KIN 346 Exam 2
Types of forces
contact and non-contact
Viscosity
fluids resistance to flow (influences coefficient of drag)
Elastic force - K
k refers to stiffness, or the ability of a material to be compressed or stretched -Pre-stretch (countermovement, landing, etc.) allows elastic force to be used
5 degrees to 60 degrees during walking
knee angle fluctuates between 5-60
Projectile Motion refers to
motion in the air
Collinear
points that lie on the same line (can go either direction on this line) -FORCE is rarely collinear
a
(acceleration) gravity (-9.81 m/s/s)
Static analysis
(∑F = 0) Object at one point in time- No motion -used when a system is at rest (not moving) or at constant velocity (no accelerating)
Law of Action-Reaction (Newton's 3rd Law)
-"To every action there is an equal and opposite reaction" -If I step with 100N of force, the ground/Earth opposes with 100N of force -Forces are never in isolation; always act in pairs -∑𝐹_(𝐴 𝑜𝑛 𝐵)=−∑𝐹_(𝐵 𝑜𝑛 𝐴) -This law has different effects on different objects jumper lands, exerts force on the Earth, Earth exerts a force back ^Earth is much larger... it does not accelerate, but human does.
Elastic Force
"When force is applied to a material, the material undergoes a change in its length" written as: 𝐹=𝑘∆𝑠 k = constant of proportionality ∆𝑠 = the change in length of the material
Revolution
# of rotations about a circle -Not common- more for skating, diving, gymnastics ..rotation type movement
What is a force?
-A push/pull on an object that tends to produce motion -Force produces & stops motion -Force is a vector quantity (magnitude & direction). -
Degree
-A unit of measurement for angles -MOST common
Negative Angle
-Add 180 degrees so it can be relative to the horizontal
Static vs Kinetic Friction
-An object must initially overcome the static (standing still) force of friction to move across a surface -Once the block is moving, its motion is easier to maintain -𝜇_𝑘𝑖𝑛𝑒𝑡𝑖𝑐< 𝜇_𝑠𝑡𝑎𝑡𝑖𝑐 𝜇_𝑘 is dependent on the relative speed of the object
absolute angle
-Angle of the inclination of a body segment relative to a fixed reference in the environment -Used to determine orientation in space
Angular Velocity
-Angular velocity and Angular speed are analogous with linear speed and velocity -
Solving for Linear Displacement
-Arm segment length of 0.17 m -Rotate about the elbow angular distance of 0.28 radians -∆s = r ∆θ = 0.17 m* 0.28rad ∆s = 0.048m
Internal work
-Calculation of the total work done resulting from the motion of all the body's segments is called internal work -𝑊_𝑠𝑒𝑔𝑚𝑒𝑛𝑡= ∆𝐾𝐸+ ∆𝑃𝐸 -𝑊_𝑠= ∆(1/2 𝑚𝑣^2 )+ ∆(𝑚𝑔ℎ) Ws work done on a segment; ∆KE is change in linear KE of the center of mass of the segment; ∆PE is change in potential energy of the segment center of mass
Drag
-Component of fluid resistance that always acts to oppose movement *Drag and air resistance are synonymous -Depending on an object's mass/size and velocity it can create 2 different types of flows around itself -Laminar -Separated Flow
JFR components
-Compression force -Shear force -Together equal JFR
Projection Angle
-Determines the shape of its trajectory, or the slope of the parabola -Generally measured from 0o (parallel to ground) to 90o (perpendicular to the ground) -What happens if we project something at 0 degrees? 90 degrees? --Optimal angle of projection depends on the activity
Drag force/ fluid resistance equation
-F_drag = (1/2)(Coefficient of Drag)(Frontal Surface Area)(Fluid Density)(Velocity2) -F_D= 1/2CDAPV2 -Acts in opposite direction of movement (braking force)
Friction
-Force that acts parallel to the surface between two bodies in contact -Force of friction is proportional to the normal force between the surfaces -𝐹_𝑓= 𝜇𝑁 -𝜇: coefficient of friction -N: normal force, force perpendicular to the surface (weight of the block)
contact forces are
-Ground reaction force (GFR) -Joint reaction to force (JFR) -Friction -Fluid resistance (drag force) -inertial force -muscle force -elastic force
Strain energy
-If an object is deformed, it may also store PE -PE due to elastic energy -SE = ½ (k * ∆x2) where k is proportionality constant (depends on the material); and ∆x is distance over which the object deformed
Pressure
-Imagine a 25lb weight plate on your chest... -Now, imagine the pointy end of a 25lb sword placed on your chest? -Pressure is how forces, generally impact forces are distributed -Defined as P = F/A -F = force; A = area the force is being applied over -Units of N/m2 or the pascal (Pa)
Dynamic CoF
-In gait, walking or running, CoF is calculated as 𝜇= 𝐹_𝑦/𝐹_𝑧 -Fz = vertical force, normal force (vertical GRF) -Fy = shear force component
Relationship between linear and angular displacement
-Linear displacement = Radius of rotation * Angular displacement -∆s = r ∆θ
FCDM: Angular Velocity
-Measures instantaneous velocity -Same formula as linear kinematics -ωi = (θ^i+1 - θ^i-1) / (t^i+1 - t^i-1)
Muscle Force
-Muscles can only produce a pulling or tensile force -unidirectional force...can only pull 1 direction -Movements are accomplished/controlled by opposing pairs of muscles biceps brachii and triceps brachii -All muscle groups acting across a joint need to be accounted for -Net Force across the joint
Ground reaction to force (GFR)
-Nearly everything we do creates a GRF -GRF has 3 components x, y, z -x = mediolateral; see it as Fx -y = anterior/posterior; Fy -z = vertical; Fz -Sometimes we'll only examine the z component
Joint Reaction Force (JRF)
-Net force acting across a joint *JRFs do not reflect bone on bone force/contact -Bone on bone force the sum of JRF and the muscle forces pulling the bones together
Vacuum
-No air resistance -2 things that may weigh differently (feather/ball) will fall at the same speed
Inertial Force
-One segment exerts force on another segment and causes the other segment to move -Not caused by muscle contraction -Generally happens in a proximal-to-distal manner in human movement -During swing phase of running the foot begins in a plantarflexed position -As we get to terminal swing the foot is now dorsiflexed -Muscle activity is almost non-existent during swing at the ankle (arms as well)
Lift Force Component
-Perpendicular to drag during movement through fluid -Important in swimming, curveballs, airplanes... -Produced by any break in the flow of an object -Creates pressure differences which cause either a rise or a fall -Small increase in life =max drag -great increase in lift=very small increase in drag
Important characteristics of Force
-Point of application -Line of action
Power equations
-Power= W/T (work = F*D) -Power= (∆(𝐹∗𝐷))/∆𝑡 -Rearranged: P = F * (∆𝐷/∆𝑡) < (D/T = average velocity) -Power = FORCE * VELOCITY
Factors influencing Projectiles
-Projection angle -Projection Velocity -Projection Height
Projection Height
-Projection height is the difference in height between the vertical takeoff position and the vertical landing position -Different vertical takeoff and landing positions influence the shape of the trajectory
Parabolic Trajectory
-Symmetrical paths around the apex -Without air resistance
Falling (surviving) massive heights
-Terminal Velocity due to air resistance. -F_drag = F_weight > No acceleration... Top speed (Terminal Velocity) -Mass and body configuration determine terminal velocity
Inverse Dynamics
-This method begins analysis at the most distal segment (foot) then proceeds proximally -Allows us to determine forces/torques at each joint -Challenges using ID is with assumptions/estimations segment masses; joint center locations, moment of inertia
Projection Velocity
-Velocity at the instant of release determines the height and distance of the trajectory -There are vertical and horizontal aspects to PV, --Both influence height and distance -Horizontal velocity is constant throughout the flight of the projectile (we're ignoring air resistance) Vertical velocity accelerates at -9.81m/s/s -At the apex (top), vertical velocity = 0 since the object is changing direction Does horizontal velocity = 0?
Change of Momentum using F=MA
-When multiplying mass and velocity, we're actually calculating an object's momentum -Momentum is represented by the letter p -Units of kg-m/s -p = m*v -We can rewrite our initial equation of ∑F = ma to: ∑𝐹= ∆𝑝/∆𝑡
BW
-body weight is force so it has LoA and PoA -The BW vector has PoA at the center of mass (COM)(also the point of origin), and LoA between the center of mass (COM) and the center of the Earth
Coefficient of Friction (CoF)
-calculated by 𝜇= 𝐹𝑓/𝑁 -Dependent on what types of surfaces are interacting
Measuring individual muscle forces
-difficult to accurately measure -done by mathematical models or through implanted force transducers
angular displacement
-displacement is the difference between initial and final positions ∆θ = θfinal - θinitial
Law of Gravitational Force (non-contact)
-force of gravity is inversely proportional to the square of the distance between attracting objects and proportional to the product of their masses -𝐹= (𝐺𝑚_1 (𝑚_2))/𝑟^2 -G = gravitational constant (6.674x10^-11 M^3 kg^-1 s^-2) -m1 = mass of one object -m2 = mass of the other object -r^2 = (distance between the mass centers of the objects)^2
Density
-mass per unit of volume -greater density=greater resistance for movement
Rotation
-measured positive or negative -"right hand rule" -Positive rotation - counterclockwise -Negative rotation - clockwise
Two components of acceleration produced by rotation of a segment
-one is tangential to the path of the segment, -one along the segment towards the axis of rotation
Law of Acceleration (Newton's 2nd Law)
-reaction is equal and opposite reaction "The change of motion is proportional to the force impressed and is made in the direction of the straight line in which that force is impressed" ∑F = ma ∑𝐹= (𝑚(∆𝑣))/∆𝑡
Force: Point of Application (PoA)
-specific point where the force is being applied to an object -Often determines whether it will result in angular or linear motion, or both -PoA of muscular force is at the insertion point -In gait, PoA for non-muscular force is the contact point of the foot with the ground
Force: Line of Application(LoA)
-straight line of infinite length in the direction in which the force is acting -LoA has horizontal and vertical components -LoA is especially important to understand angular motion -Angle of pull
Weight of the object
-the attractive force of the earth on an object -𝑊𝑒𝑖𝑔ℎ𝑡= 𝐹_𝑔=(𝐺𝑚_𝑜𝑏𝑗𝑒𝑐𝑡 (𝑀_𝑒𝑎𝑟𝑡ℎ))/𝑟^2 -Weight = mass*gravity -BW is measured in N ... body mass is measured in kg
Trajectory of a Projectile
-the flight path of a projectile - Constant acceleration of (-9.81m/s/s)
Radian
-the measure of an angle at the center of a circle described by an arc to the length of the radius of the circle -considered most appropriate unit of measurement for angular movement - 1 radian= 57.3 degrees -6.28 radians= 360 degrees -3.14 radians = 180 degrees
Object moving through liquid
-the object disturbs the fluid -the amount/magnitude of the disturbance is based on density and viscosity -greater the disturbance greater the amount of braking (energy transfer) of the object/individual
Linear and Angular Acceleration
-αt = α * r -αt = tangential acceleration; -α = angular acceleration; -r = radius -Tangential is perpendicular to the rotating segment -Acceleration: rate of change in the tangential velocity of an object along its curve path
tangential velocity
-αt = α * r Velocity that is parallel (tangent) to a curved path.
Lower extremity joint angles
-𝐻𝑖𝑝: 𝜃_ℎ𝑖𝑝= 𝜃_𝑡𝑟𝑢𝑛𝑘 − 𝜃_𝑡ℎ𝑖𝑔ℎ -𝐾𝑛𝑒𝑒: 𝜃_𝑘𝑛𝑒𝑒= 𝜃_𝑡ℎ𝑖𝑔ℎ − 𝜃_𝑠ℎ𝑎𝑛𝑘 -𝐴𝑛𝑘𝑙𝑒: 𝜃_𝑎𝑛𝑘𝑙𝑒= 𝜃_𝑓𝑜𝑜𝑡− 𝜃_𝑠ℎ𝑎𝑛𝑘−90^𝑜 (+ dorsi, - plantar) --𝐴𝑛𝑘𝑙𝑒: 𝜃_𝑎𝑛𝑘𝑙𝑒= 𝜃_𝑠ℎ𝑎𝑛𝑘 − 𝜃_𝑓𝑜𝑜𝑡+〖90〗^𝑜 (Book) -Reference line is now the segment angle
Two methods of calculating the angle degree
1. Placing a coordinate system at the proximal end of the segment 2. Placing a coordinate system at the distal end of the segment
Projectile Motion Solution steps
1.Draw a picture! 2.Resolve the resultant velocity into components 3.Determine flight time 4.Determine range 5.Determine height
Projectile Motion is acted on by
1.gravity 2.air resistance
Newton (measurement)
1kg =9.81N , 1lb= 4.45N
Kinetic Energy (KE)
=Kinetic Energy = ½ Mass*Velocity2 Energy from Motion
Terminal force of drag
=force of gravity
Centripetal acceleration
Acceleration towards the axis/center of rotation -also known as radial acceleration -αc = ω^2 * r
Angular Motion
All parts of a body move through the same angle, but NOT the same linear displacement -rotation about an axis
energy and types of energy
Capacity to do work -kinetic and potential
How can cats survive 50 Foot Falls?
Cats are much lighter, resulting in much slower terminal velocity! Therefore, after a few stories, the cat will hit the ground at the same despite increasing fall heights.
Which way is an angle measured?
Counterclockwise from reference linef
External Work
Defined by the work done by a body on an object Work done by the body to elevate the total body CoM while walking up an incline Generally calculated on a treadmill as: External work = BW * treadmill speed * % grade * duration If % grade = 0, no external work (vertical) has been done...
Work
Force applied over displacement F*D -If we push a 100 N block a distance of 10 meters, our product is 100 N * 10 m = 1,000 N-m -Units for work is the joule (J) -1 Newton-Meter (N*m)
Impulse
Force applied over period of time F*T Apply Newton's second Law F = MA; --A = ∆V/∆T F = M(∆V/∆T) F∆T = M∆V ; [IMPULSE]=∆[MOMENTUM] Relationship --F∆T = ∆(MV) if system mass is changing(collisions, etc.) - N*s -- kg*m/s
Angles
Made of two lines, two planes or a combination that intersects at a point called VERTEX of AXIS (origin)
Potential Energy (PE)
Potential Energy = Mass*Gravity*Height Energy from position
Special Force Applications: Centripetal Force
Radial force occurring along a curved path that generates acceleration -If a runner is moving around a curved path, there will be a mediolateral frictional GRF centripetal force
Lines
Represented by body segments
Axis
Represented by joint center
Why do we solve for s (arc length)?
Since we probably know the angle the segment moved through, and the radius, we need to solve for the arc length, or the amount of distance traveled at a point
Work energy theorem
The work done on an object equals the change in kinetic energy of the object -Work = ∆KE = ½ MVfinal2 - ½ MVinitial2
Laws of Motion
Three laws formulated by Sir Issac Newton that describe how objects move in relation to the forces acting on them.
Graphic Representation
Y angle changed due to X angle
Law of Inertia; Newton's First Law
an object at rest stays at rest AND an object in motion stays in motion unless acted on by an outside force -Inertia of an object is used to describe its resistance to motion -Inertia-directly related to objects mass -Greater mass= greater inertia>greater force needed to move the object -For an object to move inertia of object must be overcome -Motion does not change without acceleration -∑F = 0 then ∆v = 0 -Object will move at constant velocity in strait line
Dynamic analysis
analysis (∑F = ma; a ≠ 0) -Object in motion -Time is present -need at account for acceleration and inertial properties of segments
Radians
are dimensionless
Angular speed
distance traveled per unit of time
s_f
final position
V_f
final velocity
t
half of total flight time
separated flow
higher velocities cause the flow of fluid around it to separate causing turbulence
V_i
initial velocity
Gravity
is a variable force resulting in a constant acceleration towards the center of the Earth of 9.81 m/s/s -Newton/Einstein (theory of relativity)
Static friction
is greater than kinetic force
Air resistance
is inconsistent
Rotational Friction
resistance to rotational or twisting movements
Most important aspect of impulse
the amount of time the force is applied to an object's momentum
Relative Angles
the angle between the longitudinal axis of two ore more segments ... also referred to as the joint angle -determine the amount of flexion/extension of a segment about a joint -DOES NOT give indication of orientation in space like absolute angles
Terminal Velocity
the constant velocity of a falling object when the force of air resistance is equal in magnitude and opposite in direction to the force of gravity
laminar flow
uniform flow around the object not creating any turbulence
Linear and Angular velocity
v=rw -LV vector is instantaneously tangential to the path of the object (tangential velocity) -Linear velocity = radius x angular velocity -(∆s / ∆t) = r (∆θ / ∆t) -vt = r * ω
s
vertical displacement
Equations of constant acceleration motion
vf = vi + at sf = vit + 0.5at2 vf2 = vi2 + 2as sf = abs((0.5*a*t)2)
Origin
where the lines meet
Power
work performed per unit of time W/T -Power accounts for the time it takes for work to be completed --Watts (W) 892 N barbell moved 1.85m overhead Work = 1650.2 J What happens if this movement happens in 0.5 seconds? Power output of 3300.4 J/s (watts) -DECREASE time - INCREASE Power
Angular Acceleration
α = ∆ω / ∆t -Units of deg/s
Conversion of angle to radian
θ = s/r s = arc length r = radius Rearranged to s = rθ
angular velocity
ω = ∆θ/∆time
