L22 - Angular Impulse-Momentum

• But how does one generate angular rotation in the first place?

  • Recall that in order to accelerate an object we need to apply a torque or off centre (CoM) force
  • The resistance to change in motion we have introduced as inertia
  • a = angular acceleration = angular velocity per time vector
• Mass Moment of Inertia:

  • Resistance of body to move, can come from how much mass & how far from axis of rotation
• Moment of inertia is dependent upon the axis of rotation:

  • Moment of inertia (I) is the distribution of mass about the axis of rotation
  • With a small moment of inertia the mass is tightly collected around the axis of rotation (through the CoG)
  • When the mass is distributed away from the axis of rotation, the whole body moment of inertia is much larger
• Moment of inertia is dependent upon the axis of rotation:

  • Axis of rotation from our mass
  • Further from mass, more resistance to rotation
• We can think of our limbs in the same manner:

  • Certain amount of resistance in rotation
  • Total inertia have to look at location CoM + outside of it
• How do we explain the change in rotation velocity?

  • Mass moment of inertia only explains how difficult or easy a movement may be considering the distribution of mass about the axis of rotation
• How do we explain the change in rotation velocity?

  • In order to explain speed of rotation
  • Greater resistance has effect on velocity
  • eg a finger skater
  • Heaps mass distributed (arms/legs out) vs close to CoM
  • How initiates rotation by redistributing mass (nothing else acting to change rotation) → as changes axes
  • Change resistant to rotate can change rotation velocity without any external force
• Linear quality of motion = Momentum:

  • To fully describe the property of inertia with units we can measure, we must quantify an object’s speed of motion, direction of motion, & resistance to change of motion
  • Momentum (L) = Mass x Velocity
  • Angular Momentum (H) = mr^2 x ω
  • mr^2 = resistance to rotate
  • ω = angular velocity
• Angular Momentum:

  • Quantity of angular motion possessed by a body; measured as the product of moment of inertia & angular velocity
  • H = I * ω
  • Or
  • H = m * k^2 * ω
• Angular Momentum:

  • Angular momentum, can either increase resistance or angular velocity to maintain momentum (check)
  • Mass x radius (distance from axis)squared x angular velocity
• Angular Momentum:

  • H = mk^2ω
  • 3 factors affect the magnitude of body’s angular momentum:

  1. Its mass (m)
  2. The distribution of that mass with respect to the axis of rotation (k or r), &
  3. Significant as is a squared value
  4. The angular velocity of the body (ω)

  • Units: kg m^2 rad/s
  • Usually expressed as: kg m^2/s or kg m^2 s^-1
• What we need to know about angular momentum:

  1. Once airborne, it is impossible for the athlete to alter his/her angular velocity by changing his/her moment of inertia

  • H = I * ω
• What we need to know about angular momentum (1):

  • No other forces acting on you
  • While on ground can change moment of inertia
  • Once in air all forces you gonna have & is conserved
  • Why is it conserved in the air?
  • No other external forces acting
  • Gravity acts thru CoM, as forces act thru CoM doesn’t cause rotation
  • So angular momentum always conserved when in the air
• What we need to know about angular momentum:

  • 2. When a body is rotating in the air, momentum is conserved
  • Why?
  • No external forces, &
  • Because gravity acts through the CoM
  • Recall that forces acting through the CoM can only cause linear motion not rotation
  • Gravity can only cause translation
• Recall:

  • L initial + I∑ F dt = L final ∴ (therefore) H inital + I∑ T dt = H final
• Recall:

  • Angular version of impulse momentum equation = basically the same as the linear equation
  • Have to apply external torque to change momentum
  • Otherwise total momentum of system is conserved (if no more external torque is applied)
• What we need to know about angular momentum:

  • Can be transferred from one axis of rotation to another
• What we need to know about angular momentum:

  • Apply a rotational torque of CoM when take off (jump)
  • Rotate arms & legs to transfer momentum from CoM to keep body upright = hitch kick technique
  • Transfer momentum to maintain forward progression
  • Someone tried somersaulting as a technique for long jump as goes with CoM rotation - but got banned as a LJ technique
  • However did achieve a great jump with never even somersaulting before trying in competition
• What we need to know about angular momentum:

  • Parallel axis theorem & transfer conservation of momentum
  • Direction of rotation = RHR (RIght Hand Rule)
  • Fingers curl in direction of ω & the thumb points in the direction of H
  • Momentum transferred from wheel to person without changing the system
• What we need to know about angular momentum:

  1. What happens to the momentum of the system if an internal torque is applied?
  2. What happens to the momentum of the system if we change the direction of the momentum vector of the wheel?
• Spinning Wheel Problem (complex) pt 1:

  • Transfer momentum from wheel to person when torque is applied to system causing rotation (wheel)
  • L initial = momentum of spinning wheel + momentum of person
  • Person = 0, so initial momentum = to the wheel
  • L final = whatever wheel + person doing, rotates it 90°, wheel now up person rotating down so cancel each other our
  • So total momentum = initial momentum
  • Reason we have movement is because have initial torque applied externally at start
  • Only reason see this change is because of external torque/impulse → something to conserve
• Spinning Wheel Problem (complex):

  • Then flips wheel 180°, rotation goes in opposite direction, rotation is faster
  • This might be because:
  • Now just talking about internal torques (cancel each other out), ignore initial impulse
  • Initial = wheel + person; Initial = wheel
  • Final = equal/opposite of initial of wheel (as flip it over 180°)
  • Final momentum of person should be 2x whatever wheel was initially
  • Initial related to external torque (without no difference)
  • Without = handed to you spinning = no external torque acting
• What we need to know about angular momentum:

  • What could this person do to slow their fall?
  • Swing arms in that/same direction (backwards) - natural instinct
  • Take up some momentum from CoM to control rotation
• Consider:

  • Why do we swing our arms when we run?
  • Can you run without arm swing?
  • What does that feel like?
• What we need to know about angular momentum:

  • We can also alter the axis of rotation in the air…
  • Tip axis, change momentum axis to change rotation in air
  • Move body off centre by distributing limbs (eg 1 arm up, 2 arms or none)
  • Off centre axis to twist
  • Transferring momentum from 1 axis to another
• Practically what does this mean for athletic performance?

  • Applies initial torque that starts rotation and then can conserve it to then alter/control rotation by transferring momentum/redistributing limbs
  • High Jump: J run up to initiate twist to apply torque to get rotation & redistribute limbs to get momentum/CoM over bar
• Recall the mass moment of inertia of a segmented mass system:

  • The total inertia (ICM) = ICM + mk^2
  • Essentially the Angular momentum of the system just adds the velocity term
  • (Ht) = (ICM + mk^2) ω
  • If we work through the brackets we get…
  • Angular momentum vector is parallel to angular velocity
  • Inertia around 2 axes & the velocity
• Multi-segmented bodies have both remote & local terms (parallel axis theorem):

  • Any limb rotating in air
  • Important in sports performance
  • Things we can utilise to help performance