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# Angular Kinematics Learning Objectives Define angular kinematics Understand compute absolute relative angles angular displacements velocities accelerations Estimate instantaneous angular velocity PDF document - DocSlides

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### Presentations text content in Angular Kinematics Learning Objectives Define angular kinematics Understand compute absolute relative angles angular displacements velocities accelerations Estimate instantaneous angular velocity

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Angular Kinematics Learning Objectives: Define angular kinematics Understand & compute absolute & relative angles, angular displacements, velocities & accelerations Estimate instantaneous angular velocity & accel. Be able to identify phases of movement & infer sources of propulsion and braking Be able to use the laws of constant angular accel. Grasp the applications to human movement Questions to Think About Why would tight calf muscles restrict the ability to run more than the ability to walk? Which muscles are used to speed up the extension of the elbow during a jump shot in basketball? As the knee flexes after landing from a jump, why are the knee extensors active? If you are trying to increase a baseball players bat speed at impact, what kinematic variables should you consider? Angular Kinematics Kinematics The description of motion as a function of space and time Forces causing the motion are not considered Angular Motion (Rotation) All points in an object move in a circle about a single, fixed axis of rotation All points move through the same angle in the same time Angular Kinematics The kinematics of particles, objects, or systems undergoing angular motion Angular Kinematics & Motion Volitional movement performed through rotation of the body segments The body is often analyzed as a collection of rigid, rotating segments linked at the joint centers This is a rough approximation ANKLE KNEE HIP ELBOW SHOULDER NECK LUMBAR

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Measuring Angles 0, 2 /2 /2 = 3.14159 0, 360 90 180 270 Degrees: Radians: 1 radian 57.3° 1 radian = 57.3° 1 revolution = 360° = 2 radians Positive vs. Negative Angles 0,+360° +90° +180° +270° Positive: Negative: +57.3° Typical convention: Positive angles Counterclockwise rotation Negative angles Clockwise rotation 0,-360° -270° -180° -90° -57.3° Absolute Angle (or Inclination Angle) Orientation of a line segment with respect to a fixed line of reference Use absolute angles for equations relating torques to motion Trunk angle from vertical Trunk angle from horizontal Computing Absolute Angles in 2-D Use trigonometry to compute absolute angles from , coordinates of two points , , atan

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Relative Angle Angle between two line segments Can compute relative angle by subtracting absolute angles of segments: 2/1 = axis of rotation 2/1 segment 1 segment 2 Joint Angles Joint angles are relative angles between adjacent body segments ankle elbow knee shoulder hip Can think of as: Rotation of distal segment relative to proximal and/or Rotation of proximal segment relative to distal Joint angle of zero = anatomical position elbow knee Joint Angles in 2-D Flexion & Abduction : between longitudinal axes External rotation: between AP or ML axes Sagittal View Frontal View AP PELVIS Transverse View ML PELVIS flexion abduct KNEE SHOULDER AP THIGH ML THIGH HIP external Measuring Joint Angles Devices for directly measuring joint angles: Goniometer Electrogoniometer

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Computing Joint Angles Involves subtracting absolute angles of segments Exact formula and order of subtraction depends on the joint and the convention chosen knee ANKLE HIP KNEE thigh =25° leg =70° trunk =60° hip hip = trunk thigh knee = leg thigh If facing left and flexion > 0: Range of Motion Hip ROM Flexion ROM Extension ROM Can measure for person or task as: Maximum joint angle Difference between maximum and minimum joint angles Restrictions in range of motion can impair performance Exceeding functional range of motion can result in injury Excessive or restricted range of motion can indicate injury or other disorder Angular Displacement ( Dq Change in the absolute or relative angle of an object between two instants in time Doesnt depend on the path between orientations Has angular units ( e.g. degrees, radians) angular displacement axis of rotation final orientation initial orientation Computing Angular Displacement Compute angular displacement ( Dq ) by subtracting initial from final orientation angle: Dq = final initial axis of rotation Dq final initial initial orientation final orientation

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Computing Displacement When computing displacement, must continuously increase or decrease over the range of motion # of full rotations thus included in displacement = 320° = 390° (not 30 °) Dq = = 70° = -40° (not 320 °) = 30° Dq = -70° crossing the x-axis in the + direction crossing the x-axis in the direction Angular Velocity ( Rate of change of the angle of an object Dq angular velocity change in angle change in time angular displacement change in time final initial final t initial Can compute for an absolute or a relative angle Symbolic notation: Has units of (angular units)/time ( e.g. radians/s, °/s) gives average angular velocity from t initial to t final Example Problem #1 At stride foot contact of a baseball pitch, a pitchers shoulder is in 88° of external rotation In the arm cocking phase, the shoulder externally rotates through a displacement of 86°. When the ball is released 37 ms later, at the end of the acceleration phase, the shoulder is in 64° of external rotation What was a) the shoulder angle at the end of the arm cocking phase? b) the average shoulder angular velocity during the acceleration phase? Average vs. Instantaneous Velocity Previous formula gives average velocity between an initial time and a final time Instantaneous angular velocity = angular velocity at a single instant in time Instantaneous angular velocity often more important Estimate using the central difference method: at t = at (t 1 + t)] [ at (t 1 t)] (t 1 + t) (t 1 t) where t is a very small change in time

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Relative Angular Velocity Rate of change of the angle between two segments If segment 2 is rotating at velocity 2/1 relative to segment 1, and segment 1 is rotating at velocity , the angular velocity of segment 2 is: 2/1 2/1 Angular velocity of segment 2 relative to segment 1 Segment 1 Segment 2 = + 2/1 Example Problem #2 During a forehand tennis stroke, a player is rotating her pelvis towards the ball at 200 /s, horizontally adducting her shoulder at 540 /s, and extending her wrist at 150 /s. What are the absolute angular velocities of: Her pelvis-and-torso? Her upper limb? Her hand-and-racquet? Angular Velocity as a Slope Given a graph of angular position vs. time: slope = average from t 1 to t time ( degrees slope = instantaneous at t (21) Dq (21) Can estimate vs. time from slope, as done previously Angular Acceleration Rate of change of angular velocity Dw angular acceleration change in angular velocity change in time t Symbolic notation: Has units of (angular units)/time e.g. rad/s , °/s gives average angular accel. from t to t

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Instantaneous Angular Accel. Previous formula gives average angular acceleration between an initial time and a final time Instantaneous angular acceleration = angular acceleration at a single instant in time Estimate using the central difference method: at t = at (t 1 + t)] [ at (t 1 t)] (t 1 + t) (t 1 t) where t is a very small change in time Effects of Angular Acceleration Velocity Acceleration Change in Velocity (+) (+) (+) () () () () (+) Velocity and acceleration In same direction: velocity increases magnitude Opposite directions: velocity decreases magnitude Larger accel. magnitude faster change in velocity Increase in + dir. Decrease in + dir. Increase in dir. Decrease in dir. Angular Acceleration as a Slope Given a graph of angular velocity vs. time: time ( deg/s (21) Dw (21) slope = instantaneous at t slope = average from t 1 to t Can estimate vs. time from slope, as done previously Example Problem #3 A volleyball player spikes the ball Starting with her shoulder flexed, she begins to extend her shoulder to bring her arm forward She contacts the ball 120 ms later, with her shoulder extending at 700°/s After another 100 ms, at the end of follow-through, her shoulder stops extending What was the average acceleration at the shoulder before ball contact and after ball contact?

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Propulsive & Braking Phases Propulsive Phase Magnitude of velocity increases Velocity and acceleration in same direction Propulsion produced by: Agonist muscles (concentric contraction) External torques in direction of motion Braking Phase Magnitude of velocity decreases Velocity and acceleration in opposite directions Braking produced by: Antagonist muscles (eccentric contraction) External torques opposite to direction of motion 40 80 120 0 0.5 1 1.5 2 2.5 3 Elbow Angle (deg) Example: Biceps Curl -800 -400 400 800 0 0.5 1 1.5 2 2.5 3 Acceleration (deg/s -250 -150 -50 50 150 250 0 0.5 1 1.5 2 2.5 3 Velocity (deg/s) -60 -40 -20 20 40 60 80 0 0.2 0.4 0.6 0.8 1 Time (s) Stick angle (deg) Example Problem #4a Pictured is the absolute angle of a hockey stick during a slap shot. Sketch the angular velocity and angular acceleration during the shot and identify its phases. How would the solution differ if the player pauses at the end of the backswing? How and why might this affect the speed of the shot? -60 -40 -20 20 40 60 80 0 0.2 0.4 0.6 0.8 1 1.2 Stick Angle (deg) Example Problem #4b -200 -100 100 200 300 400 0 0.2 0.4 0.6 0.8 1 1.2 Stick Velocity (deg/s) P B P B P B P B -2000 -1000 1000 2000 0 0.2 0.4 0.6 0.8 1 1.2 Stick Accel. (deg/s Time (s)

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Laws of Constant Angular Accel. where: = angular acceleration 0 = angular velocity at initial time t = angular velocity at final time t Dq = angular displacement between t and t t = change in time (= t t Use + values for + direction, values for direction = + a D Dq = ½ ( Dq = 0 t + (½) t) = + 2 Dq When angular acceleration is constant: Example Problem #5 A discus thrower stands facing the back of the circle and starts to spin. He releases the discus 2 seconds later after spinning 540° (1.5 revolutions) to his left. Assume that he accelerates at a constant rate. What was his angular acceleration as he spun? How fast was he spinning after the first 180°? How fast was he spinning at the time of release?

Be able to identify phases of movement infer sources of propulsion and braking Be able to use the laws of constant angular accel Grasp the applications to human movement Questions to Think About Why would tight calf muscles restrict the ability to ID: 22200

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Angular Kinematics Learning Objectives: Define angular kinematics Understand & compute absolute & relative angles, angular displacements, velocities & accelerations Estimate instantaneous angular velocity & accel. Be able to identify phases of movement & infer sources of propulsion and braking Be able to use the laws of constant angular accel. Grasp the applications to human movement Questions to Think About Why would tight calf muscles restrict the ability to run more than the ability to walk? Which muscles are used to speed up the extension of the elbow during a jump shot in basketball? As the knee flexes after landing from a jump, why are the knee extensors active? If you are trying to increase a baseball players bat speed at impact, what kinematic variables should you consider? Angular Kinematics Kinematics The description of motion as a function of space and time Forces causing the motion are not considered Angular Motion (Rotation) All points in an object move in a circle about a single, fixed axis of rotation All points move through the same angle in the same time Angular Kinematics The kinematics of particles, objects, or systems undergoing angular motion Angular Kinematics & Motion Volitional movement performed through rotation of the body segments The body is often analyzed as a collection of rigid, rotating segments linked at the joint centers This is a rough approximation ANKLE KNEE HIP ELBOW SHOULDER NECK LUMBAR

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Measuring Angles 0, 2 /2 /2 = 3.14159 0, 360 90 180 270 Degrees: Radians: 1 radian 57.3° 1 radian = 57.3° 1 revolution = 360° = 2 radians Positive vs. Negative Angles 0,+360° +90° +180° +270° Positive: Negative: +57.3° Typical convention: Positive angles Counterclockwise rotation Negative angles Clockwise rotation 0,-360° -270° -180° -90° -57.3° Absolute Angle (or Inclination Angle) Orientation of a line segment with respect to a fixed line of reference Use absolute angles for equations relating torques to motion Trunk angle from vertical Trunk angle from horizontal Computing Absolute Angles in 2-D Use trigonometry to compute absolute angles from , coordinates of two points , , atan

Page 3

Relative Angle Angle between two line segments Can compute relative angle by subtracting absolute angles of segments: 2/1 = axis of rotation 2/1 segment 1 segment 2 Joint Angles Joint angles are relative angles between adjacent body segments ankle elbow knee shoulder hip Can think of as: Rotation of distal segment relative to proximal and/or Rotation of proximal segment relative to distal Joint angle of zero = anatomical position elbow knee Joint Angles in 2-D Flexion & Abduction : between longitudinal axes External rotation: between AP or ML axes Sagittal View Frontal View AP PELVIS Transverse View ML PELVIS flexion abduct KNEE SHOULDER AP THIGH ML THIGH HIP external Measuring Joint Angles Devices for directly measuring joint angles: Goniometer Electrogoniometer

Page 4

Computing Joint Angles Involves subtracting absolute angles of segments Exact formula and order of subtraction depends on the joint and the convention chosen knee ANKLE HIP KNEE thigh =25° leg =70° trunk =60° hip hip = trunk thigh knee = leg thigh If facing left and flexion > 0: Range of Motion Hip ROM Flexion ROM Extension ROM Can measure for person or task as: Maximum joint angle Difference between maximum and minimum joint angles Restrictions in range of motion can impair performance Exceeding functional range of motion can result in injury Excessive or restricted range of motion can indicate injury or other disorder Angular Displacement ( Dq Change in the absolute or relative angle of an object between two instants in time Doesnt depend on the path between orientations Has angular units ( e.g. degrees, radians) angular displacement axis of rotation final orientation initial orientation Computing Angular Displacement Compute angular displacement ( Dq ) by subtracting initial from final orientation angle: Dq = final initial axis of rotation Dq final initial initial orientation final orientation

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Computing Displacement When computing displacement, must continuously increase or decrease over the range of motion # of full rotations thus included in displacement = 320° = 390° (not 30 °) Dq = = 70° = -40° (not 320 °) = 30° Dq = -70° crossing the x-axis in the + direction crossing the x-axis in the direction Angular Velocity ( Rate of change of the angle of an object Dq angular velocity change in angle change in time angular displacement change in time final initial final t initial Can compute for an absolute or a relative angle Symbolic notation: Has units of (angular units)/time ( e.g. radians/s, °/s) gives average angular velocity from t initial to t final Example Problem #1 At stride foot contact of a baseball pitch, a pitchers shoulder is in 88° of external rotation In the arm cocking phase, the shoulder externally rotates through a displacement of 86°. When the ball is released 37 ms later, at the end of the acceleration phase, the shoulder is in 64° of external rotation What was a) the shoulder angle at the end of the arm cocking phase? b) the average shoulder angular velocity during the acceleration phase? Average vs. Instantaneous Velocity Previous formula gives average velocity between an initial time and a final time Instantaneous angular velocity = angular velocity at a single instant in time Instantaneous angular velocity often more important Estimate using the central difference method: at t = at (t 1 + t)] [ at (t 1 t)] (t 1 + t) (t 1 t) where t is a very small change in time

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Relative Angular Velocity Rate of change of the angle between two segments If segment 2 is rotating at velocity 2/1 relative to segment 1, and segment 1 is rotating at velocity , the angular velocity of segment 2 is: 2/1 2/1 Angular velocity of segment 2 relative to segment 1 Segment 1 Segment 2 = + 2/1 Example Problem #2 During a forehand tennis stroke, a player is rotating her pelvis towards the ball at 200 /s, horizontally adducting her shoulder at 540 /s, and extending her wrist at 150 /s. What are the absolute angular velocities of: Her pelvis-and-torso? Her upper limb? Her hand-and-racquet? Angular Velocity as a Slope Given a graph of angular position vs. time: slope = average from t 1 to t time ( degrees slope = instantaneous at t (21) Dq (21) Can estimate vs. time from slope, as done previously Angular Acceleration Rate of change of angular velocity Dw angular acceleration change in angular velocity change in time t Symbolic notation: Has units of (angular units)/time e.g. rad/s , °/s gives average angular accel. from t to t

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Instantaneous Angular Accel. Previous formula gives average angular acceleration between an initial time and a final time Instantaneous angular acceleration = angular acceleration at a single instant in time Estimate using the central difference method: at t = at (t 1 + t)] [ at (t 1 t)] (t 1 + t) (t 1 t) where t is a very small change in time Effects of Angular Acceleration Velocity Acceleration Change in Velocity (+) (+) (+) () () () () (+) Velocity and acceleration In same direction: velocity increases magnitude Opposite directions: velocity decreases magnitude Larger accel. magnitude faster change in velocity Increase in + dir. Decrease in + dir. Increase in dir. Decrease in dir. Angular Acceleration as a Slope Given a graph of angular velocity vs. time: time ( deg/s (21) Dw (21) slope = instantaneous at t slope = average from t 1 to t Can estimate vs. time from slope, as done previously Example Problem #3 A volleyball player spikes the ball Starting with her shoulder flexed, she begins to extend her shoulder to bring her arm forward She contacts the ball 120 ms later, with her shoulder extending at 700°/s After another 100 ms, at the end of follow-through, her shoulder stops extending What was the average acceleration at the shoulder before ball contact and after ball contact?

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Propulsive & Braking Phases Propulsive Phase Magnitude of velocity increases Velocity and acceleration in same direction Propulsion produced by: Agonist muscles (concentric contraction) External torques in direction of motion Braking Phase Magnitude of velocity decreases Velocity and acceleration in opposite directions Braking produced by: Antagonist muscles (eccentric contraction) External torques opposite to direction of motion 40 80 120 0 0.5 1 1.5 2 2.5 3 Elbow Angle (deg) Example: Biceps Curl -800 -400 400 800 0 0.5 1 1.5 2 2.5 3 Acceleration (deg/s -250 -150 -50 50 150 250 0 0.5 1 1.5 2 2.5 3 Velocity (deg/s) -60 -40 -20 20 40 60 80 0 0.2 0.4 0.6 0.8 1 Time (s) Stick angle (deg) Example Problem #4a Pictured is the absolute angle of a hockey stick during a slap shot. Sketch the angular velocity and angular acceleration during the shot and identify its phases. How would the solution differ if the player pauses at the end of the backswing? How and why might this affect the speed of the shot? -60 -40 -20 20 40 60 80 0 0.2 0.4 0.6 0.8 1 1.2 Stick Angle (deg) Example Problem #4b -200 -100 100 200 300 400 0 0.2 0.4 0.6 0.8 1 1.2 Stick Velocity (deg/s) P B P B P B P B -2000 -1000 1000 2000 0 0.2 0.4 0.6 0.8 1 1.2 Stick Accel. (deg/s Time (s)

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Laws of Constant Angular Accel. where: = angular acceleration 0 = angular velocity at initial time t = angular velocity at final time t Dq = angular displacement between t and t t = change in time (= t t Use + values for + direction, values for direction = + a D Dq = ½ ( Dq = 0 t + (½) t) = + 2 Dq When angular acceleration is constant: Example Problem #5 A discus thrower stands facing the back of the circle and starts to spin. He releases the discus 2 seconds later after spinning 540° (1.5 revolutions) to his left. Assume that he accelerates at a constant rate. What was his angular acceleration as he spun? How fast was he spinning after the first 180°? How fast was he spinning at the time of release?

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