Biomechanics Notes PDF: Forces, Kinematics, Kinetics

Biomechanics – Notes Forces, Linear Kinematics, and Linear Kinetics Branches of (rigid body) mechanics: - Statics: The mechanics of objects at rest or moving at constant velocity - Dynamics: The mechanics of objects at accelerated motion - Kinematics:...

Biomechanics – Notes Forces, Linear Kinematics, and Linear Kinetics Branches of (rigid body) mechanics: - Statics: The mechanics of objects at rest or moving at constant velocity - Dynamics: The mechanics of objects at accelerated motion - Kinematics: The description of motion - Kinetics: The forces that cause changes in motion Force: - Definition: o A push or pull o Something that can cause an object to start, stop, speed up, slow down, or change direction - SI unit of force is the Newton (N) - 1N is the force required to accelerate a 1kg mass at 1m/s2 o i.e. Force = mass x acceleration (F = ma) - Forces have: o A point of application o A direction o A magnitude Internal vs. External Forces: - Internal: Forces that act within the object whose motion is being investigated - External: Forces that act on an object as a result of its interaction with the environment Internal Forces: - Pulling forces are tensile forces (e.g. a muscle) - Pushing forces are comprehensive forces (e.g. bones) - In the human body, internal forces can change the motion of a body part but cannot cause a change in the motion of the body’s center of mass o An external force is required to change the motion of the center of mass External Forces: - Non-contact forces occur even when objects aren’t touching each other (e.g. gravity) - Contact forces occur between objects touching each other (e.g. friction) Non-contact Force: - Weight: The force of gravity acting on an object - Recall: o Force = mass x acceleration (F = ma) o Weight = mass x acceleration due to gravity (W = mg) Special note on direction: - For simplicity, you should always use the following orientation when answering questions o  + R = (+) o  + L = (-) - Thus, we will “always” treat gravity as a negative o Exception: to solve for potential energy Weight, Mass, and Inertia: - Weight: The force of gravity acting on an object - Mass: The quantity of matter composing an object - Inertia: Property of matter related to the difficulty in changing an objects motion. An object will exist in a state of rest or uniform motion in a straight line, unless acted upon by an external force → Mass is the key: ▪ It is the measure of inertia of a body ▪ It is used to determine weight (F = ma) ▪ It is constant at any location → Weight is dependent on the force of gravity (-9.81, except chapter 4 where gravity is positive) Forces in Pairs: - Forces are paired such that the force exerted by one object on another (action) is matched by an equal and opposite directed force by the second object on the first (reaction) o Newton’s Third Law of Motion Contact Forces: - Forces perpendicular to the surfaces of the objects in contact is called normal contact (reaction) force - In a simple system with a horizontal surface, normal contact force (reaction force; R) is equal and opposite to the weight of the object (+ value) - Force parallel to the contact between two surfaces is called friction Friction: - Friction is the result of the interaction between molecules of the surfaces in contact - Static friction occurs when two surfaces are not moving relative to one another - Dynamic friction occurs when two surfaces are moving relative to one another - Limiting friction refers to the maximal amount of static friction just before the two surfaces begin to move Calculating Friction Force: - Friction force is proportional to the reaction force - Static friction force (FS) = S x R o S = coefficient of static friction (value between 0-1) o R = reaction force - Dynamic friction force (FD) = D x R o D = coefficient of dynamic friction o R = reaction force - Static friction will always be greater than dynamic friction Static vs. Dynamic Friction: - Static friction is greater than dynamic friction o Once something starts moving, it requires less force to keep it moving Coefficients of Friction: - Coefficients of friction () vary by material and can be solved if friction force and reaction force are known: o (FS) = S x R → S = FS / R Friction and Surface Area: - Friction force in unchanged by the surface area in contact (self-ex 1.3/1.4) Self-experiment 1.3/1.4: - The materials in contact haven’t changed so the coefficient of friction won’t change - The weight of the book hasn’t changed so the force from the desk is the same Forces and Linear Kinematics Vector vs. Scalar Quantities: - A vector is a quantity that has a magnitude (size) and a directions (e.g. velocity, acceleration) - In contrast, a scalar is a quantity that has only a magnitude (e.g. time, mass, speed) o Has numbers + size but NO direction Force: A vector - Recall that a force has: o A point of application = tip of arrowhead o A direction = orientation of arrowhead o A magnitude = length of the arrow ▪ Direction and magnitude are represented graphically as a vector (arrow) Net Force: - Movement of an object depends on the sum of all external forces acting on the object, i.e. the net force (F) o If F = 0, the object is at rest or in a state of constant velocity (zero acceleration) and the external force are in equilibrium o If F  0, the object will accelerate in the direction of the net force Vector Addition (Composition): - To sketch a resultant force vector, we add the component vectors tip-to-tail - The net-force on an object is the vector sum of all external forces acting on it o However, the numbers cannot simply be added up because direction must be considered* - The addition of  2 force vectors is called vector composition and will produce a resultant force Vector Addition: - Net force = resultant force, only if all forces are included in the vector addition Adding Co-linear Forces: - Simplest scenario: All forces acting on an object have the same line of action (co- linear forces) o When the vectors are all acting on the same line - Forces may act in the same direction or in the opposite direction o Could go either direction - *The only case of vector addition where forces can simply be added together - However, one has to pay attention to the direction o Choose one direction to represent a positive force o All forces acting in this direction can be added o All forces acting in the opposite direction are added as negative numbers (i.e., subtracted) Adding Concurrent Forces (same spot, different lines of action): - Another scenario of vector addition involves forces which do not have the same line of action but act through the same point - If concurrent forces are directed only horizontally and vertically, we can sum horizontal and vertical forces (separately) to determine the horizontal (FX) and vertical (FY) components of the resultant force - Then, we can use trigonometry to determine the magnitude (Pythagorean theorem) and direction (tan ratio) of the resultant force Static Equilibrium: - If the object is at rest, we know that the net force is equal to zero and the object is described as being in a state of static equilibrium o F = 0 - Knowing all horizontal and vertical forces sum to zero (FX = 0, FY = 0) allows us to solve for unknown forces Free-Body Diagram: - Definition: A mechanical representation (drawing) of the object (body) you are analyzing with all external forces acting on it represented by arrows (internal forces are not shown) o Arrows indicate the point of application, direction, and size of the forces - Always go with the simplest drawing possible (squares, circle, stick figure, etc.) Constructing a Free-Body Diagram: 1. From the question, select the appropriate body or body segment to draw and create your simple drawing 2. Draw all known external forces at their points of application a. Weight vector is drawn as a downward arrow at the center of gravity) 3. Draw all unknown external forces at their point of application a. Unknowns still need to be included in the drawing Steps for Solving Mechanics Problems: 1. Draw free-body diagram of the object of interest (sometimes you’ll need to draw >1) 2. Draw an appropriate axis system that defines the positive directions a. Y/X axis or axis in degrees (0-270) 3. Write out the equation(s) of motion applicable to the problem 4. Expand equation(s) using information from the free-body diagram and solve for unknown(s) 5. Write out the solution(s) with: a. The direction and sign (+/-) based on axes drawn in step 2 b. The correct level of accuracy (1 or 2 decimal places) c. Appropriate units Vector Composition: - Recall that vector composition involved creating a single vector from  2 vectors There are three basic scenarios: 1. The vectors are co-linear and can be added a. Direction of the vectors matter 2. The vectors are only horizontal and vertical (90 to each other) a. Magnitude is found by the Pythagorean theorem b. Direction is found by tangent ratio 3. One or more of the vectors is on an angle a. Angled vectors must be split into horizontal and vertical components by vector resolution (more than one triangle) b. Angled vectors must be split into horizontal and vertical components c. Separately, add all horizontal and vertical components (they are now co- linear) d. Solve magnitude and direction in the same way as #2 Motion: - Definition: A change in position with respect to space and time o I.e., an object moves in a given space and the movement requires a given amount of time (linear, angular, general) Linear Motion (translation): - Definition: All points on an object move the same distance, in the same direction, and at the same time o Rectilinear translation: Motion along a straight line o Curvilinear translation: Motion along a curved path (an arc) Angular Motion (rotation): - Definition: All points on an object move in a circular path about the same fixed axis o E.g., elbow flexion and extension General Motion: - Definition: A combination of linear and angular movements o Most human movements (e.g. walking, running) o When possible, linear and angular motions are analyzed separately o Different mechanical laws govern linear and angular motion Characteristics of Linear Kinematics: - Linear motion is often described in terms of the object’s: o Position o Speed o Distance travelled o Velocity o Displacement o Acceleration Position: - Definition: location in place o Often given relative to a fixed point - Can be described using Cartesian coordinates Distance Travelled: - Definition: Length of the path followed by the object from start to end position - A scalar quantity (no reference to direction) Displacement: - Definition: The straight-line distance in a specific direction from the start to end position - A vector quantity - Like force in the way that: o It is represented with an arrow o The resultant vector (or component vectors) can be determined with trig Distance Travelled vs. Displacement: - May be numerically equivalent (100m race) or wildly different (400m, 800m, 10k races) - One is likely to be more relevant than the other for a given scenario Speed: - Definition: The rate of motion (SI units are m/s) - A scalar quantity (no reference to direction) o E.g., 50km/h Average Speed: - S = distance travelled/change in time = l /t = l / t2 – t1 o So, S = l / t2 – t1 o t1 = starting time (usually 0) o t2 = second time - Winner of a race will always be the athlete/team with the fastest average speed - Doesn’t provide much insight into performance within the race Instantaneous Speed: - Definition: Speed of an object at a given instant - Will fluctuate above and below average speed Velocity: - Definition: The rate of motion in a specific direction - E.g., 50km.h North (SI units still m/s) - A vector quantity - Like force in the way that: o Represented by an arrow o The resultant vector (or component vectors) can be determined with trigonometry Average Velocity: - V = change in position (displacement) / change in time = d / t = p2 – p1 / t2 – t1 - So, V = p2 – p1 / t2 – t1 Acceleration: - Definition: The rate of change of velocity o E.g., 2m/s2 north (SI units are m/s2) o For every second that passes, the velocity will increase 2m/s north, i.e., an object with a velocity of 10m/s [N] will have a velocity of 12m/s [N] after 1s, 14m/s [N] after 2s, etc. - A vector quantity (magnitude and direction) - Like force in the way that: o Represented by an arrow o The resultant vector (or component vectors) can be determined by trig - When an object speeds up, slows down, starts, stops, or changes direction, it is accelerating - “Deceleration” is more accurately called negative acceleration - Direction of motion is not necessarily the same as the direction of acceleration Average Acceleration: - a = change in velocity / change in time = v / t = v2 – v1 / t2 – t2 - So, a = v2 – v1 / t2 – t2 Instantaneous Acceleration: - Definition: Acceleration of an object at a given instant Uniform Acceleration: - Definition: Acceleration of an object is constant - Occurs when net external force is constant o E.g., acceleration due to the force of gravity does not change (g = -9.81m/s2) - Describes the motion of a projectile Projectile Motion: - Definition: Motion of an object which is propelled into the air or dropped and has only the forces of gravity and air resistance acting on it o E.g., any ball in flight - Air resistance is often negligible and ignored so o A = g = -9.81m/s2 - Motion can be described by various equations The parabolic path of a ball is symmetrical about the peak, so vertical velocity has the same value but opposite sign at a given height v2 = -v1 o At peak height (apex), v = 0 Vertical Motion of a Projectile: - Vertical position (y) of a projectile: o y2 = y1 + v1t + ½ g(t)2 - Vertical velocity (v) of a projectile: o v2 = v1 + gt o v2 = v12 + 2gy Vertical Motion of a Dropped Object: - Start position (y1) and start velocity (v1) are = 0, so the equations may be simplified: o y2 = y1 + v1t + ½ g(t)2 → y2 = ½ g(t)2 o v2 = v1 + gt → v2 = gt o v2 = v12 + 2gy → v2 = 2gy Vertical Motion of a Launched Object: - At the highest point (apex), velocity (v 2) = 0, so the velocity equations may be simplified if calculating max height or time to max height: o v2 = v1 + gt → 0 = v1 + gt o v2 = v12 + 2gy → 0 = v12 + 2gy Time to Peak vs. Flight Time: - Time to peak (max height): 0 = v1 + gt - To find flight time, you would just multiply time to peak x2 - OR recall that the parabolic path followed by a ball is symmetrical about the peak so v2 = -v1 o Flight time: v2 = v1 + gt (without making v2 = 0) Choosing the correct equation: Horizontal Motion of a Projectile: - Horizontal velocity of a projectile is constant o v = v2 = v1 = constant - As velocity is constant, then horizontal acceleration must equal zero o a=0 - Horizontal position of a projectile o x2 = x1 + vt - Generally, we make the start position (x1) = 0, so the equation is often simplified to: o x = vt Newton’s First Law of Motion: - Law of Inertia o An object stays in a state of rest, or constant velocity in a straight line, unless acted on by an external force o i.e., v = constant IF F = 0 and vice versa o Applies to the resultant motion (F) but also to the components of motion (Fx and Fy) - Interpretation of the law: 1. If an object is at rest and F = 0, the object must remain at rest 2. If an object is in motion and F = 0, the object must continue moving at a constant velocity in a straight line 3. If an object is at rest, the F must = 0 4. If an object is moving at a constant velocity in a straight line, the F must = 0 - Provides the basis for static equilibrium and also the principle of the conservation of momentum Linear Momentum (L): - Definition: The product of an object’s mass (m) and its instantaneous linear velocity (v) o x2 L = mv x2 o L and v are directionally proportionate - Vector quantity; SI units are kgm/s - The greater the mass of an object or the faster it moves, the greater the momentum - For our purposes, mass is constant so linear momentum is proportional to linear velocity o  if F = 0, both velocity and momentum are constant Conservation of Momentum Principle: - Definition: Total momentum of a system of objects is constant if the F = 0 o Linitial = Lfinal …so… (m1v1)i + (m2v2)i = (m1v1)f + (m2v2)f - Used in the study of collisions between objects Types of Collisions: - Elasticity of collision determined by the coefficient of restitution (e) - Definition: Ratio of the relative velocity after vs. before a collision - e = (v1 – v2)final / (v1 – v2)initial o Where v1 and v2 represent the velocities of objects 1 and 2 Coefficient of Restitution (e): - Reported as an absolute value, i.e., the number without reference to + or – sign - Ranges in value between 0 (perfectly inelastic collision) and 1 (perfectly elastic collision) - Like any coefficient, it has no units - In ball sports, the bounciness of the ball (e) is often regulated by the league - To calculate ‘e’, the ball is dropped onto a specific surface 𝑏𝑜𝑢𝑛𝑐𝑒 ℎ𝑒𝑖𝑔ℎ𝑡 o 𝑒=√ 𝑑𝑟𝑜𝑝 ℎ𝑒𝑖𝑔ℎ𝑡 Perfectly Elastic and Inelastic Collisions: - Perfectly elastic (e = 1) o Objects collide, bounce off each other and their combined momentum and kinetic energy are conserved (e.g., curling rocks, pool table balls) o (m1v1)i + (m2v2)i = (m1v1)f + (m2v2)f - Perfectly inelastic (e = 0) o Objects collide, stay together and move with the same velocity so that their combined momentum is still conserved but kinetic energy is lost (e.g., football players colliding midair) o (m1v1)i + (m2v2)i = (m1 + m2)vf - So, in a closed system, momentum is conserved in all collisions, but kinetic energy is not - Virtually all collision in real life are neither perfectly elastic not perfectly inelastic (plain old elastic) o Some kinetic energy is lost as heat, sound, etc. o Use this equation: (m1v1)i + (m2v2)i = (m1 + m2)vf Collisions thus far: - Colliding objects were considered part of the same system - Net external force (F) = 0 - When F  0, Newton’s First Law no longer holds Newton’s Second Law of Motion: - The Law of Acceleration: o The change of motion of an object is directly proportional to the (net external) force impressed (in the object) and is made in the direction in which the force is impressed (in the object) o F = ma ▪ Where: F = net external force m = mass of the object a = instantaneous acceleration of the object o Acceleration will be directly proportional to the net external force and inversely proportional to its mass ▪ F = ma o Net external force is a vector, so the equation also applies to the component vectors: ▪ Fx = max ▪ Fy = may o Represents a cause-and-effect relationship ▪ Force causes acceleration ▪ Acceleration is the effect of forces o Newton’s 1st law is actually a special case of the 2 nd law where F = 0, so a = 0 To summarize: - Newton’s Second Law (law of acceleration): o Acceleration of an object is directly proportional to the net external force and is in the direction the force is applied o Forces cause acceleration // acceleration is the effect of forces Calculating Vertical Acceleration: - Fy = may - Recall for co-linear forces that F = F1 + F2 +… - If gravity and reaction force are the only forces… o Fy = ground reaction force + object weight negative o So, Fy = R + (-W) - Thus, Fy can be written as: R + (-W) = may - If there is another vertical force (or forces), Fy  0 and a  0, then o Fy = R + (-W) +/- Y and R + (-W) +/- Y = may Calculating Horizontal Acceleration: - Fx = max - If a push or pull & friction are the only forces… o Fx = push or pull force + friction force negative o Fx = Px + (-Ff) - Thus, Fx can be written as: Px + (-Ff) = max - If there is another horizontal force (or forces), Fx  0 and a  0, then o Fx Px + (-Ff) +/- X and Px + (-Ff) +/- X = max Newton’s Second Law of Motion: - To accelerate (move) an object upward, you need to exceed the force of gravity (W) - To accelerate (move) an object sideways, you need to exceed the force of friction (F f; Fs or Fd) - To accelerate an object, Fx < Fy because Ff < W o Recall that FS/d = s/d x R o R is equal and opposite to W (in the absence of other vertical forces) and  has a value of less than 1 o Therefore, Ff will always be less than R (W) Impulse and Momentum: - Newton’s Second Law (F = ma) gives the net external force or acceleration at an instant in time - In sport/human movement, we are more interested in the final outcome of external forces acting over some period of time o i.e., average acceleration caused by an average net force o F = ma *remember to put the line above the F and a to show it is an ‘average’ - Average acceleration is the change in velocity over time o a = v / t → Ft = m(v2 – v1) *F is an average ▪ This equation describes the impulse-momentum relationship ▪ Momentum (L) = mass x velocity ▪ Impulse = applied force x time that force acts (change in momentum) ▪ This is a more useful expression of Newton’s Second Law because an impulse will cause a change in an object’s momentum (i.e., velocity because mass is constant) Impulse to Increase Momentum: - Ft = m(v2 – v1) - To maximise velocity in sports (e.g., throwing or jumping), we try to increase the average net force and/or the time it is applied o Technique: (t; duration of force application) is most important with light objects o Strength: (Faverage; force that can be applied) is more important with heavy objects Impulse to Decrease Momentum: - Ft = m(v2 – v1)  - Two possible ways to minimize impulse: o Ft = m(v2 – v1) or  Ft = m(v2 – v1) - To safely decrease velocity of an object in sports (e.g., landing or catching), we try to minimize the average net force by maximizing the impact time Newton’s Third Law of Motion: - Law of Action-reaction o To every action there is always opposed an equal reaction ▪ I.e., forces exist in pairs - BUT, the effects of these forces are rarely the same because the objects are usually different o i.e., have different masses (measures of inertia) so have different frictional forces (FS = S x R) - When you push or pull on an object, the force you feel is equal but opposite of the reaction force that is pushing or pulling on you  Work, Power, and Energy Mechanical Work (U): - Definition: Product of force (F) and displacement (d) in the direction of that force o U = Fd o The means by which energy is transferred from one object to another - SI unit is the Joule (J) o 1J=1Nm - The equation U = Fd describes the work done by a constant force - In most cases, the force applied is not constant, so average force is used o U = Fd *there is meant to be a line about the F, indicating it is an average - To determine the amount of work done on an object we need to know: o The average force exerted on the object o The direction of the force o The displacement of the object while the force acts Mechanical Work of Muscles: - There are three types of muscle contraction: o Isometric (static) – zero work o Concentric (shortening) – positive work o Eccentric (lengthening) – negative work Mechanical Energy: - Definition: The capacity to do work - Kinetic energy: Object’s capacity to do work due to the motion of the object - Potential energy: Object’s capacity to do work due to the position of the object o Total mechanical energy is KE + PE o SI unit is the Joule (J) o Energy is always positive Kinetic Energy (KE): - Is affected by the mass and velocity of an object o KE = ½ mv2 - A stationary object has no KE (v = 0) - Velocity is squared, so changes in velocity have large effects on KE Momentum vs. Kinetic Energy: - Recall that, in a closed system, momentum is conserved in all collisions - Not necessarily true for kinetic energy as during impact, some kinetic energy is converted to other forms of energy (e.g., sound, heat) Potential Energy (PE): - Definition: Object’s capacity to do work due to the position of the object 1. Gravitation potential energy 2. Strain energy Gravitational PE: - Definition: PE due to an object’s position relative to the Earth - Related to the object’s: o Weight (W) o Height (h) above reference point (usually the ground) - PE = Wh or PE = mgh o Energy must be positive, so gravity is positive Strain Energy (SE): - Definition: PE due to the deformation of an object o E.g., bending pole in pole vault, stretched muscle or tendon - Related to an object’s: o Stiffness constant (k) o Deformation or change of length in meters (x) - SE = ½ kx2 Work-Energy Relationship: - Definition: The work done by external forces (other than gravity) acting on an object causes a change in the energy of the object - U = E → U = KE + PE → U = (KEf – KEi) + (PEf – PEi) o Strain energy won’t be included in a question with KE and PE Doing Work to Increase Energy: - In sports, we often try to change the velocity (KE) of an object - Recall that work = force x displacement - To maximize the increase in KE, a large force needs to be applied over a long distance - In sports, we utilize techniques (flexing lower limb joints on landing) and materials (pole vault landing mat) to perform negative work on an object to absorb some of the kinetic energy by increasing the displacement of the object (athlete) to minimize the impact force o Force being applied is the opposite of the displacement to limit force/stress Conservation of Mechanical Energy: - In the absence of external forces (other than gravity), no work is done, and total mechanical energy is conserved (constant) - U = 0 = E so 0 = (KEf – KEi) + (PEf – PEi) so (KEi + PEi) = (KEf + PEf) o Ei = Ef, initial energy = the final energy o Make sure to put all the initial energy on the outside of the equation (0) - Useful in the motion of projectiles (e.g., dropped objects) o Before being dropped, an object has PE but no KE o As it falls, PE decreased because the height decreases but KE increases because it accelerates due to gravity o The decrease in PE is equal to the increase in KE o The instant before the object strikes the ground, it has no PE because the height is zero ▪ PE decrease = KE increase ▪ KE decrease = PE increase Power (P): - Definition: Rate of doing work - Related to: o Work done on an object (U) o Time taken to do work (t) - P = U / t - SI unit is the Watt (W) Another Definition of (Linear) Power: - Definition: Product of average force and average velocity - P = U / t → P = Fd / t → P = F d/t → P = Fv Muscle Power (P = Fv): - As the velocity of a contraction increases, the maximal force the muscle can generate decreases - Use the next section as an example of this Background: Why study aging? - We all want to get old (and be as healthy as possible) - Based on the 2021 census, there were ~7 million Canadians aged 65 and over, with ~2.5 millions of these 80 years of age - Statistics Canada predicts these numbers will increase by 75-100% during the next 25 years - Many seniors will have a partial or complete loss of independence due to a decline of muscle performance - It is imperative to understand the neuromuscular changes that accompany healthy aging and develop the means to mitigate or reverse such declines Background: Why study power? - It is well established that isometric muscle strength decreases with age - However, muscle power is more functionally relevant because: o Most daily tasks are dynamic o Power is affected by an age-related change in either strength or velocity Dorsiflexor Power in Aging: 1. Measured isometric strength (i.e., force production) a. Peak effort is referred to as a maximal voluntary contraction (MVC) Muscle Strength: - Strength decreases between 7th and 9th decades Dorsiflexor Power in Aging: 2. Measured maximal angular velocity at different loads Angular Velocity: - Measured “isotonically” o I.e., a fixed load is moved through a range of motion as fast as possible - Velocity decreases between 3rd and 7th decades Dorsiflexor Power in Aging: 3. Calculate muscle power at various forces a. P = Fv (linear motion) b. Angular power = torque x angular velocity so P = T Muscle Power: - Power decreases between 3rd and 7th decades Loss of Strength vs. Power: - Power deficits > strength deficits at both ages Maximal Muscle Power: - Maximal power is achieved not at the fastest velocity or the largest force possible but at a compromise between the two variables o Usually ~1/3 of the maximal velocity Sustaining Muscle Power: - As the duration of an activity increases, the power output which can be sustained decreases exponentially Torque and Moments of Force Lever Systems: - Consists of 4 elements: o Lever o Fulcrum (axis of rotation – usually drawn as a circle) o Load o Force - 3 classes that differ by the relative position of the elements above Classes of Levers: Torque (Moment of Force): - Definition: The turning effect produced by a force o Causes a change in angular (rotary) motion o Is a vector quantity Types of External Forces: - Centric Force (linear motion) o External force directed through the centre of gravity of an object o Effect of force is a change in the linear motion of the object as per Newton’s second law o No angular motion - Eccentric force (linear + angular motion) o External force not directed through the centre of gravity of an object o Effect of force is a change in the linear and angular motions of the object (linear as per Newton’s second law) - Force couple (angular motion) o A pair of non-colinear external forces equal in size but opposite in direction o Effect of force couple is a change in the angular motion of the object ▪ i.e., the object rotates ▪ No linear motion o Net resultant force is zero (F = 0) so there is no change in linear motion as per Newton’s first and second laws Moment (Force or Lever) Arm (r): - Definition: The perpendicular (⊥) distance between the line of action of a force and a parallel line passing through the axis of rotation Mathematical Expression of Torque: - Torque (T) is equal to the product of the size of the force (f) and the moment arm (r) o T = Fr - T = 0 for centric forces because the force acts through the axis of rotation (i.e., r = 0) o Reason for the no turning effect - Torque (T) is equal to the product of the size of the force (F) and the moment arm (r) o You can achieve the same T with a large F and a small r or a small F and a large r - Force production in humans is limited so we use large moment arms to increase torque - SI unit is the Nm - Torque acting about the same axis can be added or subtracted algebraically Describing a Torque: - To fully characterize a torque, one must describe o The size of the torque (magnitude or size) o The axis about which the turning effect is created o The direction of the turning effect (clockwise/counterclockwise) - Direction of torques can be: o Positive = counterclockwise o Negative = clockwise Direction of a Torque Vector: - Although torque involves rotation, we still use a straight arrow to represent the vector - Direction of the torque vector is determined with the right-hand thumb rule o Right-hand thumb rule: 1. Identify axis of rotation 2. Imagine you wrap your right hand around the axis of rotation with your curled fingers pointing in the direction of rotation (application force) 3. Your extended thumb points in the direction of the torque vector (perpendicular to the plane of rotation) Torque in Sports: - In any sport where we turn, spin, or swing something (including your body), torque must be created to initiate these turns, spins, and swings Muscular Torque: - Torque produced by muscles is responsible for movement of our limbs about our joints - Line of action (or pull) for a muscle force is generally along the line between the tendons - The moment arm of a muscle changes as the joint angle changes o Maximal at 90 degrees - When muscle force pulls at an angle, only the component of force perpendicular to the bone (lever) produces torque, i.e., the rotary component (same as vertical component from earlier in the term) - The component of force parallel to the bone acts to stabilize or dislocate the joint (i.e., stabilizing or dislocating component – horizontal force) - It is rare for muscle force to dislocate a joint but it can increase or decrease joint stability depending on the particular position Forces and Torques in Equilibrium: - A net force = 0 ensures no change in the linear motion of an object but does not limit the objects angular motion Static Equilibrium: - For an object to be in static equilibrium, external forces must sum to zero (F = 0) AND external torques must sum to zero (T = 0) o T = T1 + T2 + T3 +… o F = (Fr)1 + (Fr)2 + (Fr)3 +… Center of Gravity (CG): - Definition: The point in a body around which its mass or weight is balanced and through which the force of gravity acts - Point at which the entire mass or weight is considered to be concentrated Finding an Object’s CG: - Every object can be considered to be made of smaller elements o E.g., limbs, trunk, head for the human body - Force of gravity acts on all elements and the sum is the object’s weight - Object’s weight acts through the point where the torques created by each elemental weight sum to zero (the CG) - T = (Wr) = (W)rcg o Where: o W = weight of one element o r = moment arm of that element o W = weight of the object o rcg = moment arm of weight of the object (relative to the axis about which the torques are being measured) - If the axis of rotation is not identified, choose any point you like (just keep it the same for both sides of the equation and pay attention to the sign for the moment arm) Center of Gravity of the Human Body: - Easy to estimate in anatomical position - Left to Right: Midline of the body - Back to Front: Through shoulders and hips and just forward of the ankle joints - Top and Bottom: o 55% of standing height for females o 57% of standing height for males o Higher still for infants and toddlers because of their relatively large heads and short legs - CG shifts in the direction of movement of a body part o E.g., abduct (raise to side) your right arm and the CG will move to the right o Size of shift of CG depends on the weight of the body part and the distance it moves - To estimate CG in body positions other than anatomical position… 1. Imagine the body in anatomical position 2. Consider movement of each body segment (limb, head, etc.) separately - In sports, the CG often lies outside the body Stability: - Definition: The capacity of an object to return to equilibrium or to its original position after being displaced - Object stability is affected by: o Height of CG (lower is better) o Size of the base of support (larger the better) o Weight (heavier the better) - Stability is directional, i.e., an object can be more or less stable depending on the direction of the toppling force - If the CG is below the base of support the object will return to the original position after a displacement (stable equilibrium) - If the CG is above the base of support, stability is only maintained if the line of gravity acting on CG is within the base of support Stability and Potential Energy (PE): - CG height is raised as an object is toppled - Increase in CG height leads to an increase in PE - Recall that work (U) is = E, so work is done - The lower the original CG height, the greater the increase in height of CG as the object is toppled - Hence, the increase in PE is greater and more work must be done - The object with the lower CG is more stable - The most stable position for a person or object is the one that minimizes PE - Object is more stable if CG is below rather than above the points of support Angular Kinematics Angular Kinematics: - Definition: The description of angular motion - Important because most human movements are the result of angular motions of limbs about joints Angular Position: - Definition: Orientation of a line relative to another line or plane - Absolute angular position: o The reference line/plane is fixed - Relative angular position: o The reference line/plane is moveable Absolute vs. Relative Angular Position: - Angular positions of forearm relative to horizontal plane? A = 0; B = 90 (absolute) - Angular positions of forearm relative to arm? A & B = 90 (relative) Units of Angles: - To this point we’ve always used degrees - Recall angular motion definition from L5 o All points on an object move in a circular path about the same fixed axis - Using a circle, another measure of an angle is called radians or radius units (rad) o  = arc length / radius so =l/r What’s the deal with Radians? - Radians are related to the movement around a circle, but they are a dimensionless unit o E.g.,  = arc length / radius = 8m / 2m = 4rad - Radians can/will appear in quantities of circular motion (e.g., angular displacement) - With linear motion around a circle, radians drop out when you multiply by the radius o E.g., 30rad/s x 1.25m = 37.5 m/s not 37.5 radm/s Units of Angles: - Relationship between degrees and radians o 360 / circle = 2 rad / circle o Recall   3.14 o 360 = 2 rad 2 rad → rad = 360 / 2(3.14) → rad = 57.3 Converting between Degrees and Radians: - Degrees to radians: Multiply by  / 180 o E.g., 120 x  / 180 = 2.09 rad - Radians to degrees: Multiply by 180 /  o E.g., 5.26 rad x 180 /  = 301.4 Angular Displacement (): - Definition: The change in absolute angular position of a rotating line - The angle formed between the final and initial position of a rotating line o  = final - initial - Like torque, direction is described as clockwise (-) or counterclockwise (+) - Unlike linear motion, we will consider distance travelled and displacement the same o E.g., two clockwise revolutions = -720 (-4 rad) - Number of twists/rotations is critical in sports with judges (diving, gymnastics, etc.) - Angular displacement is the same for all points on a line Linear Distance & Displacement: - Linear distance (arc length) travelled by any point on a line is directly proportional to the distance from the axis of rotation (radius, r) and the angular displacement () in radians o  = arc length / radius → l = r - Linear displacement (chord length, d) travelled by any point on a line is directly proportional to the distance from the axis of rotation (radius, r) but not the angular displacement () o Chord = line joining any 2 points on a curve o ra / rb = da / db Muscle Moment Arms: - Last week, we saw that large muscle forces are needed to produce small torques because muscles attach close to joints (small moment arm) o Mechanical disadvantage for torque production - However, we just saw that a small movement near the joint produces a large linear displacement at the distal end of the limb o Mechanical advantage for linear displacement o Useful with sports equipment as small movements produce large linear displacements of a badminton racquet, putter, hockey stick, etc. Angular Velocity (): - Definition: The rate of change of angular displacement () o  =  / t = f - I / t - A vector quantity o Counterclockwise (+) or clockwise (-) direction - Most common units are rad/s, /s or revolutions per minute (rpm) - Average  is important if we want to know how long it takes for something to rotate from A to B o Relevant to sports with twists/somersaults such as diving, gymnastics, etc. - Instantaneous  is important if we want to know how fast something is rotating at a given time o Relevant to sports where a bat/stick/club strikes an object such as baseball, hockey, golf, etc. Angular Velocity (): - Recall, angular displacement () is the same for all points on a line - Average angular velocity is also the same for all points on a line Average Linear Speed (s *with line above, indicating average): - Recall, linear distance (arc length; l) increases with distance from the axis of rotation (radius length) o l = r - So, for a distal point (A) to undergo the same angular displacement in the same time as point B, it must have faster average linear speed (s) o s = r o Note that  and average  must be in rad and rad/s Instantaneous Linear Velocity (vT): o s = r - At an instant in time this relationship becomes: o VT = r o Where: VT is the instantaneous tangential linear velocity  is the instantaneous angular velocity (rad/s) r is the radius length - A tangent is a line which touches a circle at only one point and is perpendicular to the radius Muscle Moment Arms (cont.): - Small movement near the joint produced 8x greater linear displacement at the wrist - Linear velocity would also be 8x greater o Mechanical advantage for linear velocity o Useful with sports equipment as small movements near axis of rotation produce large linear velocities at business end of bat, stick, club, etc. Angular Acceleration (): - Definition: Rate of change of angular velocity () o  =  / t = f - I / t - A vector quantity (ccw or cw) - Most common units are rad/s2 or /s2 - Angular acceleration occurs when an object: o Spins faster or slower o The axis of spin changes direction Linear Acceleration: - There are two types of instantaneous linear acceleration associated with angular motion: 1. Tangential acceleration (at tangent to circular path) 2. Centripetal acceleration (toward the axis of rotation/towards center) Tangential Acceleration (aT): - aT = r - Where: o  = the instantaneous angular acceleration (rad/s2) o r = is the radius length Centripetal (radial) Acceleration (ar): - To maintain an object on a circular path, centripetal (center-seeking) force is needed o Without it, the object would travel in a straight line at a tangent to the circle (e.g. hammer throw) - Centripetal force is always directed to the axis of rotation, so the direction is constantly changing - Recall, any change in direction = acceleration o ar = VT2 / r (tangential linear velocity) o ar = 2r (angular velocity) o Where: ▪ VT = the instantaneous tangential linear velocity ▪ r = radius length, from center of circle to edge/half diameter ▪  = angular velocity Centripetal Acceleration and Radius Size: - ar = VT2 / r - Sprinters in lanes 1 & 4 of a 200m race have the same tangential linear velocity (V T), but not the same centripetal acceleration (due to smaller radius) - If VT is constant, ar is inversely proportional to r o I.e., smaller radius = greater centripetal acceleration - a r =  2r - Hammer throwers with the same angular velocity () use hammers on chains of 0.75m and 1m, which means centripetal acceleration is not the same (due to larger radius) - If  is constant, ar is proportional to r o I.e., larger radius = greater centripetal acceleration Angular Kinetics Angular Kinetics: - Definition: The causes of angular motion Angular (Rotary) Inertia: - Definition: Property of an object to resist changes in its angular motion - Quantified as moment of inertia (I) - Directly proportional to the mass of an object (e.g., harder to swing a heavy than light bat) - Also affected by distribution of mass relative to axis of rotation (e.g., harder to start or stop swinging a long than short bat) Moment of Inertia (I): - An object is composed of many particles of mass - Thus, mass and distance from the axis of rotation must be considered for each particle - Ia = miri2 o Where: ▪ Ia = moment of inertia about axis ‘a’ through the CG ▪ mi = mass of particle i ▪ ri = radius (distance) from particle i to axis of rotation ‘a’ - Scalar quantity - SI unit is the kgm2 Moment of Inertia (I) Example: Moment of Inertia (cont.): - Measuring a typical object particle by particle is obviously not an option so there is another formula o Ia = mka2 o Where: ▪ Ia = moment of inertia about an axis ‘a’ ▪ m = mass of the object ▪ ka = radius of gyration about axis ‘a’ Radius of Gyration: - Definition: Distance from the axis of rotation to the point at which an object’s mass could be concentrated without changing the angular inertia of the object Linear vs. Angular Inertia: - Linear inertia is dependent only on mass - Angular inertia is dependent on mass (m) and distribution of the mass (r or k) o Ia = miri2 or Ia = mka2 - Distribution of mass has the greater influence o i.e., m x 2 → I x 2 (heavier bat) or r or k x 2 → I x 4 (longer bat) Moment of Inertia – Eccentric Axis: - An object free to rotate about any axis will rotate about an axis through its CG - Objects that we swing aren’t free to rotate and have an axis of rotation thrust upon them o This axis does not pass through the object’s CG so is termed an eccentric axis o Where: ▪ Ib = moment of inertia about axis ‘b’ ▪ Icg = moment of inertia about axis through the CG and parallel to axis ‘b’ ▪ m = mass of the object ▪ r = distance between axis ‘b’ and axis through CG ▪ Ib > Icg Moments of Inertia about Different Axes: - Objects can have multiple moments of inertia (I) because they can rotate about many different axes - Principle axes: o Non-symmetrical objects will have an axis about which the ‘I’ is largest and perpendicular to this is an axis about which ‘I’ is the smallest o 3rd principal axis is perpendicular to these two axes Manipulating Moments of Inertia: - Each axis of rotation has only 1 moment of inertia - The human body isn’t rigid so movement of our body can change the moment of inertia about an axis o Divers, gymnasts, skaters, dancers, etc. regularly manipulate their moment of inertia to perform skills more effectively Moments of Inertia and Linear Velocity: - Recall that linear velocity (VT) increases as distance from the axis of rotation (r) increases o VT = r o VTa > VTb - So, linear velocity (VT) will be greater for a long compared to short implement (greater r) if angular velocity is the same o However, as implement length increases so does its moment of inertia meaning it is more difficult to accelerate the object to achieve the same  o  It is necessary to try and account for this fact (e.g. decreased mass) when increasing implement length Angular Momentum (H): - Recall that linear momentum (L = mv) quantifies the linear motion of an object - Angular momentum quantifies the angular motion of an object - Ha = Iaa o Where: ▪ Ha = angular momentum about axis ‘a’ ▪ Ia = moment of inertia about axis ‘a’ ▪ a = angular velocity about axis ‘a’ - Vector quantity (direction is the same as ) - SI unit is the kgm2/s - Recall that mass (m) doesn’t change so linear momentum (L) is dependent only on velocity - Similarly, for rigid objects, the moment of inertia (I) doesn’t change so angular momentum (H) is dependent only on angular velocity () - For non-rigid objects, both the moment of inertia (I) and angular velocity () can influence angular momentum (H) - Angular momentum (H) about an axis through CG of a multi-segment body can be approximated o Ha  (Hi) so  (Ii/cgi) *total sum of all body parts around center of gravity o Where: ▪ Ha = angular momentum about axis ‘a’ through CG of body ▪ Hi = angular momentum of segment ‘i’ about CG of body ▪ Ii/cg = I = moment of inertia of segment ‘i’ about CG of body ▪ I = angular velocity of segment ‘i’ Angular Momentum of Human Body: - In a symmetrical activity like running, the total angular momentum of the body = 0 o Equal and opposite momenta for left/right arms o Equal and opposite momenta for left/right legs o Zero momentum for the trunk because it is not rotating - Law of Inertia o Angular momentum of an object remains constant unless a net external torque acts on it - For a rigid body (where I is constant), the angular velocity must stay the same - For a non-rigid body, it isn’t necessary for angular velocity to be constant – only that I x  remains constant if no external torques act o i.e., If ‘I’ increases, then  decreases and if ‘I’ decreases, then  increases o Conservation of Angular momentum (T = 0) Angular Momentum and Newton’s 2nd Law: - Law of acceleration (linear: F = ma) o Change in angular momentum of an object is proportional to net external torque acting on it and in the direction of the torque - For a rigid body (constant ‘I’): Ta = Iaa o Where: ▪ Ta = net externa torque about axis ‘a’ ▪ Ia = moment of inertia about axis ‘a’ ▪ a = angular acceleration about axis ‘a’ - Angular acceleration will be directionally proportional to the net external torque and inversely proportional to its moment of inertia o a = Ta / Ia - Law of acceleration o Net external torque exerted on an object is proportional to the rate of change in angular momentum o For a non-rigid body (variable ‘I’): Ta = (H2 – H1)a / t ▪ t = time to change in momentum Angular Momentum and Newton’s 2nd Law (cont.): - Change in angular momentum may be seen as: o A speeding up/slowing down of angular velocity () o A change in the direction of the axis of rotation o A change in the moment of inertia (I) Angular Impulse and Angular Momentum: - Angular impulse-momentum relationship enables assessment of final outcome of external torques acting on an object over time (now using torque) - Linear impulse: Ft = m(v2 – v1) - Angular impulse (applied impulse) = change in angular momentum o Tat = (H2 – H1)a - A large external torque over a long period of time will create change in angular momentum (greatly increase) o Large torques can be created with long moment arms as per ch.5 o Torque application can be prolonged by starting with a large moment of inertia and reducing it to increase angular velocity Angular Momentum and Newton’s 3rd Law: - Law of Action-Reaction o For every torque exerted by one object on another, the other object exerts an equal and opposite torque back on the first object - Effects of the torques are often different because the objects have different moments of inertia o E.g., in elbow flexion/extension, muscles of the arm (biceps/triceps brachii) exert a torque on the forearm which moves the forearm, but the reaction torque causes no/little movement of the arm  Biological Materials and the Neuromuscular System Rigid vs. Deformable Bodies: - We have treated the body as a system of linked rigid segments and used rigid-body mechanics - Now, we will consider the body as deformable with internal forces that resist external forces Stress (): - Definition: The internal force (F) divided by the cross-sectional area (CSA) of the surface (A) on which the internal force acts o =F/A - Units are N/m2 which is the same as a Pascal (Pa) - Large values are often expressed in larger units o MegaPa = 1 million Pa o GigaPa = 1 billion Pa - Stress is inversely related to area over which the force is applied (i.e., A → ; A → ) Self-experiment 1.3: - Friction force is unchanged by the surface area in contact - With the book flat, the area of contact is 10x greater - Pressure = Force  Area - The pressure is 10x less when the book is flat Principal Stresses: 1. Tension () 2. Compression () a. Both are Axial (longitudinal) stresses i. Directed along the long axis of a body (perpendicular to plane of CSA) 3. Shear () a. Transverse stress i. Directed across the long axis of a body (parallel to plane of CSA) - =F/A o CSA of the bone varies along its length and so does stress Tension (Tensile Stress): - Definition: Axial stress as a result of a force that tends to pull apart the molecules that bond an object together - Tensile stress is produced within the object to stop from being pulled apart - An object under tension (forces pulling on both ends) tends to deform by stretching in the direction of the external forces - Stretch is generally directly proportional to the magnitude of the stress - In the human body, tensile loads may cause: o Sprain/rupture of ligaments and tendons o Tears of muscles and cartilage o Fracture of bones Compression (Compressive Stress): - Definition: Axial stress as a result of a force that tends to push molecules of an object more tightly together - Compressive stress is produced within the object to stop from being crushed - An object under compression (forces pushing on both ends) tends to deform by shortening in the direction of the external forces - In the human body, compressive loads may cause: o Bruising of soft tissues o Crushing fractures of bones Shear Stress: - Definition: Transverse stress as a result of a force that tends to slide molecules of the object past each other o =F/A - Shear stress is produced within the object to stop from being skewed - In the human body, shear loads may cause: o Blisters o Joint dislocations o Shear fractures of bones Special Cases of Loading: - Bending: o Creates tensile and compressive stresses on opposite sides of the long axis o Objects will curve as the tension side stretches while the compressive side shortens o In the human body, bending loads may cause: ▪ Fractures of bones (typically on the tension side) - Torsion: o Torques in opposite directions act at each end of an object about the longitudinal axis to create shear stress o Object deforms by twisting o In the human body, torsion may cause: ▪ Spiral fractures of bones Strain: - Definition: Quantification of deformation of a material - Linear strain: o Caused by a change in an object’s length o Produced by tensile or compressive loads Linear Strain (): -  = change in length / original length o = l2 – l1 / l1 o Where: ▪ l2 = is the stretched or compressed length ▪ l1 = is the original (undeformed) length - No units as it is a dimensionless quality (ratio of length to length so units cancel out) - Often given as a % - Negative if l2 < l1 Stress-Strain Relationship: - Explains the behaviour of a material under load - Elastic materials return to their original shape when the load is removed - Elasticity (elastic modulus; E) of a material can be quantified as: o E =  /  o Where: ▪  = the change in stress (Pa) ▪  = the change in strain (unitless ratio) - Expressed in units of pressure, Pa (MPa or GPa) - Elasticity (elastic modulus; E) of a material can be shown graphically as a slope of the stress-strain curve o E =  /  - If the load is large enough, a material remains deformed after the load is removed (plastic behaviour) Mechanical Failure: - Yield strength is the stress level beyond which permanent deformation will occur - Ultimate strength is the maximal stress a material can withstand before failure - Failure strength is the stress art which rupture or breakage occurs o < ultimate strength if there is tearing and then rupture o = ultimate strength if the rupture occurs suddenly Anisotropic Behaviour: - Anisotropic materials have different mechanical properties depending on the direction of the load Mechanical Properties (MP): Bone - Bones carry nearly all the compressive loads experienced by the human body but can resist large shear and tensile loads as well - Bones are strongest in compression and weakest in shear (tension in the middle) MP: Cartilage - Cartilage can withstand compressive, tensile and shear loads - Articular cartilage transmits compressive loads from bone and absorbs some of the force because the elastic modulus is much less than that of bone MP: Tendons and Ligaments - Have high tensile strength but little resistance to compression or shear - Ligaments are less stiff and slightly weaker than tendons MP: Muscle - Can actively contract to produce tension within itself and the structures to which it attaches - Muscle stiffness varies as a function of the number of active contractile elements