Week 11 – Linear Kinetics – Relationship between force and motion

• Read Chapter 12 in text• Classification of forces• Types of forces encountered by humans• Force and motion relationships

– Instantaneous effect – Newton’s law of acceleration (F=ma)– Force applied through time (Impulse-momentum)

• Conservation of Momentum

– Force applied through distance (work-energy) • Conservation of Energy

• Problems– Introductory problems, p 411: 1,3,5,7,8,10– Additional problems, p 412: 6,8,9

Classification of Forces

• Action vs reaction

• Internal vs external

• Motive vs resistive

• Force resolution – horizontal and vertical components

• Simultaneous application of forces – determining the net force through vector summation

Types of external forces encountered by humans

• Gravitational force (weight = mg)

• Ground Reaction Force (GRF)(Figure 12-4, p 386)– Vertical– Horizontal (frictional)

• Frictional force (coefficient of friction) (pp 389-395)

• Elastic force (coefficient of restitution) (pp 399-402)

• Free body diagram - force graph (p 63)

Force Plates – Measurement of ground

reaction forces

While walking

Cfr = Frf /Nof

Sample Prob# 2, p 392

Coefficient of restitution: Sample problem #5, p 402

Free body diagrams:

Instantaneous Effect of Force on an Object

• Remember the concept of net force?• Need to combine, or add forces, to

determine net force • Newton’s third law of motion (F = ma)• Inverse dynamics – estimating net forces

from the acceleration of an object• Illustrations from Kreighbaum: Figures F.4,

F.5, and F.6 (pp 283-284)

Force Applied Through a Time: Impulse-Momentum Relationship (pp 295-399)

• Force applied through a time • Impulse - the area under the force-time curve• Momentum - total amount of movement (mass x velocity)• An impulse applied to an object will cause a change in its

momentum (Ft = mv)• Conservation of momentum (collisions, or impacts)

– in a closed system, momentum will not change

– what is a closed system?

Impulse: areaunder force-time curve

Impulse produces a change in momentum (mV)

Sample problem #4, p 397

Vertical impulse While Running: Area underForce-timecurve

Anterioposterior(frictional) component of GRF: impulseIs area under Force-time curvePositive andNegative impulseAre equal ifHorizontal compOf velocity isconstant

Conservation of momentum: when net impulse is zero (i.e. the system is closed), momentum does not change

Conservation of momentum: is this a closed system?

Sample prob#3, p 396

Force Applied Through a Distance: Work, Power, Energy (pp 403-409)

• Work - force X distance (Newton-meters, or Joules)– On a bicycle: Work = F (2r X N)– On a treadmill: Work = Weightd X per cent grade

• Power - work rate, or combination of strength and speed (Newton-meters/second, or watts)– On a treadmill: P = Weightd X per cent grade/ time– On a bicycle: P = F (2r X N) / time

• What about kilogram-meters/min?• Energy - capacity to do work

– kinetic, the energy by virtue of movement (KE = 1/2 mv2 ) – gravitational potential, energy of position (PE = Weight x height)– elastic potential, or strain, energy of condition (PE = Fd)

Work while pedaling on bicycle:

From McArdle and Katch.Exercise Physiology

Work while running on treadmill:

Note that %grade = tan θ X 100,and tan θ and sin θ are very similar below 20% grade

From McArdle and Katch. Exercise Physiology

Homework: Calculating Power on a Treadmill

• Problem: What is workload (power) of a 100 kg man running on a treadmill at 10% grade at 4 m/s?

• Solution:– Power = force x velocity– Force is simply body weight, or 100 x 9.8 = 980 N– Velocity is vertical velocity, or rate of climbing

• Rate of climbing = treadmill speed x percent grade = 4 m/s x .1 = .4 m/s

– Workload, workrate, or power = 980N X .4 m/s = 392 Watts• Note: 4 m/s = 9 mph, or a 6 min, 40 sec mile

• Calculate your workload if you are running on a treadmill set at 5% grade and 5 m/s.– Answer for 200 lb wt is: 223 Watts

Power running up stairs: Work rate = (weight X vertical dist) ÷ time

Sample prob#6, p 405

Conservation of Energy• In some situations, total amount of mechanical energy

(potential + kinetic) does not change– Stored elastic energy converted to kinetic energy

• diving board• bow (archery)• bending of pole in pole vault• landing on an elastic object (trampoline)

– Gravitational potential energy converted to kinetic energy• Falling objects

• Videodisk on pole vault

Energy conservation – Case I : elastic potential (strain) and kinetic

Potential energy (FD) + Kinetic energy (1/2mv2) remains constant

Energy conservation – Case II : gravitational potential and kinetic

Potential energy(Wh) + kineticenergy (1/2mv2) remains constant

Linear Kinetics Formulae