Biomechanics Lecture Notes (1) PDF - Fall 2024 Lecture Notes

KNPE 153: Introductory Biomechanics Lecture 2: Scalars and Vectors Dr. Pouya Amiri Fall 2024 September 5, 2024 1 Update Office Hours 1. Wednesdays 10-11 am 2. Thursdays 2-3 pm Book in advance...

KNPE 153: Introductory Biomechanics Lecture 2: Scalars and Vectors Dr. Pouya Amiri Fall 2024 September 5, 2024 1 Update Office Hours 1. Wednesdays 10-11 am 2. Thursdays 2-3 pm Book in advance here. September 5, 2024 2 Learning Objectives Reference (coordinate) system Scalars Vectors Vector resolution (Trigonometry) Adding and subtracting vectors Dot product of vectors September 5, 2024 3 Scalar vs. Vectors Scalar: defined only by its magnitude It is just one number Examples Mass => m=20 kg Length => l=2m Temperature => θ=30⁰c Vector: defined by its magnitude and its direction Example Push force => Magnitude=100 N ; Direction=Left September 5, 2024 4 Coordinate (or Reference) System 1D x gin 2D x y 3D 2 September 5, 2024 5 Vectors: Methods to Represent Magnitude and direction 𝑉 = 𝑚𝑎𝑔 𝑢𝑛𝑖𝑡 𝑣𝑒𝑐𝑡𝑜𝑟 Y Example 𝑒̂ Magnitude: 10 m vector unit Direction: 𝑒̂ 𝑉 = 10𝑒̂ magnitude µ X September 5, 2024 6 Vectors: Methods to Represent Coordinates in a coordinate system Easier operations Y 𝑒̂ 𝑥 𝑉 = 𝑥, 𝑦 = 𝑦 = 𝑥 𝚤̂ + 𝑦𝚥̂ it i j n y T X September 5, 2024 7 Vector Equality Vectors are equal if: Their magnitude and directions are the same Y Or All their coordinates are the same X September 5, 2024 8 Vector Equality Vectors are equal if: Their components are the same 𝑉 = 𝑥 ,𝑦 Y 𝑉 = 𝑥 ,𝑦 𝑉 = 𝑥 ,𝑦 𝑉 = 𝑥 ,𝑦 equal if the vectors are E X September 5, 2024 9 Vector Resolution Knowing the magnitude and direction of a vector, calculate its individual directional components (coordinates) The component vectors are usually in direction of the main axes of the reference system Y 𝑒̂ 𝑉 = 𝑥, 𝑦 = 𝑥𝚤̂ + 𝑦𝚥̂ X September 5, 2024 10 Trigonometry 𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 𝑠𝑖𝑑𝑒 𝐶𝑜𝑠 𝜃 = → 𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 𝑠𝑖𝑑𝑒 = 𝑚𝑎𝑔𝑛𝑖𝑡𝑢𝑑𝑒 × Cos(𝜃) If ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒 hypotenuse 𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝑠𝑖𝑑𝑒 𝑆𝑖𝑛 𝜃 = → 𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝑠𝑖𝑑𝑒 = 𝑚𝑎𝑔𝑛𝑖𝑡𝑢𝑑𝑒 × 𝑆𝑖𝑛(𝜃) ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒 hypotenuse September 5, 2024 11 Vector Resolution cost 𝑥 = 𝑙 × Cos(𝜃) 𝑉 = 𝑥, 𝑦 → 𝑦 = 𝑙 × 𝑆𝑖𝑛(𝜃) magnitud at i IF Sino a 19 Y 𝑒̂ e y 0 I X justRayyan Y September 5, 2024 12  Vector Resolution: Example Find the coordinates of the vector I xy Y Find e 𝜃 = 35° I X September 5, 2024 13 Vector Resolution: Example Find the coordinates of the vector J Xy LB cosα Y 35cm y 𝜃 = 125° 0 0 180 180125550 35cm cossso 20.1 sin α y esin α X 35cm sinsso 28.7 September 5, 2024 14  2nd Method COSβ Y cost cos Y P 90 B 3g e 𝜃Y = 125° sine X September 5, 2024 15 3rd Method use the angle director writ the positive of X axis y ecoso Y y l sin 0 𝜃 = 125° C X September 5, 2024 16 Vector Resolution: Example Find the coordinates of the vector HI l Y l cos 0 y x e sino 𝜃 = 55° E X September 5, 2024 17 2nd Method β1 35 T i September 5, 2024 18 Vector Resolution: Example Find the coordinates of the vector in the shown coordinate system. ecosa 10in y Lsin α Isoin sin40 y 𝜃 = 65° to β paralell 𝛽 = 25° September 5, 2024 19 Vector Resolution: Example Find the coordinates of the vector in the shown coordinate system. α angle of andvector 0 7 lcos α 1 156in ye esin α in 𝜃 = 65° 𝛽 = 65° September 5, 2024 20 Vector Operations: Scalar Multiplication Mathematical Representation Visual Representation 𝑉 = (𝑥, 𝑦) Y 2 𝑉 = 2 447 4 14 −1.5 𝑉 = 1 Sx 1 Sy X September 5, 2024 21 Vector Operations: Addition Mathematical Representation C 62 𝑉 = (𝑥 , 𝑦 ) 92 𝑉 = (𝑥 , 𝑦 ) 𝑉 + 𝑉 = x tXz Y ty w̅ September 5, 2024 22 Vector Operations: Addition Visual Representation i Fairer September 5, 2024 23 Vector Operations: Subtraction Mathematical Representation 𝑉 = (𝑥 , 𝑦 ) 𝑉 = (𝑥 , 𝑦 ) 𝑉 − 𝑉 = x X2 41 42 I iz September 5, 2024 24 Vector Operations: Subtraction Visual Representation i i fit September 5, 2024 25 Example: 4 2.4 w̅ 16 𝑉 = (−2,3) 1.3 𝑉 = (−3, −3) 82 0.8 Calculate and draw 𝑉 + 𝑉 2.7 w̅ 82 2 3,3 3 S O Y X September 5, 2024 26 Example: 𝑉 = (−2,3) 𝑉 = (−3, −3) Calculate and draw 𝑉 − 𝑉 Y X September 5, 2024 27 Vector Operations: Dot Product Mathematical 𝑉 = (𝑥 , 𝑦 ) 𝑉 = (𝑥 , 𝑦 ) 𝑉.𝑉 = September 5, 2024 28 Example: 𝑉 = (−2,4) 𝑉 = (−3, −3) Calculate 𝑉. 𝑉 September 5, 2024 29 Example In the following figure, calculate 𝑉 + 𝑉 , 𝑉 − 𝑉 , 𝑉. 𝑉 Y 𝜃 = 75° 𝑉 𝑉 𝜃 = 45° X September 5, 2024 30 Continued 2000N FN September 5, 2024 31 Continued 412 w̅ 1.6 1.3 oz 5 6 66 September 5, 2024 32  Questions? September 5, 2024 33  KNPE 153: Introductory Biomechanics Lecture 3: Force Fundamentals Dr. Pouya Amiri Fall 2023 September 10, 2024 1 Update Office Hours 1. Wednesdays 10-11 am 2. Thursdays 10-11am Book in advance here. September 10, 2024 2 Vector Operations: Addition Mathematical Representation 𝑉 = (𝑥 , 𝑦 ) 𝑉 = (𝑥 , 𝑦 ) 𝑉 +𝑉 = September 10, 2024 3 Vector Operations: Addition Visual Representation better Ilike or A September 10, 2024 4 Vector Operations: Subtraction Mathematical Representation 𝑉 = (𝑥 , 𝑦 ) 𝑉 = (𝑥 , 𝑦 ) 𝑉 −𝑉 = September 10, 2024 5 Vector Operations: Subtraction Visual Representation V Cia EE f if September 10, 2024 6 Vector Operations: Dot Product Mathematical dot product makes a scalar 𝑉 = (𝑥 , 𝑦 ) 𝑉 = (𝑥 , 𝑦 ) 𝑉. 𝑉 = XX 4142 September 10, 2024 7 Example: 𝑉 = (−2,4) w̅ 412.4 𝑉 = (−3, −3) 1.6 1.3 Calculate 𝑉. 𝑉 2 38 vi is 1 2,555,3 5.2 12 9.04 s y 30 18 30 41 1 September 10, 2024 8 Example In the following figure, calculate 𝑉 + 𝑉 , 𝑉 − 𝑉 , 𝑉. 𝑉 i so tvicoso.s Y cost 𝜃 = 75° units add 𝑉 in CUsing Icoso 𝑉 𝜃 = 45° 70sin 75 70cos 14 67.6 18.1 X is oppoi.sn 30 3 185 2 September 10, 2024 8 9 2 60  I Example V2 99 4m 13.7m Vi Vj 31.82 31.82 67 6 18.1 1358m 49.9m Vi V2 31.82 31.82 67.6 18.1 1574.1m September 10, 2024 10 Learning Objectives What is force? How do we represent forces? Types of the forces Net force September 10, 2024 11 What is a force? A push or pull exerted by one object on another object SI unit is newton, shown by N British unit is pound, shown by lbf 1 pound = 4.45N not lb September 10, 2024 12 A force can cause 1. No movement balance offorce Deformation 2. Acceleration change in speed, e.g. starts, stop, speed up, slow down change in movement direction September 10, 2024 13 Assumption in Rigid Body Mechanics The forces do not cause any deformation on the body The effect of the forces acting on the body are not affected by changes in the dimensions or shape Common assumption in Biomechanics of Movement Example Two forces that are equal but in opposite direction No movement worrisome Deformation is not considered forces two September 10, 2024 14 Force Characteristics It is a vector, so it has Magnitude: the size of the push or pull Direction: line of action September 10, 2024 15 Types of Forces 1. Contact forces Objects touch each other to exert force Example: Shot putter 2. Noncontact forces Objects do not touch each other Example: magnetic force 3. Gravitational force A type of noncontact forces, from the earth ummmm Basically, weight of the object September 10, 2024 16 Weight Weight: force of gravity acting on an object W = mg W = weight in newtons (N) m = mass in kilograms (kg) g = acceleration due to gravity = 9.81 m/s2 downward m September 10, 2024 17 Calculating Weight What is your weight if your mass is 70 kg? W my 70kg 9.81 m s 686-7 N September 10, 2024 18 Calculating Mass What is the mass of an object weighing 25000 N? W mg me me 2 mi 2548.4kg September 10, 2024 19 Calculating Weight What is your weight on Mars if your mass is 70 kg if 𝑔𝑚𝑎𝑟𝑠 = 3.73 𝑚/𝑠2? 𝑊 = 𝑚𝑔 On Mars Wmors mgmars 70kg 3.73m s 281.1W On earth Weart myearn 70kg asims 681.7W September 10, 2024 20 Contact Forces Forces occur between objects touching each other Contact forces can be resolved into 2 components: 1. Normal component: perpendicular to the two objects in contact 2. Parallel component: parallel to contacting surfaces of two objects September 10, 2024 21 Example: Contact Forces What are the forces between the sled and the floor? F=250 N September 10, 2024 22 Example: Contact Forces Isolate the object Identify the forces applied to it w enema September 10, 2024 23 Example: Contact Forces Isolate the object Identify the forces 1. Contact forces September 10, 2024 24 Example: Contact Forces Isolate the object Identify the forces 1. Contact forces 2. Non-Contact forces September 10, 2024 25 Friction: What is it? Present when two surfaces are in contact Opposes sliding or motion between objects in contact Parallel to the contact surfaces Two types of friction Static friction Dynamic friction September 10, 2024 26 Friction Static friction Two surfaces not moving on each other Exist between the two surfaces when there is no movement How much is it? in w F m m In EEE.IE September 10, 2024 28 Friction Static friction Reaches its maximum equal to: 𝐹 = static friction force 𝐹 =𝜇 𝐹 𝜇 = coefficient of static friction give Iudto calculate 𝐹 = normal force E F I m F m Ismax In F Fsmax FN W Fsmax MsFn Msw September 10, 2024 30 Friction Static friction is not always equal to the maximum static friction force! Only when the object is about to move! 2000W SEEN September 10, 2024 31 Friction Dynamic friction Two surfaces are moving on each other Exist between the two surfaces when there is relative movement between them I m 𝐹 =𝜇 𝐹 𝐹 = dynamic friction force 𝜇 = coefficient of dynamic friction 𝐹 = normal force Im 1s no movement in the y direction FN W Fd An subineq Fd September 10, 2024 33 Friction: Example A 100 kg box sits on the warehouse floor. The coefficient of static friction is 0.25. If a dock worker applies a horizontal force of 200N, will the box slide? 𝜇 = 0.25 F1=200N object starts to move if static friction becomesmaximum 100kg Fsmax MsFN FEW 90kg 9.81m if f F may objectstarts Emax Fw tomove September 10, 2024 34 Friction: Example A 100 kg box sits on the warehouse floor. The coefficient of static friction is 0.25. If a dock worker applies a horizontal force of 200N, will the box slide? 𝜇 = 0.25 F1=200N September 10, 2024 35 Friction: Example If the box is not moving, how much is the value of the static friction? 𝜇 = 0.25 F1=200N Fs F 200N Fn September 10, 2024 36 Friction: Example If the box is sliding and the dynamic friction coefficient is 0.1, how much friction is applied to the box? 𝜇 =0.1 F1=400N September 10, 2024 37 Friction: Example If the box is sliding and the dynamic friction coefficient is 0.1, how much friction is applied to the box? bK movingdynamic F Fd 𝜇 =0.1 Fd 0.1 looks iims F1=400N Fd TdFm units movement in w no Fw TEN September 10, 2024 38 Net Forces Net force Sum of all external forces acting on one object Also called resultant force Not simply summing all external force magnitudes Must account for the direction of each external force Follows vector summation rules What is the effect of two 100 N forces on an object? Depends on the specified direction of the two forces September 10, 2024 39 Net Forces: Colinear Forces tw Barbell i I sum of the forces in direction or net forces orresultant forces y Éf Fez w September 10, 2024 40  Questions? September 10, 2024 41  KNPE 153: Introductory Biomechanics Lecture 4: Free-Body Diagrams Dr. Pouya Amiri Fall 2024 September 12, 2024 1 Updates Midterm- Will be held in Kingston Hall​ 201 and Etherington Auditorium You will be divided into two groups. Office hours If you book and don’t show up, there will be 0.5% deducted from your final grade per booking. Do not send email to my personal email. Send your emails to: [email protected] Tutorial 1 answer key is available in onQ. September 12, 2024 2 Net Forces Net force Sum of all external forces acting on one object Also called resultant force Not simply summing all external force magnitudes Must account for the direction of each external force Follows vector summation rules What is the effect of two 100 N forces on an object? Depends on the specified direction of the two forces September 12, 2024 3 Net Forces: Colinear Forces Barbell September 12, 2024 4 Net Forces: General Case Block B weighs 10 kg. 𝐹 , 𝐹 , and 𝐹 are 100,150, 50 N. What is the resultant force acting on B? net forces Fx F F 𝐹 1 𝐹 Sumi offorefan 100N SON B Et er Im 𝐹 10kg 9.81ms If I Net force magnitude EFFEY EF.EE 51.9T 502 Direction 72 I N instrof EF September 12, 2024 lady o 5 Net Forces: General Case Example: forces acting on a shot during the shot-put 100 N @ 60° from shot-putter’s hand Mass of shot = 4 kg What is the resultant force on the shot? in iii iii soon 2000W EEf.f.in too 1264 September 12, 2024 taken  Force Components on the Shot agitate frontact Fx Fsncoso 188 60 1mg Fy Fsnsino Msn9 60 4199.81miss 100Nsin amino 47.4N mater EF FEET 68.7N T.EE tail 43.50 NOT 0 September 12, 2024 7 Net Forces on the Shot ming Fx Fsncost 500 Fsnsin many Fy 1000560 4.9.81 y 39.24 826 7 September 12, 2024 8 Free-Body Diagram First step toward analysis of movement of objects It provides a graphical representation of the object/system of interest with all the external forces acting on it 1. It allows to investigate an object in static equilibrium using Newton’s 1st law 1. Static 2. Constant velocity 2. It allows to investigate accelerated movement of an object using Newton’s 2nd law September 12, 2024 9 Method to Construct Free-Body Diagrams 1. Draw the object of interest in isolation 2. Include all the forces acting on the object as vectors 1. Contact forces: Count the number of the objects in contact with your object. There are two components from each external object: a) Normal component is touched b) Parallel component 3 persurface 1. Noncontact forces: Weight 3. Based on the movement direction and forces, choose a coordinate system that makes the calculations easier 4. Find the sum of all forces in all directions September 12, 2024 10 Example 1: A Sled on Ice Only a horizontal force is applied. Draw the FBD of the sled. astatic Fs 𝐹 Wsomsg is assu.EE EFx FtF FN msg HF EFy September 12, 2024 11 Example 2: A Sled on the Ground Only a horizontal force is applied, and the sled is on the ground. Draw the FBD of the sled. freeftp.agnotmatter 𝐹 a grope death is coordinatesystem EFFF Ff Efy Fw Msg September 12, 2024 12 Example 3: A Sled on the Ground A general force is applied, and the sled is on the ground. Draw the sled FBD. 𝐹 1surface 2forces 𝜃 Msg t.EE FcosO Ii EFx F EFy FN msgtFsinO FN September 12, 2024 13 Example 4: A force is applied to block A which is placed on top of the block B. Draw the FBD of A. A 𝜃 𝐹 AS B A c he Fx Faso Er Fsino Mag Fy FN September 12, 2024 14 Example 4: A force is applied to block A which is placed on top of the block B. Draw the FBD of B. reversf A 1 A 𝜃 𝐹 mB9 B B EFx Ff FE Fm EFy Fn Fn Mbg September 12, 2024 15 Example 5: no Draw the FBD of block A. 𝐹 III in 𝛼 y.fi e sin.IE EFx FcosO Fr Mag FNtFsinO MAgcosX EFy September 12, 2024 16  Questions? September 12, 2024 17  KNPE 153: Introductory Biomechanics Lecture 5: Free-Body Diagrams Dr. Pouya Amiri Fall 2024 September 17, 2024 1 Review: Free-Body Diagrams First step toward analysis of movement of objects It provides a graphical representation of the object/system of interest with all the external forces acting on it 1. It allows to investigate an object in static equilibrium using Newton’s 1st law 1. Static 2. Constant velocity 2. It allows to investigate accelerated movement of an object using Newton’s 2nd law September 17, 2024 2 Method to Construct Free-Body Diagrams 1. Draw the object of interest in isolation 2. Include all the forces acting on the object as vectors 1. Contact forces: Count the number of the objects in contact with your object. There are two components from each external object: a) Normal component b) Parallel component 1. Noncontact forces: Weight 3. Based on the movement direction and forces, choose a coordinate system that makes the calculations easier 4. Write the equations for static equilibrium or accelerated motion September 17, 2024 3 Example 1 A force is applied to block B which is in contact with block A. Draw the FBD of block A. ma parallelground 𝐹 9 𝜃 B A µg 1 A FN FFA 9 Efy Ff mag FNA September 17, 2024 4 Example 2 A force is applied to block B which is in contact with block A. Draw the FBD of block B. MBS I 𝐹 𝜃 B A I B it FNB Fx Fino FEB FN Fy FNB Ff mpg Fos o September 17, 2024 5 Example 3 A force is applied to block B which is in contact with block A. Draw the FBD of blocks A and B as a system. assure object 𝐹 7 mAtmBg 𝜃 B A f f B Eg A CFT prime Fy Fn Matmpg Foso September 17, 2024 6 Example 4 An egg is free-falling from a nest in a tree. Draw its FBD. Neglect air resistance. 1ms Efy mg v y September 17, 2024 7 Example 5 Draw the FBD of a figure skater. EFy Fw mg 1mg iii September 17, 2024 8 Example 6 me Draw the FBD of a figure skater spinning in the air. neglect Efy mg Img September 17, 2024 9 Example 7 horizontal direction Draw an FBD of a wingsuit flyer. Fixed speed in the Efx Farag 1mg 1 Efy Fair mg Fair resistance September 17, 2024 10 Example 8 Draw the FBD of the runner. my EFX Fs Efy FN my For Yiannis E.EE September 17, 2024 11  Quiz 1 Solution September 17, 2024 12 Question 1 What is the length of x in the following right-angled triangle? Answer: 5 97th C 132 92 122 a 5 September 17, 2024 13 Question 2 What is 𝑉 + 𝑉 ? Answer: (2.7,0.8) w̅ 4,29 Bz 1.3 1.6 82 4 1.3 24 1 6 2.7 0.87 September 17, 2024 14 Question 3 What is 𝑉 − 2𝑉 ? Answer: (6.6,5.6) w̅ 2V2 4 2.9 2 t 3 1.6 6 56 6 September 17, 2024 15 Question 4 What is 𝑉. 𝑉A? scalar Answer: -9.04 B 82 LIFE 1a 9.04 September 17, 2024 16 Question 5 The length of 𝑉 is 8 m and its angle with the horizontal is 30⁰. The length of 𝑉 is 5 and its angle with the horizontal is -60⁰. What is 𝑉 − 𝑉 ? Answer: (4.43,8.33) sindi5 w̅ Iiicoso fi µ s.no ñ CÑ w̅ 82 4.43 8.33 September 17, 2024 17 Question 6 𝑉 is shown below and 𝑉 = (−3.1,4.2). Wha is 𝑉. 𝑉 ? Answer: 5.18 w̅ s 30 cos300 so 0.5 0.87 VI Iz 0.5 3 1 t 0.87 4 2 5.18 September 17, 2024 18 Question 7 As the surface area of two objects in contact increases, what happens to the force of friction? Answer: It remains constant. September 17, 2024 19 Question 8 As the normal contact force increases, what happens to the dynamic friction force? Answer: It increases. Fdp Fd NdFN lionstat c September 17, 2024 20 Question 9 A 100 kg shot-putter pushes forward and upward on a 4 kg shot with a force of 1000 N. This force acts at an angle of 60° above horizontal. How much of this 1000 N force acts in the forward direction (i.e., what is the horizontal component of this 1000 N force)? Is Answer: 500 N Ex September 17, 2024 21 Question 10 If a normal force of 500 N is acting on two surfaces in contact with a coefficient of friction equal to 0.50, what would be the maximum force of static friction? Answer: 250N Fsmax MsFN 0.5 500N 250N September 17, 2024 22 Question 11 An 80 kg volleyball player lands from a jump and tries to stop her forward motion. The normal contact force exerted by the floor on her shoes is 2000 N. The friction force exerted by the floor on her shoes is 600 N acting backward on the player. What is the magnitude and angle of resultant force on the volleyball player? Answer: Magnitude: 1355 N, Angle: 63.7⁰ with the left horizontal direction Enfant EI.EE mg yooo 2000 809.9imsz 1215N Net force magnitude FFN FET 1355N www.casieo i.si September 17, 2024 23  KNPE 153: Introductory Biomechanics Lecture 6: Static Equilibrium Dr. Pouya Amiri Fall 2024 September 19, 2024 1 Update Office hours and location Office: SKHS Building, KINE 301B E-mail: [email protected] If you don’t want to show up, you must remove your name by 9 AM. Quiz 1 grading Both answers are correct. The quiz grades have been fixed. Forward Forward September 19, 2024 2 Learning Objectives Understand the Newton’s first Law of Motion Develop free-body diagrams of objects in static equilibrium and compute the forces applied to an object of interest September 19, 2024 3 Newton’s First Law of Motion If no net external force acts on an object, that object will not move if it was not moving, or it will continue moving at constant speed in a straight line if it was already moving. September 19, 2024 4 Newton’s First Law of Motion The sum of all forces acting on the object is zero In 2D X direction Y direction 𝐹 =0 𝐹⃗ = 0 → 𝐹 =0 September 19, 2024 5 Application: Static Equilibrium What are the forces that keep an object stationary? September 19, 2024 6 Application: Static Equilibrium 1. Construct the free-body diagram of the object of interest I. Isolate the object of interest II. Include all the forces acting on the object as vectors a) Contact : count the number of the objects in contact with the object. There will be the followings for each contact: a) Normal component b) Friction component b) Non-contact: Weight 2. Select a coordinate system that best suits the problem. 3. Write the sum of the forces in all directions. 4. Count the number of unknowns and equations and solve the equations to find the unknowns. September 19, 2024 7 Example 1 𝐹⃗ is horizontal and has a magnitude of 20 N. The sled is not moving. Calculate the friction and the normal contact force between the sled and the ground? 𝐹 Lf 119 FEET An EFx 0 F Is 0 Fs msg 15kg 9 81m s msg 0 FN Efy 0 Fj 147.2N September 19, 2024 8 Example 2 𝐹⃗ is horizontal and has a magnitude of 30 N. The sled is about to move. How much is the coefficient of static friction between the sled and the ground? 𝐹 1 FIMUXG.FI Fsmax FN Fx 0 F Fsmax O Fsmay F 30N Efy 0 Fn mg For msg 14792 As g o.fi September 19, 2024 9 Example 3 If 𝐹⃗ is vertical and 100 N, calculate the friction and the normal contact force between the sled and the ground? station 𝐹 is FN EEO Fs on Fy 0 FN F my O For mg F 100N 15kg 9.81nF 47.2N September 19, 2024 10 Example 4 If 𝐹⃗ has an angle of 30⁰ and a magnitude 100 N, and the sled is not moving, calculate the friction and the normal contact force between the sled and the ground? 𝐹 it no 𝜃 If TFW Fx 0 Fioso Fs Food lowcos30 0 direction needs for means 86.6MW way Fy 0 In Fit ing0 FNIY.IN www.qq.gg September 19, 2024 11 Example 5 𝐹⃗ has an angle of 30⁰ and magnitude of 100 N. The sled is about to move. How much is coefficient of static friction between the sled and the ground? 𝐹 Find 𝜃 19 Emma FeosO t Fsmax 0 Fsman abutte Fx Faso 86.6N FN my Fsino 0 Fw mg Fsino notneed Efy 97.12N Fsmax Nsfw As F ff.fi September 19, 2024 12 Example 6 John is pulling on the rope with a horizontal force equal to 100 N. What are the normal contact and friction forces applied from B to A if the sleds are stationary? mA=10 kg mB=20 kg I mA=10 kg Fx o F Fib 0 Fsb F LOON EFy 0 In _MAY 0 For Mag 10kg 9.81m s 298 IN September 19, 2024 13 Example 7 John is pulling on the rope with a force equal to 100 N. What are the normal contact and friction forces applied from the ground to B if the sleds are not moving? Opp b c winter 3rdlaw FNb mA=10 kg y mB=20 kg mB9 mB=20 kg adto sowcfoaff.a F FW EFx 0 Fsb Fs 0 Fs Fsb 100N O FN FNB Mpg O FN Fab Mpg 98 IN 20 9 EFy 294.3N Frev ex September 19, 2024 14 Example 7 John is pulling on the rope with a force equal to 100 N. What are the normal contact and friction forces applied from the ground to B if the sleds are not moving? mA=10 kg 2nd method 1 mA=10 kg AtB are a system mB=20May mB=20 kg or can be assured kg other 1 Object stacktoeach FN EF 0 ÉÉ 0 Fs F IOON Fy 0 In mis mg 0 FN September 19, 2024 15 Matt P Example 8 John is pulling on a rope and A is moving with a constant speed while B stays stationary. The coefficient of dynamic friction between A and B is 0.1. What is the friction force applied from the ground to B? E FI 0 Efx 0 IIF mA=10 kg mA=10 kg mB=20 kg EFy 0 Fib m 5o FNB 98 N ffffmanenn ra É f mB=20 kg Efy Ff Msg FI O FN MBg Fnb FN 20 9.81 98.1 2 state September 19, 2024 16 bicbisnfy.gg Example 9 John is pulling on the rope with a force equal to 100 N with an angle 60⁰. If the sleds stay stationary, what are the forces between A and B? mA=10 kg mA=10 kg mB=20 kg September 19, 2024 17 Example 10 John is pulling on the rope with a force equal to 100 N with an angle 60⁰. If the sleds stay stationary, what are the forces applied from the ground to B? mA=10 kg mB=20 kg mB=20 kg September 19, 2024 18  Questions? September 19, 2024 19  KNPE 153: Introductory Biomechanics Lecture 7: Static Equilibrium Dr. Pouya Amiri Fall 2024 September 24, 2024 1 Updates Assignment 1 The assignment will be posted on September 27th The assignment is due on Midnight of October 4th pdf September 24, 2024 2 Newton’s First Law of Motion If no net external force acts on an object, that object will not move if it was not moving, or it will continue moving at constant speed in a straight line if it was already moving. September 24, 2024 3 Newton’s First Law of Motion The sum of all forces acting on the object is zero In 2D X direction Y direction 𝐹 =0 𝐹⃗ = 0 → 𝐹 =0 September 24, 2024 4 Application: Static Equilibrium Steps to find the unknown forces 1. Construct the free-body diagram of the object of interest I. Isolate the object of interest II. Include all the forces acting on the object as vectors a) Contact : count the number of the objects in contact with the object. There will be the followings for each contact: a) Normal component b) Friction component b) Non-contact: Weight 2. Decide about your coordinate system 3. Write the sum of the forces in each direction 4. Count the number of unknowns and equations and solve the equations if possible September 24, 2024 5  Examples From Previous Week September 24, 2024 6 Example 9 John is pulling on a rope with a force equal to 100 N and an angle of 60⁰. If the sleds stay stationary, what are the forces between A and B? mA=10 kg 1mA mA=10 kg 𝜃 mB=20 kg FB FNB Is FN MAY 9.8m s Efy FN mag 0 0 19 September 24, 2024 7 Example 10 John is pulling on a rope with a force equal to 100 N and an angle of 60⁰. If the sleds stay stationary, what are the forces applied from the ground to B? mA=10 kg no m =20 kg it I 𝜃 B mB=20 kg FN Fx 0 FIFO E 9100cos60 7m EFy0 For Is.io ÉiEin gt FN 1o.sin60 98.1 209.81 207.7W September 24, 2024 8  New Examples September 24, 2024 9 Example 1 A skater standing on ice with a mass of 50 kg. How much is the force from the ground to her skate? E FN mg 0 FN M9 Efy O 50kg 9.81ms 490.5N FN September 24, 2024 10 Example 2 An 80 kg weightlifter has lifted 100 kg over his head and is holding it still. What is the reaction force from the floor that acts on the weightlifter’s feet? mB9 EFX 0 Fs 0 mug Efy 0 FN muy MBS 0 FN 9 41765.8N 19 a.amsz FN September 24, 2024 11 Example 3 Bryan is applying a force of 200 N to the wall with angle of 30⁰. He weighs 75 kg. If he is stationary, how much are the friction and normal contact forces applied to his feet from the ground? wallapplies are foropp direction 𝜃 t Fs Écoso 0 FSE2,91 30 Fx 0 FÉO 0 FN EFy 0 7 Fi mj FN 75kg9.81ms 200sin30 8357N i September 24, 2024 12 Example 4 Bryan is applying a force of 200 N to the wall with angle of 30⁰. He weighs 75 kg. How much is the minimum required coefficient of static friction between his feet and the ground to prevent him from sliding? Fsmux Fine six kftothsipfe.rs Nsfw 𝜃 qo EFX 0 Fsmax FOSO 0 IsFN Fcoso Us ÉTÉ FIFI Esma FN Fy mg Fsino sub µ's Fcos0 myFino o.fi 0.21 µ September 24, 2024 13  Example 5 John is holding onto two ropes. He weighs 70 kg. If the angle of the left and right ropes with the vertical are 10⁰ and 30⁰, respectively, how much are the tension forces in each rope (or the forces applied to his arms)? Litre Fisintor Fesino Efx e 0 Frcosor ing 30° I E Fecosoe o Efy 0 10° from Frsinor Fesinoe Fr s feg.si d7Fffcosortfecosoe mg s Fe cosor cos e s my Fl my cosor taste Fe 534.2N subto Fr ftp.f September 24, 2024 14 L8S 5N Example 6 John is pulling on a rope attached to a sled with a force equal to 300 N. The masses of the sled and John are 20 kg and 65 kg, respectively. The coefficient of static friction between the ground and the sled is 0.2. How much is the static friction force applied to John’s feet if they are both stationary? 𝜃 = 20° Efx 0 Efy o Is missin FÉÉo Fi mycosis Éi Ii reverse 𝛼 = 30° ifwasnegatie Fs 600.7N don'tneedtoanswerhatcan FN 6548N September 24, 2024 15  Questions? September 24, 2024 16  KNPE 153: Introductory Biomechanics Lecture 8: Static Equilibrium & Linear Kinematics Dr. Pouya Amiri Fall 2024 September 26, 2024 1 Newton’s First Law of Motion If no net external force acts on an object, that object will not move if it was not moving, or it will continue moving at constant speed in a straight line if it was already moving. September 26, 2024 2 Newton’s First Law of Motion The sum of all forces acting on the object is zero In 2D X direction Y direction 𝐹 =0 𝐹⃗ = 0 → 𝐹 =0 September 26, 2024 3 Application: Static Equilibrium Steps to find the unknown forces 1. Construct the free-body diagram of the object of interest I. Isolate the object of interest II. Include all the forces acting on the object as vectors a) Contact : count the number of the objects in contact with the object. There will be the followings for each contact: a) Normal component b) Friction component b) Non-contact: Weight 2. Decide about your coordinate system 3. Write the sum of the forces in each direction 4. Count the number of unknowns and equations and solve the equations if possible September 26, 2024 4 Example 1 John is pulling on a rope to move a sled, and the sled is about to move. The masses of the sled and John are 20 kg and 65 kg, respectively. The coefficient of static friction between the ground and the sled is 0.2. How much is the static friction force applied to John’s feet if he’s stationary? 1m59 9 x 1 a Case John as asystem ftp for untsolvable unknowns afarFstFna Equations 2 September 26, 2024 6  Example 1 John 1m59 case Aareseparate it NsFÑA EE EEE 1050 fauf 0 FNA 0 ii li in w drown answerquestion Efy 0 Fny my9 Fsino0 September 26, 2024 7 Example 2: Practice John is pulling on a rope to move a sled, and the sled is about to move. The masses of the sled and John are 20 kg and 65 kg, respectively. The coefficient of static friction between the ground and the sled is 0.2. How much is the static friction force applied to John’s feet if he’s stationary? 𝜃 = 20° 𝛼 = 30° September 26, 2024 8 Example 3 Brian, a 200 kg strongman competitor, is competing in the truck pull event in a strongman competition. Brian wears a harness attached to a cable, and the cable is attached to a truck and trailer. Brian is also using his hands to pull on a rope that is attached to an immovable object in front of him. Brian and truck are stationary. Brian pulls forward and slightly upward on the cable with a force of 2200 N acting at an angle of 14° above horizontal. Brian also pulls on the rope with a horizontal force of 650 N. How much force does the ground exert against Brian’s feet? InardFs notjustmasri table one if make rightf triangle angle September 26, 2024 9 Example 3 Y 0 140 EFx É Fs Écoso 0 1mg Fin my no ÉsiÑ 0 Efy E Fs Fecost Fr 1485N FN mg Fesino 2494N FINE September 26, 2024 10 Example 4 What is the minimum required coefficient of static friction between Brian’s shoes and the ground to make sure he doesn’t slip? Fsmax force is Fs Fsmax NsFN Ns E 0.6 if Ns 0.6 hewon'tslip larger September 26, 2024 11  Linear Kinematics no rotation September 26, 2024 12 Learning Objectives Distinguish between linear, angular, and general motion Define distance traveled and displacement and distinguish between the two Define average speed September 26, 2024 13 Categories of Motion Linear motion (or translation): all points on the body move The same distance In the same direction At the same time 1. Rectilinear translation Example: Figure skater gliding across the ice September 26, 2024 14 Categories of Motion Linear motion (or translation): all points on the body move The same distance w In the same direction At the same time 2. Curvilinear translation Example: Sitting in a car that goes up and down the hill September 26, 2024 15 Categories of Motion Angular motion (rotation) It occurs when all points on a body or object move in circles (or parts of circles) about the same fixed central line or axis Example: Spinning & a child on a swing September 26, 2024 16 Categories of Motion General Motion (mixed) Combines angular and linear motion Examples Walking A rolling ball Pedaling a bike In Biomechanics, most of the time we deal with general motion. September 26, 2024 17 Linear Kinematics Concerned with the description of linear motion component. It could be only translation or a general motion. Examples: Speed Distance Direction September 26, 2024 18 Position Is the location in space First step is to define a coordinate system, i.e. origin and directions Example: Runner: 1D or September 26, 2024 19 Position Is the location in space First step is to define a coordinate system, i.e. origin and directions consistent Example: stay Football player: 2D R 15yd80yd record15yd r ñ y September 26, 2024 20 Distance Travelled and Displacement Both measure change in position (motion) Distance (scalar): Length of path traveled coordinatesystem Displacement (vector) Linear distance in a specified direction Vector connecting the initial (start) to final (end) position atline Distancetravelled length scalar pan f y fatement vector i x September 26, 2024 21 Example: Distance Travelled Fi a if if fi X Distance travelled = 69yd 50 40 Position at finish: 30 Position at start: (20, 60) a 20 (30, 10) if ri 10 20 Y 0 60 September 26, 2024 22 Example: Displacement X Distance travelled = 69yd 50 40 Position at finish: 30 Position at start: (20, 60) (30, 10) 20 10 Y September 26, 2024 23 Example Initial position: (30, 10) Final position: (20, 60) Find the displacement vector and the resultant displacement. September 26, 2024 24 Example After running one lap on a 400m track what is the distance and what is the displacement? September 26, 2024 25 Speed and Velocity How fast an object is moving Speed (scalar): Rate of distance travelled Compute average speed ℓ 𝑠̅ = ∆ 𝑆̅ : average speed (m/s) ℓ : distance travelled ∆𝑡: time taken to finish the movement Units: meters/second = m/s September 26, 2024 26 Example What is the average speed of the football player if the movement happens in 6 seconds? Distance travelled = 69yd X 50 40 30 Position at Position at start: 20 finish: (30, 10) (20, 60) 10 Y September 26, 2024 27 Problem with Average Speed Doesn’t tell much about what happened during the movement Distance travelled = 69yd X 50 40 30 Position at Position at start: 20 finish: (30, 10) (20, 60) 10 Y September 26, 2024 28  Questions? September 26, 2024 29  KNPE 153: Introductory Biomechanics Lecture 9: Linear Kinematics Dr. Pouya Amiri Fall 2024 October 3, 2024 1 Update Quiz 3: Linear kinematics Original date was Oct 4 New date is Oct 11 midterm Quiz 4: Linear kinetics not on Original date was Oct 11 New date is Oct 25 Assignment 2 (Linear kinematics and linear kinetics) Will be posted on Oct 11th Due on Oct 17th October 3, 2024 2 Update Midterm exam October 24th from 13:30 to15:00 You will be divided to two groups taking the exam at Etherington Hall AUD Kingston Hall AUD October 3, 2024 3 Example 1 John is pulling on a rope to move a sled, and the sled is about to move. The masses of the sled and John are 20 kg and 65 kg, respectively. The coefficient of static friction between the ground and the sled is 0.2. How much is the static friction force applied to John’s feet if he’s stationary? October 3, 2024 4 Example 1 October 3, 2024 5  Example 1 0 947 0 to FNA a F 38 93N 18 s 3 0.2FNA 0 FNA 18295N John Fis Fios 0 Fss Fcoso 36 58N Fns Mtg Find 0 solve for 650.97N October 3, 2024 infgy award 6  Linear Kinematics October 3, 2024 7 Learning Objectives Review of different types of motion Review distance traveled and displacement Define average speed Define acceleration Define average acceleration October 3, 2024 8 Categories of Motion Linear motion (or translation): all points on the body move The same distance In the same direction At the same time 1. Rectilinear translation addthis.si Example: Figure skater gliding across the ice October 3, 2024 9 Categories of Motion Linear motion (or translation): all points on the body move The same distance In the same direction At the same time 2. Curvilinear translation Example: Sitting in a car that goes up and down the hill October 3, 2024 10 Categories of Motion Angular motion (rotation) It occurs when all points on a body or object move in circles (or parts of circles) about the same fixed central line or axis Example: Spinning & a child on a swing October 3, 2024 11 Categories of Motion General Motion (mixed) Combines angular and linear motion Examples Walking A rolling ball Pedaling a bike In Biomechanics, most of the time we deal with general motion. October 3, 2024 12 Linear Kinematics Concerned with the description of linear motion component. It could be only translation or a general motion. Examples: Speed Distance Direction October 3, 2024 13 Position Is the location in space First step is to define a coordinate system, i.e. origin and directions Example: Runner: 1D October 3, 2024 14 Position Is the location in space First step is to define a coordinate system, i.e. origin and directions Example: Football player: 2D October 3, 2024 15 Distance Travelled and Displacement Both measure change in position (motion) Distance (scalar): Length of path traveled Displacement (vector) Linear distance in a specified direction Vector connecting the initial (start) to final (end) position y x October 3, 2024 16 Example: Distance Travelled X Distance travelled = 69yd 50 40 Position at finish: 30 Position at start: (20, 60) (30, 10) 20 10 Y October 3, 2024 17 Example: Displacement X Distance travelled = 69yd 50 40 Position at finish: 30 Position at start: (20, 60) (30, 10) 20 10 Y October 3, 2024 18 Example Initial position: (30, 10) ñ length angle Final position: (20, 60) If p Find the displacement vector and the resultant displacement. if fi 30 10 20,60 10 so yd a 559 0 17 October 3, 2024 11.30 19  Example After running one lap on a 400m track what is the distance and what is the displacement? distance L 400m dismlacement d October 3, 2024 20 Speed and Velocity How fast an object is moving Speed (scalar): Rate of distance travelled Compute average speed ℓ average 𝑠̅ = stanous ∆ 𝑆̅ : average speed (m/s) 7.5ms ℓ : distance travelled 41.25 e ∆𝑡: time taken to finish the movement Units: meters/second = m/s October 3, 2024 21 Example What is the average speed of the football player if the movement happens in 6 seconds? Distance travelled = 69yd 5 69 11.41s X 10 52m s 50 40 30 Position at Position at start: 20 finish: (30, 10) (20, 60) 10 Y October 3, 2024 22 Problem with Average Speed Doesn’t tell much about what happened during the movement To understand more details, we can look at the average speed in Distance travelled = 69yd X shorter time intervals 50 40 30 Position at Position at start: 20 finish: (30, 10) (20, 60) 10 Y October 3, 2024 23  Usain Bolt’s Speed POSITION (M) ELAPSED TIME INTERVAL TIME (S) (S) 0 0 - 10 1.89 1.89 RunningVictory.jpg 20 2.88 0.99 30 3.78 0.90 40 4.64 0.86 50 5.47 0.83 60 6.29 0.82 70 7.10 0.81 80 7.92 0.82 90 8.75 0.83 100 9.58 0.83 October 3, 2024 24 Usain Bolt’s Speed Average speed for 0-100m POSITION (M) ELAPSED TIME INTERVAL (S) TIME (S) s̅ = = 10.44 𝑚/𝑠. 0 0 - 10 1.89 1.89 20 2.88 0.99 Average speed for the 1st 50 m 30 3.78 0.90 s̅ = = 9.14 𝑚/𝑠 40 4.64 0.86. 50 5.47 0.83 60 6.29 0.82 Average speed for the 2nd 50 m 70 7.10 0.81 s̅ = = 12.17 𝑚/𝑠 80 7.92 0.82.. 90 8.75 0.83 100 9.58 0.83 reset October 3, 2024 25 Usain Bolt’s Speed POSITION (M) ELAPSED TIME INTERVAL TIME Distance (M) AVERAGE (s) (s) SPEED (m/s) 0 0 - - 10 1.89 1.89 1.89 5.29 20 2.88 0.99 0.99 10.10 30 3.78 0.90 0.90 11.11 40 4.64 0.86 0.86 11.63 50 5.47 0.83 0.83 12.05 60 6.29 0.82 0.82 12.20 70 7.10 0.81 0.81 12.35 80 7.92 0.82 0.82 12.20 90 8.75 0.83 0.83 12.05 100 9.58 0.83 0.83 12.05 October 3, 2024 26 Instantaneous Speed Speed of an object at a specific instant of time. To determine the instantaneous speed, reduce the time interval and calculate the average speed When the time interval becomes very small, the average speed over that time interval is basically the Instantaneous Speed Speed means Instantaneous Speed Average Speed specifically needs to be mentioned aus instantaneous derivative ℓ 𝑠̅ = En ∆ October 3, 2024 27  Example Is the speedometer in your car a measure of instantaneous speed or average speed? instrous that Speed time October 3, 2024 28 Velocity Velocity (vector): Rate of displacement 6 19 Compute average velocity y averase 1 𝑑⃗ 𝑣⃗̅ = x ∆𝑡 ⃗̅ average velocity (m/s) 𝑣: 𝑑⃗ : vector of the displacement ∆𝑡: time taken to finish the movement Units: m/s October 3, 2024 29 Example What is the average velocity vector of the football player if the movement happens in 6 seconds? Distance travelled = 69yd X E 50 40 1 67 8 3 yds 30 Position at Position at start: 20 finish: (30, 10) (20, 60) 10 Y October 3, 2024 30 Instantaneous Velocity If the time interval for average velocity calculation becomes very small, average velocity becomes instantaneous velocity. Direction of instantaneous velocity: tangent to the path first 𝑑⃗ 𝑣⃗̅ = y ∆𝑡 toypath VxVg x If the term velocity is used, it refers to instantaneous velocity. October 3, 2024 31 Speed and Velocity In Biomechanics and mechanics in general, we use velocity, because we are dealing with vectors and directions are important. October 3, 2024 32 Acceleration The rate of change in velocity 1. Starts, stops, speeds up, slows down 2. Changes direction A vector quantity October 3, 2024 33 Average Acceleration The change in velocity divided by the time it took for that velocity change: 𝑣 −𝑣 ∆𝑣⃗ m 𝑎⃗ = = m 5 𝑡 −𝑡 ∆𝑡 𝑎⃗ is average acceleration ∆𝑣⃗ is change in velocity 𝑣 is the instantaneous velocity at the end of an interval 𝑣 is the instantaneous velocity at the beginning of an interval ∆𝑡 is the time taken or change in time / 𝑢𝑛𝑖𝑡: = m/s October 3, 2024 34 Example A car accelerates from 0 to 10 m/s in x and 0 to 7 m/s in y directions in 7 seconds, what is the average acceleration? x̅ 4.43 1 m 82 October 3, 2024 35 Acceleration Acceleration refers to instantaneous acceleration Similar to velocity, if the time interval is very small, average acceleration turns into (instantaneous) acceleration Acceleration is the derivative of velocity with respect to time ∆𝑣⃗ 3m12 ⃗ 𝑎= ∆𝑡 a d October 3, 2024 36 Zero Acceleration When acceleration is zero (meaning zero external forces) The velocity does not change. Velocity is a vector. Thus, both amplitude and direction of the velocity stay the same. October 3, 2024 37  Quiz 2 Solution October 3, 2024 38 Question 1 John is applying a horizontal force of 100 N to sled A. If the mass of A is 30 kg, and mass of B is 53.2 kg, and sled B is about to move (sled A is not moving relative to sled B), what is the friction force applied from the ground to sled B? Answer: 100 N stationary y Fx F Fs 0 fs F 7100N Fw Efy Fw mating 9 0 October 3, 2024 39 Question 2 Assume you're pushing a book to the wall and holding it in place. If the mass of the book is 800 grams and the coefficient of static friction is 0.1, what is the value of the static friction force applied to the book from the wall? Answer: 7.85 N I F EFx FN F O 1mg Efy Fs mg IF 9.814 c 7.85N October 3, 2024 40 Question 3 The force that acts parallel to the surfaces of two objects in contact is: Answer: Friction force October 3, 2024 41 Question 4 The coefficient of dynamic friction between the grass and a 200 kg football blocking sled is 0.20. If a 100 kg football player only pushes horizontally against the sled, how large must the horizontal force be to keep the sled moving at constant velocity? d Answer: 392 N I 1 EFF O E FP FEET F FN EF 0 Fw mg FN mg October 3, 2024 FI mfg 0.2 42 200 9.81 392N Question 5 The force F with a magnitude of 100 N and angle of 60⁰ is applied to a stationary sled. If the sled is about to move, how much is the friction force between the sled and the ground? Answer: 50 N it is statinary Fino Efx Faso Fs Fs Rose FN 88 60 October 3, 2024 43 Question 6 Katie pulls upward with a 400 N force on a 700 N barbell that is resting on the floor. The barbell does not move. How large is the normal contact force exerted by the floor on the barbell? Answer: 300 N Fx 8 0 FN Efy FN W FK FN 700 400 300N October 3, 2024 44 Question 7 John is pulling on a rope attached to the block A with a force equal to 100 N and angle of 30⁰ with horizontal. If both blocks are stationary, how much is the normal contact force, applied from ground to block B? Answer: 244.3 N no FN Efg FN MAtmBg Fsino 0 FN 0 2074.81 1008in30 244.3N October 3, 2024 45 Question 8 The coefficient of static friction between an object and the surface is 0.5. If the object starts to slide when a horizontal force of 1200N is applied, what is the object's mass? Answer: 244.6 kg F EFA F FEYENEN EmIFN aboutTove Efy FN mg D FN mg F Ns mg October 3, 2024 9581m 46 244.6 Question 9 Bryan is pulling on a rope with a force equal to 80 N (figure below). The sled is not moving. What is the value of the friction force applied to sled from the ground? Answer: 28.46 N October 3, 2024 47 Question 10 Bryan is standing on one foot and applying a 140 N force with angle of 50⁰ to the sled A. If both Bryan and sled are stationary, how much is the friction force applied to his foot from the ground and what is its direction? Answer: 90 N, toward right October 3, 2024 48 Question 11 The gymnast in the figure below is stationary. He weighs 75 kg and the angles of both ropes with vertical are 10⁰. What is the magnitude of the force in each rope? Answer: 373 N October 3, 2024 49 Question 12 Paul is standing on one foot and applying a 200 N force with angle of 60⁰ to the sled A. Both the sled and Paul are stationary. If his mass is 66 kg, what is the minimum required coefficient of static friction between the foot and the ground to prevent the foot from sliding? Answer: 0.12 October 3, 2024 50  Questions? October 3, 2024 51  KNPE 153: Introductory Biomechanics Lecture 10: Linear Kinematics II Dr. Pouya Amiri Fall 2024 October 8, 2024 1 Learning Objectives Review acceleration Define constant (uniform) acceleration Derive the equations when moving with constant acceleration Apply the concepts to projectile movement October 8, 2024 2 Acceleration The rate of change in velocity Starts, stops, speeds up, slows down Changes direction A vector quantity October 8, 2024 3 Average Acceleration The change in velocity divided by the time it took for that velocity change: 𝑣 −𝑣 ∆𝑣⃗ 𝑎⃗ = = 𝑡 −𝑡 ∆𝑡 𝑎⃗ is average acceleration ∆𝑣⃗ is change in velocity 𝑣 is the instantaneous velocity at the end of an interval to 17 𝑣 is the instantaneous velocity at the beginning of an interval ∆𝑡 is the time taken or change in time 7.3 / 𝑢𝑛𝑖𝑡: = m/s October 8, 2024 4 Acceleration Acceleration refers to instantaneous acceleration Similar to velocity, if the time interval is very small, average acceleration turns into (instantaneous) acceleration Acceleration is the derivative of velocity with respect to time a 1 a 4 October 8, 2024 5 Zero Acceleration When acceleration is zero (zero external force) The velocity does not change Velocity is a vector Thus, both amplitude and direction of the velocity stay the same magnitude October 8, 2024 6 Direction of Motion and Direction of Acceleration If velocity and acceleration are: 1. in the same direction Speeding up samedirector 2. in the opposite direction Slowing down 3. Zero acceleration oppsigns Constant velocity saved oppdirecta October 8, 2024 7 Constant (uniform) Acceleration When acceleration is constant p ampointsupthesame Average acceleration is equal to Constant Acceleration Welolis ininterval É A 4 forany i tf FYI 2111 3m18 This means the velocity changes linearly with respect to time 0 ΔÑ ADE October 8, 2024 8 Movement of an Object with Constant Acceleration Equations in x direction 11direction i: initial and f: final Δ𝑣 = 𝑣 − 𝑣 , Δ𝑥 = 𝑥 − 𝑥 , Δ𝑡 = 𝑡 − 𝑡 Δ𝑣 = 𝑎Δ𝑡 1 Δ𝑥 = 𝑎Δ𝑡 + 𝑣 Δ𝑡 time involved 2 dispies 𝑣 − 𝑣 = 2𝑎Δ𝑥 October 8, 2024 9 Example 1 A biker’s velocity 5 seconds after the start of his movement is 36 km/h. What is the value of their constant acceleration? Vi Up 5s oh no a Up 6kmh 2m15 a I kmh ims IET Hortons 6 October 8, 2024 10 Example 2 A car moves with a constant acceleration of 8 m/s2. If starting at a position of 10 m with a velocity of 2 m/s, what is its position after 7 seconds? u a 8m15 xp xTÉaΔÉtÉΔÉ 0m 8 7 2 7 10 Vi 2m15 220m Δt 7seconds xF October 8, 2024 11 Example 3: A car is moving with a speed of 72 km/h, and the driver hits the break suddenly, giving the car a constant acceleration of -4 m/s2, what is the distance covered by the car before it stops completely? ms -20 -713.6 - 8 Vi 72km h BVf 0 ox e itit a - 4m 5 É Δx to Y F Ox 86.6 October 8, 2024 12 Case of Constant Acceleration: Projectile Motion An object that has been projected into the air or dropped and is acted on only by the forces of gravity. October 8, 2024 13  Projectile Motion: Free-Body Diagram constat Vx Efx Max 0 Max a my Efy may mfg May ay g does the vertrial acceleration of a projectile afghan't not depend on it's mass coffish Lmg October 8, 2024 14 Projectile: Horizontal Motion Movement with Constant Speed Horizontal acceleration is 0, thus velocity in horizontal direction is constant. 𝑎 =0 displatement 𝑣 = 𝑐𝑡𝑒 = 𝑣 = 𝑣 → Δ𝑥 = 𝑣 Δ𝑡 Konstt October 8, 2024 15 Projectile: Vertical Motion a Vertical acceleration is constant and is sequal to 𝑔 = 9.81 𝑚/𝑠 Vertical velocity at the end of an interval: ΔVy got Vertical position at the end of an interval: Δy gΔE try ot I Vertical speed after a certain vertical displacement: Yf Vy 2gLy October 8, 2024 16 Example 4-Part A Oliver punts a football into the air. The football has an initial vertical velocity of 15 m/s and an initial horizontal velocity of 15 m/s when it leaves Oliver’s foot. What is the ball’s horizontal velocity 2 s after it leaves Oliver’s foot? my ismis Uxf 15m's October 8, 2024 17 Example 4-Part B What is the ball’s vertical velocity 2 s after it leaves Oliver’s foot? Vy 15ms Δt 2s Vyf Vy got Vyf Vyf gottry 9.81 2 IS 462m15 October 8, 2024 18 Example 4-Part C What is the ball’s horizontal displacement 2 s after it leaves Oliver’s foot? Ux cte I 230m October 8, 2024 19 Example 4-Part D What is the ball’s vertical displacement 2 s after it leaves Oliver’s foot? Vy Is Δy f 9.81129 1512 0 Hee Δt 25 110.38m Δy October 8, 2024 20 Example 5-Part A massdoesnoteffect direction