Chapter 9: Statics, Torque, and Biomechanics

The First Condition for Equilibrium

• Definition of Mechanical Equilibrium: A state wherein a physical system experiences zero linear acceleration and zero angular acceleration.

• The First Condition for Equilibrium:

  • To prevent translational acceleration, the vector sum of all external forces acting on the system must equal zero:     $\sum \mathbf{F} = 0 \quad \text{or} \quad \mathbf{F}_{\text{net}} = 0$

  • In component form along standard Cartesian axes, this condition is resolved into three independent scalar equations:     $\sum F_x = 0$     $\sum F_y = 0$     $\sum F_z = 0$

  • Only external forces are included in these summations; internal forces cancel in pairs via Newton's third law and cannot accelerate the center of mass.

• Static Equilibrium:

  • Occurs when an object is completely motionless relative to an inertial reference frame and remains at rest.

  • Both translational velocity and acceleration are zero ($\mathbf{v} = 0$, $\mathbf{a} = 0$).

  • Vertical force balance example: A person standing motionless on a floor. The upward normal force ($\mathbf{N}$) exerted by the floor balances the downward gravitational force (weight $\mathbf{w}$):     $\mathbf{N} + \mathbf{w} = 0 \implies N - w = 0$

  • Geological example: Massive natural rock arches, such as those in Australia's Kalbarri National Park, remain in static equilibrium over short timeframes because external gravitational and ground reaction forces balance identically to zero.

• Dynamic Equilibrium:

  • Occurs when an object moves with a non-zero, constant velocity ($\mathbf{v} = \text{constant}$, $\mathbf{a} = 0$).

  • The net external force in every spatial dimension equals zero.

  • Moving automobile example:

  • Vertical forces: Downward weight ($\mathbf{w}$) of the vehicle is balanced by the upward normal forces exerted on all four tires ($N_1 + N_2 + N_3 + N_4 = w$).

  • Horizontal forces: The forward driving traction force ($\mathbf{F}_{\text{app}}$) applied by the road surface onto the drive tires is matched in magnitude by opposing resistive forces ($\mathbf{f}$), which include air friction (drag) and rolling resistance:       $\mathbf{F}_{\text{app}} - \mathbf{f} = 0$

• Insufficiency of the First Condition:

  • Satisfying $\mathbf{F}_{\text{net}} = 0$ guarantees zero translational acceleration, but it does not prevent rotational acceleration.

  • Demonstration with a cat spinning a ball

  • Collinear forces: When two equal and opposite forces ($\mathbf{F}$ and $-\mathbf{F}$) are applied along the exact same line of action, $\mathbf{F}_{\text{net}} = 0$ and the stick remains in static equilibrium.

  • Non-collinear forces: When identical forces ($\mathbf{F}$ and $-\mathbf{F}$) act at points separated by a distance perpendicular to their direction, $\mathbf{F}_{\text{net}} = 0$, but a net turning moment is generated. The stick undergoes angular acceleration and rotates rapidly.

  • Conclusion: The first condition is necessary, but not sufficient, to guarantee complete equilibrium.

The Second Condition for Equilibrium and Torque

• Definition of Torque ($\tau$):

  • Torque (the moment of force) measures the turning or twisting effectiveness of an applied force to cause rotation about a designated pivot point or rotational axis.

  • Mathematical formulation of torque magnitude:     $\tau = r_{\perp} F = r F \sin(\theta)$

  • $F$ is the magnitude of the applied force.

  • $r$ is the straight-line distance from the chosen pivot point to the point of application of the force.

  • $\theta$ is the angle between the position vector $\mathbf{r}$ (pointing from pivot to force contact) and the force vector $\mathbf{F}$.

  • $r_{\perp} = r \sin(\theta)$ is the perpendicular lever arm (moment arm), defined as the shortest perpendicular distance between the pivot point and the line of action of the force.

  • SI Unit of Torque: Newton-meter ($\text{N}\cdot\text{m}$). Although dimensionally equivalent to Joules ($\text{J}$), torque is an energy-transfer moment and must never be expressed in Joules.

• Factors Governing Torque Magnitude:

  • Force magnitude ($F$): Directly proportional to torque; doubling the force doubles the torque.

  • Lever arm length ($r$): Increasing the distance from the pivot increases turning effectiveness for a given force.

  • Angle of application ($\theta$):

  • Maximum torque occurs when the force is perpendicular to the radial vector ($\theta = 90^\circ$, $\sin(90^\circ) = 1$, yielding $\tau = r F$).

  • Oblique angles ($\theta < 90^\circ$) yield smaller torque because only the perpendicular component $F \sin(\theta)$ induces rotation.

  • Zero torque occurs when force acts parallel to the position vector ($\theta = 0^\circ$ or $180^\circ$, $\sin(\theta) = 0$), passing directly through the pivot.

  • JUST USE THIS FORMULA: $\tau=rF\sin(\theta)$ , where $\theta$ is found by extending the r vector backwards and measuring the angle between this extended vector and the applied force vector.

  • Direction and sign conventions: Right hand rule

  • Counterclockwise (CCW) torques are conventionally defined as positive ($+\tau$).

  • Clockwise (CW) torques are defined as negative ($-\tau$).

• Dependence on Pivot Point:

  • Torque is not an absolute intrinsic quantity; its value and rotational direction depend directly on the chosen pivot location.

  • A force applied to a body can create a counterclockwise torque relative to a pivot point located at one end, but create a clockwise torque relative to a pivot located at the opposite end.

• The Second Condition for Equilibrium:

  • To prevent rotational acceleration, the vector sum of all external torques acting on the system about any arbitrary pivot point must equal zero:     $\sum \boldsymbol{\tau} = 0 \quad \text{or} \quad \boldsymbol{\tau}_{\text{net}} = 0$

  • In planar statics, this simplifies to setting clockwise torques equal to counterclockwise torques:     $\sum \tau_{\text{cw}} = \sum \tau_{\text{ccw}}$

• Application to a Two-Child Balanced Seesaw:

  • Pivot/fulcrum positioned at the midpoint supporting a board with two children of masses $m_1$ and $m_2$.

  • First condition application:     $F_p - w_1 - w_2 = 0 \implies F_p = m_1 g + m_2 g$     where $F_p$ is the upward contact force provided by the fulcrum.

  • Second condition application (pivot selected at the fulcrum):     $\tau_{\text{ccw}} - \tau_{\text{cw}} = 0 \implies w_1 r_1 = w_2 r_2 \implies m_1 g r_1 = m_2 g r_2$

  • If child 1 is lighter ($m_1 < m_2$), equilibrium requires child 1 to sit farther from the pivot ($r_1 > r_2$):     $r_1 = r_2 \left(\frac{m_2}{m_1}\right)$

Stability and Equilibrium Types

• Center of Gravity (CG):

  • The center of gravity is the single average point at which the entire gravitational weight of an extended object can be treated as acting.

  • In a segmented body (such as a human or doll), each individual limb and section possesses an individual CG; the composite CG represents the mass-weighted spatial average of all constituent centers of gravity.

• Stable Equilibrium:

  • Definition: An equilibrium configuration in which any small displacement generates net restoring forces or torques that drive the system back toward its original equilibrium state.

  • Energy and geometric mechanism: The center of gravity is at a local potential energy minimum; displacing the object raises the CG.

  • Example: A pencil standing vertically on its flat eraser end:

  • When tilted slightly counterclockwise, the normal force shifts to the outer pivot edge of the eraser.

  • The line of action of the weight remains within the eraser's base of support, creating a clockwise restoring torque ($\tau_{\text{cw}}$) that rights the pencil.

  • Threshold of stability: If displaced so far that the line of action of gravity falls outside the contact edge pivot, the gravitational torque flips direction to counterclockwise ($\tau_{\text{ccw}}$), causing the pencil to topple.

• Unstable Equilibrium:

  • Definition: An equilibrium state in which any small displacement produces net forces or torques directed in the same direction as the displacement, moving the system further from equilibrium.

  • Energy and geometric mechanism: The center of gravity is at a local potential energy maximum; any displacement lowers the CG.

  • Example: A pencil balanced vertically on its sharpened point:

  • Both conditions for equilibrium are mathematically satisfied while perfectly vertical ($\mathbf{F}_{\text{net}} = 0$, $\tau = 0$).

  • The contact base is effectively a single infinitesimal point; any minute lateral perturbation shifts the CG past the pivot point.

  • The downward force of gravity immediately generates a torque in the direction of tilt ($\tau_{\text{ccw}}$), accelerating the fall.

• Neutral Equilibrium:

  • Definition: An equilibrium state in which small displacements neither produce restoring nor destabilizing forces or torques; the system remains in equilibrium at any new position.

  • Energy and geometric mechanism: The height of the center of gravity remains completely constant as the object is displaced.

  • Examples:

  • A uniform sphere resting on a flat horizontal surface: As it rolls, its CG remains directly above the point of support at all times, maintaining $\tau = 0$.

  • A pencil resting flat on its side: Demonstrates neutral equilibrium for rolling displacements perpendicular to its cylindrical longitudinal axis.

• Stability in the Human Body:

  • Center of gravity position: Located above the hip joints (the body's central pivots) and positioned over a base of support defined by the feet.

  • Directional stability comparison:

  • Lateral stability: Stable against small sideways tilts, but because the feet are narrowly separated, moderate leaning shifts the CG beyond the base of support.

  • Anteroposterior (front-to-back) stability: Very low passive stability because human feet are relatively short. The body requires constant, active muscular contractions (calves, back, abdominal groups) to adjust balance.

  • Methods to enhance human stability:

  • Lowering the center of gravity (e.g., bending knees, crouching).

  • Expanding the base of support (e.g., spreading the feet farther apart).

  • Avian biomechanics comparison (Pendulum stability):

  • A chicken's center of gravity is located anatomically below its hip joints.

  • The chicken's torso is suspended from above by the hips, functioning mechanically like a hanging pendulum that is intrinsically self-stabilizing.

Applications of Statics and Problem-Solving Strategies

• Step-by-Step Statics Problem-Solving Protocol:

  • Step 1 (System Identification): Clearly isolate the specific physical object or interconnected assembly to be analyzed.

  • Step 2 (Free-Body Diagram Construction):

  • Draw the isolated object with all active external forces represented as vectors located at their precise physical points of application.

  • Exclude internal forces and forces exerted by the system onto external bodies.

  • Step 3 (Coordinate System and Pivot Selection):

  • Establish Cartesian axes, decomposing all non-orthogonal forces into parallel and perpendicular components.

  • Select an optimal pivot point for torque calculations. Choosing a pivot point through which one or more unknown forces pass eliminates their torques ($r_{\perp} = 0$), simplifying the torque balance equation.

  • Step 4 (Apply Equilibrium Conditions):

  • Force balance: $\sum F_x = 0$ and $\sum F_y = 0$.

  • Torque balance: $\sum \tau = 0$ about the chosen pivot point.

  • Step 5 (Algebraic Resolution): Solve the independent system of equations for the unknown quantities and verify physical units and consistency.

• Analysis of a Horizontal Pole Vaulter Grip:

  • An athlete holds a long pole horizontally using a left hand (force $\mathbf{F}_L$) and a right hand (force $\mathbf{F}_R$), supporting pole weight $\mathbf{w}$ at its center of gravity.

  • Configuration 1 (CG centered between hands): Both hands push upward ($\mathbf{F}_L > 0$, $\mathbf{F}_R > 0$), sharing the load such that $F_L + F_R = w$.

  • Configuration 2 (CG positioned near the right hand): The right hand bears almost the entirety of the load ($F_R \gg F_L$), while the left hand exerts a minimal upward stabilizing force.

  • Configuration 3 (CG located outside the grip beyond the left hand):

  • To satisfy $\sum \tau = 0$, the right hand must exert a downward force ($F_R$ downward) to act as a hold-down anchor.

  • The left hand must exert a substantial upward force ($F_L$ upward) greater than the weight of the pole itself: $F_L = w + F_R$.

Simple Machines and Mechanical Advantage

• Principles of Simple Machines:

  • Simple machines transfer and transform mechanical energy by altering the magnitude, direction, or travel distance of an applied force.

  • Work conservation in an ideal (frictionless) machine:     $W_{\text{in}} = W_{\text{out}} \implies F_i d_i = F_o d_o$     where $F_i$ is input force, $d_i$ is input displacement, $F_o$ is output load force, and $d_o$ is load displacement.

• Mechanical Advantage ($\text{MA}$):

  • The operational ratio of output force produced by the machine to the input force applied to the machine:     $\text{MA} = \frac{F_o}{F_i}$

  • Lever arm formulation under static equilibrium conditions ($F_i l_i = F_o l_o$):     $\text{MA} = \frac{F_o}{F_i} = \frac{l_i}{l_o}$     where $l_i$ is the perpendicular lever arm of the input force and $l_o$ is the perpendicular lever arm of the output force.

• Classes and Mechanics of Levers:

  • Nail puller:

  • Large input arm $l_i$ from hand to fulcrum; small output arm $l_o$ from fulcrum to nail claw.

  • Reaction force on the puller from the nail ($F_n$) acts downward, balancing upward force $F_o$ exerted by puller on nail.

  • Since $l_i \gg l_o$, $\text{MA} = \frac{l_i}{l_o} \gg 1$, generating immense extracting forces from modest hand inputs.

  • Wheelbarrow:

  • Pivot is located at the front wheel axle.

  • Load center of gravity ($F_o = w$) is positioned between the axle and the operator's hands.

  • Because the input arm extends to the handles ($l_i > l_o$), $\text{MA} > 1$, allowing heavy cargo to be lifted with reduced upward effort.

  • Shovel:

  • Pivot is located at the rear handle grip (held by the upper hand).

  • Input force ($F_i$) is applied by the forward hand along the shaft.

  • Load ($F_o = w$) is situated at the shovel blade at the distal end.

  • Because $l_i < l_o$, $\text{MA} < 1$. The machine sacrifices force to amplify the speed and displacement of the load.

• Rotational and Continuous Simple Machines:

  • Crank: A lever rotating continuously through $360^\circ$. Designed with a large handle sweep radius $r_i$ relative to axle drum radius $r_o$, yielding high $\text{MA}$.

  • Automobile wheel and drive axle: Input force applied by axle at inner radius $r_i$; output driving force delivered by tire at outer radius $r_o$. Because $r_o > r_i$, $\text{MA} < 1$, optimizing rotational input into linear road speed.

  • Single fixed pulley: Reorients the direction of line tension ($T$) without altering force magnitude; $\text{MA} = 1$.

• Pulley Systems (Block and Tackle):

  • Multiply force through combinations of fixed and movable pulleys suspended in series.

  • Assuming low-friction bearings and negligible cable mass, mechanical advantage equals approximately the integer number of cable segments directly supporting the moving load:

  • Two cable segments supporting load: $\text{MA} \approx 2$, total upward force $F_o \approx 2T$.

  • Three cable segments supporting load: $\text{MA} \approx 3$, total upward force $F_o \approx 3T$.

  • Four cable segments supporting load: $\text{MA} \approx 4$, total upward force $F_o \approx 4T$.

Forces and Torques in Muscles and Joints

• Biomechanical Anatomy and Mechanical Trade-Offs:

  • Tendons attach skeletal muscles to bones at attachment points situated very close to joint pivots.

  • Consequently, muscle perpendicular lever arms ($r_{\perp}$) are extremely short compared to the lever arms of limbs and external loads ($\text{MA} \ll 1$).

  • Functional and evolutionary significance:

  • Muscles must produce enormous internal tensile forces to balance small external weights.

  • A tiny muscular shortening distance produces large linear and angular displacements at the extremity, maximizing speed and range of motion while maintaining compact limb morphology.

• Forearm Analysis: Biceps and Elbow Statics:

  • Quantitative anatomical configuration:

  • Pivot point: Elbow joint.

  • Muscle tendon attachment: Biceps attach to the radius at distance $r_1 = 4.0\,\text{cm} = 0.04\,\text{m}$ from the joint.

  • Forearm mass: $m_a = 2.5\,\text{kg}$, weight $w_a = m_a g = (2.5\,\text{kg})(9.8\,\text{m/s}^2) = 24.5\,\text{N}$, located at forearm CG distance $r_2 = 16\,\text{cm} = 0.16\,\text{m}$.

  • Handheld load (book): Mass $m_b = 4.0\,\text{kg}$, weight $w_b = m_b g = (4.0\,\text{kg})(9.8\,\text{m/s}^2) = 39.2\,\text{N}$, located at distance $r_3 = 38\,\text{cm} = 0.38\,\text{m}$.

  • Triceps muscle: Assumed relaxed.

  • Torque balance about the elbow pivot ($\sum \tau = 0$):     $\tau_B - \tau_a - \tau_b = 0$     $F_B r_1 - w_a r_2 - w_b r_3 = 0$     $F_B = \frac{w_a r_2 + w_b r_3}{r_1}$     $F_B = \frac{(24.5\,\text{N})(0.16\,\text{m}) + (39.2\,\text{N})(0.38\,\text{m})}{0.04\,\text{m}} = \frac{3.92\,\text{N}\cdot\text{m} + 14.896\,\text{N}\cdot\text{m}}{0.04\,\text{m}} = 470.4\,\text{N}$

  • Vertical force balance ($\sum F_y = 0$) to determine elbow joint compressive force ($F_E$):     $F_B - F_E - w_a - w_b = 0$     $F_E = F_B - w_a - w_b = 470.4\,\text{N} - 24.5\,\text{N} - 39.2\,\text{N} = 406.7\,\text{N}$

  • The biceps must exert $470.4\,\text{N}$ of upward tension to support a total downward weight of only $63.7\,\text{N}$, loading the joint with over $400\,\text{N}$ of compression.

• Posture and Spinal Mechanics:

  • Good posture:

  • The center of gravity of the upper body ($\text{CG}_{\text{ub}}$) is aligned vertically over the hip joint pivots.

  • The perpendicular lever arm of the upper body's weight is zero ($r_{w\perp} = 0$).

  • The gravitational torque is zero; back muscles remain relaxed.

  • Poor posture / anterior pelvic tilt:

  • Upper body leans forward, shifting $\text{CG}_{\text{ub}}$ forward such that $r_{w\perp} > 0$.

  • Gravity exerts a continuous clockwise torque about the hips and lumbosacral disc.

  • Back muscles (erector spinae) must exert a continuous tensile force ($F_b$) to supply counterclockwise torque.

  • Because the back muscle lever arm ($r_{b\perp}$) is only a few centimeters, it must exert hundreds of Newtons of continuous tension, inducing chronic muscular fatigue and pain.

• Dynamic Postural Compensation Under Load:

  • Carrying asymmetrical loads shifts the combined system CG, requiring compensatory postural adjustments to keep the line of gravity over the feet:

  • Carrying a child piggyback: Body leans forward.

  • Carrying a heavy shoulder bag: Body leans laterally away from the bag.

  • Carrying books in arms: Body leans backward.

• Spinal Loading During Heavy Lifting:

  • Lifting with a curved, bent back exposes the lumbosacral joint (pivot at the base of the spine) to extreme biomechanical loads:

  • Upper body mass ($w_{\text{ub}}$) acts at $35\,\text{cm}$ horizontal lever arm from the pivot, with the torso inclined $29^\circ$ below horizontal ($61^\circ$ from vertical / $39^\circ$ above horizontal).

  • Heavy box weight ($w_{\text{box}}$) held in hands acts at a horizontal lever arm of $50\,\text{cm}$.

  • Deep back muscles run almost parallel to the spinal column, pulling at an angle of only $10^\circ$ relative to the spine with an effective perpendicular lever arm of just $8\,\text{cm}$.

  • Resulting force amplification:

  • Balancing the gravitational torques requires muscle tension forces ($F_B$) that frequently reach several thousand Newtons.

  • Translational equilibrium requires the lumbosacral vertebral contact force ($F_V$) to balance the vector sum of muscle tension and weights, generating destructive compressive stresses that can rupture or herniate intervertebral discs.

  • Proper lifting posture: Bending the knees while keeping the torso upright aligns the load closer to the body, drastically reducing horizontal moment arms ($r_{\perp}$) and transferring work to the large muscles of the legs.