Vector Algebra

• Vector

• Kinetic vector quantities – force weight, pressure, specific weight & torque

• Kinematic vector quantities– Displacement, velocity & acceleration

• Scalar quantities– Mass, volume, length & speed

Vector Composition

• Resultant vector

• “Tip-to-tail” vector composition

Vector #1

Vector #2

Resultant vector

Vector Resolution

Vertical

Horizontal

Example: A ball is thrown into the air

Graphic Solution of Vector Problems

• Graphic vector manipulation may yield approximate result

30 N = 3 cm

35 N = 4.5 cm

1 cm = 10 N

Trigonometric Solution of Vector Problems

• A more accurate procedure for quantitatively dealing with vector problems

Vector Resolution Problems(pp 76-81)

• Projectile motion situations– Find horizontal velocity– Find vertical velocity

• Friction problems – Find horizontal force component (Friction)– Find vertical component (Normal)

• First step in adding, or combining vectors– When more than one force is acting on an object– When adding velocity vectors

Vector resolution:

Turning comp = F•d•sinΘRadial comp = F•d•cosΘ(d = d•sinθ)

Vert comp = F•sinΘHoriz comp = F•cosΘ

Θ

Vert comp = F•sin•ΘHoriz comp = F•cos•Θ

Θ Θ

Θ

Θ

d

Vector Addition Problems

• Combining forces– Net effect of two forces applied to any object– What is maximum safe speed for a curve?

• Centrifugal force, frictional force, & gravity

– What makes a curveball curve?• Wind force and weight

• Combining velocities– In crossing a river, what direction is best?

• Velocity of water and swimmer

– In aviation, correcting for wind• air speed and ground speed

Sum of two forces: Sum of two velocities:

(May be deleted if your calculator providesresultant angle in a 0-360 deg system)

Self-test Problems for Chapter 3on Kinetic concepts

• Introductory problems, p 80– #1,2, 5

• Additional problems, p 81– #1, 4, 5, 7

Projectile Motion characteristics and analysis procedures (PP 329-347)

• Objectives– Review characteristics of projectile motion (pp 329-347)– Review factors influencing projectile motion – Present and explain calculation methods for analyzing

projectile motion

Projectile motion characteristics

• It is a freely falling body, subject to only two forces – gravity and wind resistance

• May be dropped, or projected• (1) vertically, (2) horizontally, • or (3) obliquely• Flight pattern is determined by initial velocity

– Speed (V0)– Direction ()

Projectile motion (cont’d)

Projectile motion, cont;d

Projectile motion, cont’d

Projectile motion, cont’d

• Ignoring air resistance, flight pattern is parabolic

Projectile motion, cont’d• Effect of angle or projection on flight pattern

Projectile motion, cont’d• Projectile height plays a role in optimal

projection angle for horizontal distance

Projectile motion, cont’d• If projection angle is constant, range increases

directly with magnitude of velocity

Projectile motion, cont’d

• Resolving projectile motion into vertical and horizontal components – the first step in calculating motion parameters of interest

Vv = V sin Vh = V cos

Projectile motion, cont’d

• Primary motion quantities of interest are usually:– Time of flight

– Zenith, or maximum height

– Range, or horiz dist

– If time is calculated first,

this reduces to: Dhoriz = VhorizT = V cos T

Self-test problems on Projectile motion

• Introductory problems• #6, 7, 8, 9 p 345-346

Additional Problem

#10, p 347

Hint: Solve for time of flight:

then solve for Dhoriz : Dhoriz = VhorizT = V cos T

COM Concepts and calculation procedures

• Obbjectives– To review concept of COM– To review reasons for studying COM– To review calculation method for calculating COM of

human body

Why study Center of Mass of Human Body?

• It simplifies the movement of the complex system so that we can understand its overall motion

• It allows us to easily consider the effect of net external forces (F = ma; a = F/m)

• Example of important issues related to COM• To review reasons for studying COM

– Is COM different for children and adults in a way that affects movement?

– Can an athlete really “hang in the air”– Is COM different for males and females in a way that

affects movement?

COM/COG Calculation Method (pp

436-445: Kreighbaum pp 116-123, 586-587)

• Concept of balancing segmental torques

• Segmental Calculation of COM – General calculation method– Information needed

• Proportionate mass of each segment (p 520)

• location of COM of each segment (p 521)

– Homework problem

Segmental concept of center of mass

Segmental Calculations&

Homework Problem

Segmental concept of center of mass

–Information needed

•Proportionate mass of each segment (p 520)

•location of COM of each segment (p 521)

Summary• This chapter introduced basic concepts related to kinetics• Several types of mechanical loads act on the human body.• The distribution of force within a body structure is termed

mechanical stress.• Vectors quantities have magnitude & direction• Vector problems may be solved by a graphic or a

trigonometric approach. • Kinematic concepts related to projectile motion were

introduced• Calculation methods for total body COM were presented