping; kinetic μ acts during sliding.
• Joint lubrication: Synovial cartilage gives μ ≈ 0.003–0.02, far below engineered bearings, minimising wear during articulation.
• Rolling vs sliding: Femoral condyles both roll and slide on the tibial plateau, distributing contact stress.

IV. Work, power and energy relationship

Quantifying mechanical effort and its transfer.

A. Definitions and equations

Work links force to displacement; power is its time rate; energy is the capacity to do work.

W = F · d · cosθ          (work, joules)
P = W / t = F · v         (power, watts)
KE = ½mv²                 (kinetic energy)
PE = mgh                  (gravitational potential energy)

• Symbols: d = displacement, θ = angle between force and displacement, v = velocity, g = 9.81 m/s², h = height.
• Work–energy theorem: Net work equals change in kinetic energy, W = ΔKE.
• Muscle context: Concentric contraction does positive work (shortening under load); eccentric contraction does negative work (lengthening, absorbing energy, as in landing).
• Example: Lifting a 20 kg mass 0.5 m does W = 20 × 9.81 × 0.5 ≈ 98 J; done in 0.5 s requires P = 196 W.

V. Basic concepts of Force, Moments and Torque

Turning effects of forces about an axis.

A. Moment of a force

A moment is the rotational effect of a force about a pivot.

• Definition: M = F × d, where d is the perpendicular (moment) arm from axis to line of action.
• Torque: The moment producing rotation of a segment about a joint axis; identical units (N·m).
• Sign convention: Counter-clockwise positive; opposing muscle and load moments carry opposite signs.
• Muscle moment arm: Small arm (biceps ≈ 3–4 cm at elbow) means large muscle force needed to lift modest loads.

VI. Equilibrium

The state of zero net force and zero net moment.

A. Conditions of equilibrium

A body is in static equilibrium when it neither translates nor rotates.

ΣFx = 0    ΣFy = 0    ΣM = 0

• Translational: Sum of forces in each direction is zero.
• Rotational: Sum of moments about any point is zero.
• First-law link: Equilibrium is the special case a = 0 of Newton's second law.

VII. Analysis of systems in equilibrium

Solving for unknown forces using free-body diagrams.

A. Method of free-body diagrams

Isolate one segment, replace all contacts with force vectors, then apply the three equilibrium equations.

• Steps: (1) sketch segment; (2) mark weight at centre of gravity, muscle force, joint reaction; (3) resolve into components; (4) solve ΣM = 0 first to eliminate the joint force.
• Lever classes: First (fulcrum central, e.g. atlanto-occipital neck), second (load central, e.g. standing on toes), third (effort central, most limb muscles — favours speed over force).
• Example: Holding a 50 N weight in the hand 30 cm from the elbow, biceps arm 3 cm: Fmuscle × 0.03 = 50 × 0.30, so Fmuscle = 500 N — showing the mechanical disadvantage of third-class levers.

VIII. Skeletal joints

The articulations that permit and constrain motion.

A. Structure and classification

Joints determine the degrees of freedom available to a segment.

• Fibrous: Immovable (skull sutures).
• Cartilaginous: Slightly movable (intervertebral discs, symphysis pubis).
• Synovial: Freely movable, fluid-filled — hinge (elbow), ball-and-socket (hip), pivot, gliding.
• Mechanical role: Act as fulcrums for lever systems and transmit compressive joint reaction forces.

IX. Skeletal muscle

The active force generators of the system.

A. Contraction mechanics

Muscle produces tension along its line of action, always pulling, never pushing.

• Force–length relation: Peak active tension at optimal sarcomere length (~2.0–2.2 µm); force falls when over-stretched or over-shortened.
• Contraction types: Isometric (constant length), concentric (shortening), eccentric (lengthening).
• Cross-sectional area: Maximal force ∝ physiological cross-section, roughly 20–40 N/cm².
• Line of pull: Determines the moment arm and hence the torque delivered to the joint.

X. Mechanics of the elbow, shoulder, spinal column, hip, knee and ankle

Region-specific joint loading and muscle balance.

A. Mechanics of the elbow

• Model: Third-class lever; biceps/brachialis provide effort near the fulcrum. Small moment arm yields high muscle and joint reaction forces when carrying loads.

B. Mechanics of the shoulder

• Model: Ball-and-socket with the deltoid abducting the arm. Because the deltoid's moment arm is small at low abduction angles, force can exceed 8–9× the arm's weight; the rotator cuff stabilises the humeral head against upward pull.

C. Mechanics of the spinal column

• Model: A stacked column loaded in compression and bending. Lifting bends the trunk forward, giving the erector spinae a short arm (~5 cm) against a long load arm, so lumbar (L5/S1) compressive forces can exceed several thousand newtons.

D. Mechanics of the hip

• Model: Ball-and-socket bearing body weight. In single-leg stance the abductors balance body weight across the pelvis, so joint reaction force reaches ~2.5–3× body weight; a cane in the opposite hand reduces it.

E. Mechanics of the knee

• Model: Modified hinge with rolling-sliding of condyles. Patella increases the quadriceps moment arm; patellofemoral and tibiofemoral compressive forces rise steeply with knee flexion during squatting.

F. Mechanics of the ankle

• Model: Hinge (talocrural) transmitting body weight to the foot. During push-off the gastrocnemius/soleus act through the Achilles tendon as a second-class lever, with joint forces several times body weight.

XI. Center of gravity

The point representing total body weight.

A. Location and determination

The centre of gravity (CoG) is where the whole weight can be treated as acting.

• Definition: Weighted average of segment masses; x̄ = Σ(mᵢxᵢ)/Σmᵢ.
• Position: In an erect adult, roughly at the level of S2, anterior to the sacrum — about 55% of standing height.
• Shift: Moves with posture and load; raising the arms raises the CoG.

XII. Stability and balance

Maintaining equilibrium against toppling.

A. Determinants of stability

Balance is retained while the CoG's vertical line falls within the base of support.

• Base of support: The area enclosed by contact points; wider stance increases stability.
• CoG height: Lower CoG improves stability (crouching before contact in sport).
• Line of gravity: Must project inside the base; when it exits, an uncorrected toppling moment M = W × d develops.
• Body mass: Greater weight resists disturbing forces, aiding stability.
• Dynamic balance: Standing sways continuously; ankle and hip muscles generate corrective moments to keep the line of gravity centred.