Angular Variables Radians Measuring Angles Measuring Angles
Thym, Jolene, Food & Fashion Reporter has reference to this Academic Journal, PHwiki organized this Journal Angular Variables Radians q r r q = 1 rad = 57.3 o 360 o = 2 p rad What is a radian a unitless measure of angles the SI unit as long as angular measurement 1 radian is the angular distance covered when the arclength equals the radius r
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Measuring Angles Relative Angles (joint angles) The angle between the longitudinal axis of two adjacent segments. Absolute Angles (segment angles) The angle between a segment in addition to the right horizontal of the distal end. Should be measured consistently on same side joint straight fully extended position is generally defined as 0 degrees Should be consistently measured in the same direction from a single reference – either horizontal or vertical Measuring Angles The typical data that we have to work with in biomechanics are the x in addition to y locations of the segment endpoints. These are digitized from video or film. Tools as long as Measuring Body Angles goniometers electrogoniometers (aka Elgon) potentiometers Leighton Flexometer gravity based assessment of absolute angle ICR – Instantaneous Center of Rotation often have translation of the bones as well as rotation so the exact axis moves within jt
Calculating Absolute Angles Absolute angles can be calculated from the endpoint coordinates by using the arctangent (inverse tangent) function. opp = y2-y1 adj = x2-x1 Calculating Relative Angles Relative angles can be calculated in one of two ways: 1) Law of Cosines (useful if you have the segment lengths) c2 = a2 + b2 – 2ab(cosq) Calculating Relative Angles 2) Calculated from two absolute angles. (useful if you have the absolute angles) q1 q2 q3 q3 = q1 + (180 – q2)
CSB Gait St in addition to ards segment angles joint angles Canadian Society of Biomechanics RIGHT sagittal view Anatomical position is zero degrees. CSB Gait St in addition to ards segment angles joint angles Canadian Society of Biomechanics LEFT sagittal view Anatomical position is zero degrees. CSB Gait St in addition to ards (joint angles) RH-reference frame only! qhip = qthigh – qtrunk qknee = qthigh – qleg qankle = qfoot – qleg – 90o qhip> 0: flexed position qhip< 0: (hyper-)extended position slope of qhip v. t > 0 flexing slope of qhip v. t < 0 extending dorsiflexed + plantar flexed - dorsiflexing (slope +) plantar flexing (slope -) qknee> 0: flexed position qknee< 0: (hyper-)extended position slope of qknee v. t > 0 flexing slope of qknee v. t < 0 extending Angle Example The following coordinates were digitized from the right lower extremity of a person walking. Calculate the thigh, leg in addition to knee angles from these coordinates. HIP (4,10) KNEE (6,4) ANKLE (5,0) Angle Example segment angles qthigh qleg Angle Example segment angles qthigh qleg Angle Example segment angles qthigh = 108° qleg = 76° qknee = qthigh qleg qknee = 32o qknee joint angles Angle Example alternate soln. (4,10) (6,4) (5,0) qknee a b c f a = b = c = f = Angle Example segment angles Angle Example c2 = a2 + b2 - 2ab(cosf) 10.052 = 6.322 + 4.122 - 2(6.32)(4.12)(cosf) qknee= 180o - f = 180o - 147.8o = 32.3o joint angle (1) Angle Example qknee = qthigh - qleg qknee = 108.4o - 76.0o = 32.4o (4,10) (6,4) (5,0) qknee joint angle (2) CSB Rearfoot Gait St in addition to ards qrearfoot = qleg - qcalcaneous Typical Rearfoot Angle-Time Graph Angular Motion Vectors The representation of the angular motion vector is complicated by the fact that the motion is circular while vectors are represented by straight lines. Angular Motion Vectors Right H in addition to Rule: the vector is represented by an arrow drawn so that if curled fingers of the right h in addition to point in the direction of the rotation, the direction of the vector coincides with the direction of the extended thumb.
Angular Motion Vectors A segment rotating counterclockwise (CCW) has a positive value in addition to is represented by a vector pointing out of the page. A segment rotating clockwise (CW) has a negative value in addition to is represented by a vector pointing into the page. Angular Distance vs. Displacement analogous to linear distance in addition to displacement angular distance length of the angular path taken along a path angular displacement final angular position relative to initial position q = qf – qi Angular Distance vs. Displacement
Angular Position Example – Arm Curls Consider 4 points in motion 1. Start 2. Top 3. Horiz on way down 4. End 1,4 2 3 Position 1: -90 Position 2: +75 Position 3: 0 Position 4: -90 NOTE: starting point is NOT 0 f q 1 to 2 165 +165 2 to 3 75 -75 3 to 4 90 -90 1 to 2 to 3 240 +90 1 to 2 to 3 to 4 330 0 Computing Angular Distance in addition to Displacement
By examining the components of the velocity it is clear that there is acceleration even when the angular velocity is constant. w is constant Centripetal Acceleration Even if the velocity vector is not changing magnitude, the direction of the vector is constantly changing during angular motion. There is an acceleration toward the axis of rotation that accounts as long as this change in direction of the velocity vector. This acceleration is called centripetal, axial, radial or normal acceleration. a to ac (ac = w2r or ac = v2/r) Since the tangential acceleration in addition to the centripetal acceleration are orthogonal (perpendicular), the magnitude of the resultant linear acceleration can be found using the Pythagorean Theorem: Resultant Linear Acceleration
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