L2 - Intro & Maths Primer

• Adding Muscle Forces:

  • Muscles contract & exert forces on bones
  • eg Gluteus maximus & tensor fasciae latae pull on femur
• Adding Muscle Forces:

  • Length & direction of arrows represent force magnitude & direction (vectors)
  • eg Gluteus max exerting more force (longer arrow)
  • Role of gluteus max changes the force, orientation of forces important
  • What happens to bone from these vectors/forces?
  • What is their combined effect?
• Body Segment Positions & Orientations:

  • Muscles move body segments
  • How can we describe the positions & orientations?
  • Landmark positions
  • Centre of mass (x, y)
  • Difference in coordinates show how move body
  • Angles (for orientation)
  • Different angles
  • Inside (easy to measure)
  • Outside/Supplementary angle
  • Segment angle → relative to something outside of body
• Body Segment Positions & Orientations:

  • Landmark positions
  • Centre of mass on forearm (x, y), will be more proximal
  • Elbow joint centre, talk about distal end (wrist joint)
  • Why would that be important? Difference in coordinates show how move their body
• Body Segment Positions & Orientations:

  • Angles (for orientation)
  • Elbow joint, where forearm is relative to upper arm
  • If in line = 0°
  • Different angles
  • Inside (easy to measure)
  • Outside/Supplementary angle
  • Either are fine, just say which one (need context)
  • Segment angle → relative to something outside of body, eg vertical, laying on a bench
• Use algebra & trigonometry to to calculate angles
• Background:

  • Unit circle is centred at (0,0)
  • Radius is 1
  • Trig functions give (x, y, r) ratios
  • sin(50°) = 0.77 - (vertical line)
  • If change to 60° sin will be bigger as point will be higher therefore longer
  • cos(50°) = 0.64
  • If 60° it decreases
  • True for right angle triangles
• SOH-CAH-TOA:

  • sin(𝜃) = o/h
  • cos(𝜃) = a/h
  • tan(𝜃) = o/a
  • opposite side
  • hypotenuse = longest one, opposite right angle
  • adjacent
  • Depends on what you know & what you want to find
• Calculate the Hip Angle:

  • Always a right angle
  • Know opposite length (22.7cm) & adjacent (38.9cm)
  • Want to find theta (𝜃)
  • tan(𝜃) = o/a
  • tan(𝜃)/tan = (o/a) / tan
  • 𝜃 = tan(o/a)
  • 𝜃 = tan(22.7cm/38.9cm)
  • 𝜃 = 30.3°
• Calculate the Length of the Femur:

  • Only opposite & angle given
  • o = 22.7cm, 𝜃 = 30.3°
  • Want to find hypotenuse (h)
  • Needed to get h alone
  • sin(𝜃) = o/h
  • (h*sin(𝜃 ))/sin(𝜃) = o/h*(h/sin(𝜃 ))
  • h = o/sin(𝜃 )
  • h = 22.7cm/sin(30.3°)
  • h = 0.45m
• Just the sides = Pythagorean Theorem:

  • a = Vertical distance bw/ the knee & hip joints
  • b = Horizontal distance bw/ the knee & hip joints
  • c = Hypotenuse or length of the femur
• Just the sides = Pythagorean Theorem:

  • Right angle triangles only, when working with side lengths
  • Those adjacent to angle = a & b doesn’t matter what is what
  • c = hyp, opposite right angle
• Pythagorean Theorem:

  • Calculate the length of the femur:
  • Have a & b looking for c
  • c^2 = a^2 + b^2
  • Need to get rid of square on both sides
  • So square root c (√c^2)
  • c = √a^2 + b^2
  • c = √(38.9cm^2) + (22.7cm^2)
  • c = 0.45m
  • Maths used, but applications will changed
• Vectors:

  • Vectors have magnitude & direction
  • Can be described in terms of vertical (y) & horizontal (x) components
• Vectors:

  • Force plate measures reaction force onto ground
  • Can use to determine what's gonna happen to their body?
  • Reaction force mostly up & anterior
  • Length of arrow & orientation tells you interaction with the ground
  • How much of this force is vertical (up), & horizontal (forward)
  • Make right angle triangles
• Adding & Subtracting Vectors:

  • Segments are often subject to many forces, to find the overall effect we add them
  • If forces are parallel & act along the same line, we can add them algebraically
• Solving Vector Problems:

  • Often need to resolve a resultant vector into its components
  • Decide on a reference frame for the components
  • Horizontal & vertical relative to the ground
  • May not be horizontal or vertical (eg lay on bench)
  • Anatomical planes
  • Along a segment & perpendicular to it
  • To a joint (segment) & perpendicular to rotate
• Multiple Muscle Forces:

  1. Find components of F1 & F2 that are parallel (eg to the femur)
  2. Find components of F1 & F2 that are perpendicular (eg to the femur)
  3. Find the combined (resultant) force of F1 & F2
• Multiple Muscle Forces:

  • Find the combined (resultant) force of F1 & F2
  • Add parallel components
  • Fry = F1x + F2y
  • Fry = 1033.7 N
  • Add perpendicular components
  • Frx = F1x + F2x
  • Frx = -239.4 N + 136.8 N
  • Frx = -102.6 N
  • Fr = √Fry^2 + Frx^2
  • Fr = 1038.8 N
  • 𝜃 = tan-1 Frx/Fry
  • 𝜃 = 5.7 (posterior of long axis)