Forces and Moments of Force D. Gordon E. Robertson, PhD, FCSB Biomechanics Laboratory, School of Human Kinetics, University of Ottawa, Ottawa, Canada.

Forces and Moments of Force

D. Gordon E. Robertson, PhD, FCSB

Biomechanics Laboratory,

School of Human Kinetics,

University of Ottawa, Ottawa, Canada

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Force

• a push or pull• physical property that causes a mass

to accelerate (i.e., change of speed, v, or direction, )

• vector possessing both a magnitude and a direction and adds according to the Parallelogram Law

• a resultant force is the sum of two or more forces

Fa+b = Fa + Fb

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The Resultant Force

• sum of all external forces acting on a body

• according to Newton’s Second Law the resultant force is proportional to the body’s acceleration. I.e., using a consistent system of units:

F = m a• Where, m = mass in kilograms

a = acceleration in m/s2

F = force in newtons

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Types of Forces

External forces are environmental forces which act on the body or the forces exerted by other objects that come into contact with the body.

Examples:

• gravitational forces especially the earth’s

• frictional forces of surfaces and fluids

• ground reaction forces (includes friction)

• drag (viscous) forces of air (wind) or water

• impact forces of objects

• springs (poles, cables, springboards)

• buoyant force of water

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Types of Forces

Internal forces are forces that originate and terminate within the body. Sum of all internal forces within any body is always equal to zero (zero vector).

Examples:

• muscle forces (through tendons)

• bone-on-bone forces (including cartilage)

• ligamentous forces

• joint capsular forces and skin

• fluid (viscous) forces

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Dynamometry

• measurement of force, moment of force (torque) or power

• torque is a moment of force that acts through the longitudinal axis of an object (e.g., torque wrench, screw driver, motor) but is also used as another name for moment of force

• power is force times velocity (F ∙ v) or moment of force times angular velocity (M

• Examples of power dynamometers are the KinCom, Cybex, BioDex and electrical meters

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Force Transducers

• devices for changing force into analog or digital signals suitable for recording or monitoring

• typically require power supply and output device

• types:– spring driven (tensiometry, bathroom scale)

– strain gauge (most common)

– linear variable differential transformer (LVDT)

– Hall-effect (in some AMTI force platforms)

– piezoelectric (usually in force platforms)

• Examples: tensiometer, KinCom, Cybex, Biodex, spring scale, force platform

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Tensiometer

• essentially a spring-type sensor

• measures tension (magnitude of a force)

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Strain Link

• uses strain gauges to measure tiny length changes in a material that are proporional to the applied force

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Strain Gauge Transducers

• strain link measured forces from a rowing ergometer

• S-link used during underwater weighing to compute body density and lean body mass

strain link transducer

S-link load cell (used in underwater weighing lab)

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Power Dynamometers

potentiometer

strain linklever arm

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Strain Gauge Lever

• uses strain gauges to measure bending moment, which can then be used to compute applied force (Cybex, Kincom, Biodex)

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Force Platforms

• devices usually embedded in a laboratory walkway for measuring ground reaction forces

• Examples: Kistler, AMTI, Bertek• Types:

– strain gauge (AMTI, Bertek)– piezoelectric (Kistler)– Hall-effect (AMTI)

• Typically measure at least three components of ground reaction force (Fx, Fy, Fz) and can include centre of pressure (ax, ay) and vertical (free) moment of force (Mz)

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Kistler Force Platforms

standard

in treadmill

clear topportable

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AMTI Force Platforms

small model

standard model

glass-top model

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Pressure Mapping Systems

• Pedar measures pressures from a matrix of capacitive sensors to display a pressure map

• Tekscan F-Scan measures normal forces using resistive ink sensors to display a force tensor of the pressure distribution

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Moment of Force

• turning effect of a force

• physical property which causes a rigid body to change its angular acceleration

• vector quantity with units of newton metres (N.m)

• also known as a torque or force couple depending upon its application

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The Resultant Moment of Force

• sum of all external moments of force acting on a rigid body

• according to Newton-Euler equations the resultant moment of force is proportional to the body’s angular acceleration. I.e., using a consistent system of units and a particular axis (A, usually at the centre of gravity)

MA = IA where IA = moment of inertia about A in kg.m2

= acceleration in rad/s2

MA = moment of force in N.m

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Types of Moments of Force

External moments are environmental forces which act on the body or the forces exerted by other objects which come into contact with the body.

Examples:

• vertical moment of ground reaction force

• force couple of ground reaction forces

• drill

• eccentric forces

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Types of Moments of Force

Internal moments are moments of force that originate and terminate within the body. Sum of all internal moments of force within any body is always equal to zero (zero vector).

Examples:

• muscle forces (through tendons)

• bone-on-bone forces (including cartilage)

• ligamentous forces

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Moment Transducers

• devices for changing force into analog or digital signals suitable for recording or monitoring

• typically require power supply and output device

• types:– springs

– strain gauge (most common)

– piezoelectric (usually in force platforms)

• Examples: KinCom, Cybex, Biodex, force platforms

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torque transducer(for forearm axial torque)

Strain Gauge Transducers

bending moment(rowing oar lock pin)

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Strain Gauge Lever

• uses strain gauges to measure bending moment and torque (Cybex, Kincom, Biodex)

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Force Platforms

• devices usually embedded in a laboratory walkway for measuring ground reaction forces and moments of force

• Typically measures vertical (free) moment of force (Mz) but can also be designed to measure gripping moments

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Force Platforms

Kistler AMTI

Bertec

In laboratory stairway

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Moments of Inertia

I 2.5 I 5 I

10 I

(a) (b) (c)

(d) (e) (f) (g)

10 I 2.5 I3.5 I

I 2.5 I 5 I

10 I

(a) (b) (c)

(d) (e) (f) (g)

10 I 2.5 I3.5 I

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Quick Release Experiment

• used to measure moments of inertia non-invasively

• assumes no friction in the joints

• performed in the horizontal plane to eliminate gravity effects

• angular acceleration is measured by video analysis or electrogoniometry

force transducer

30.0 cm

150.0 N

= 50.0 rad/s²

(a) before release

(b) after release

axis of rotation

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Law of Reaction

Third Law of Motion

• For every force there must be a reaction force, equal in magnitude but opposite in direction, that acts on a different body (e.g., ground)

F = – R

• to increase the size of an action force you must be able to have an object or surface that can create a large reaction

• action and reaction are arbitrary designations

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Reaction Forces

centripetalreaction

axis ofrotation

180 cm

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Reaction Forces

action

centripetalforce

track

direction ofmotion force

centre of curvature

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Banking of Tracks

• Ground reaction force (Fg) should pass through centre of gravity, otherwise the person rotates (about A/P axis).

• To run the bend, Fg must provide a radial acceleration. I.e.,

Fr = mar = –mvt2/r (recall ar = r2 = vt

2/r)

where r is radius of curvature of the bend

vt is transverse velocity (race speed)

m is mass

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Banking of Tracks

centre ofgravity

weight

normalforce = Fg

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Banking of Tracks

centre ofgravity

weight

normalforce = Fg

weight andnormal causerotation

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Banking of Tracks

centre ofgravity

weight

normalforce = Fg

weight andnormal causerotation

ankle strain

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Banking of Tracks

r

= – mvt2/r

centripetal force

+

y

add banking

Ideal angle of banking = = tan-1(vt2/rg)

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Banking of Tracks

r

= – mvt2/r

centripetal force

+

y banking permitsa normal forcethat acts throughcentre of gravitycausing norotation

Ideal angle of banking = = tan-1(vt2/rg)

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Banking of Tracks

• since = s / r thus r = s /

• for sprinting at 10.0 m/s

• 100 bend: r = 100 / metres

= tan-1 [102 / (31.8 x 9.81) ]

= 17.8 degrees

• 50 m bend: r = 50 / metres

= tan-1 [102 / (15.9 x 9.81) ]

= 32.6 degrees