L21 - Angular Kinetics: Moment of Inertia

• Recap:

  • T = Fr
  • Locating the CoM is related to torque in that, a force directed through the CoM should cause linear motion only
  • CoM is located at a point where the sum of torques = 0
  • rcm = ∑ri mi / ∑m
• Recap:

  • Intent & direction
  • How far from axis of rotation → can use to locate CoM (where sum of rotations = 0)
  • The sum of those masses that cause torque
• Newton’s Laws - acceleration:

  • Law of linear acceleration
  • F = m * a or a = F/m
• Newton’s Laws - acceleration:

  • Tells us something about the relation
  • For a given acceleration if we change the mass (increases) will need to increase force to maintain acceleration; if forced to stay same acceleration decreases
  • Mass important for resistance to accelerate
  • Why mass is equivalent to inertia of object
• Angular acceleration:

  • F = m* r *a
  • Fr = m * r^2 * a
  • T = mr^2a
  • a = T / m * r^2 (inertia)
• Angular acceleration:

  • Mass & distance will give us torque
  • Alpha symbol = rotational acceleration
  • Static is for equilibrium, when assumed system is not moving
  • This is now a dynamic equation; going to cause motion
  • What does this tell us to resistance of rotation
  • Them resistance to rotate related to mass to distance from axis of rotation (mass moment of inertia)
• Angular acceleration:

  • We know that T = Fr
  • a = T / mr^2
  • a = T / m * r ^2
• Angular acceleration:

  • How much movement going to be
  • Resistance related to mass & moment arm
  • If took mass & doubled it, will change resistance to rotation, is thinking of muscle effort, rotational acceleration (of muscle) will decrease
• Angular acceleration:

  • What if increase length of moment arm (away from axis), how hard does muscle have to work to cause acceleration, so will need more force/torque, as decreases acceleration
• Angular acceleration:

  • a = T / mr^2
  • Which one affects resistance more? Mass (m) or distance from axis of rotation (r^2)
  • The distance as is a squared term
  • Can increase resistance by adding mass or distance
  • In order to maintain acceleration muscle torque has to be larger when resistance increased
• Angular acceleration:

  • Resistance to rotate is related to mass & the distribution of mass around the axis of rotation
  • a = T / mr^2
• Angular acceleration:

  • Skating:
  • Redistribute limbs alters rate of spin
  • Increase rate of spin by pulling limbs in closet to axis (CoM - without changing torque)
  • When wants to slow down put limbs further away from CoM
• Angular acceleration:

  • Diving:
  • Change body segment in or to speed up spin, not rotate in entry
  • Change speed of rotation, rate of spin by distributing masses
• Determining Mass Moment of Inertia:

  • I = mr^2
  • I = m * r^2
  • I = inertia
  • m = mass
  • r^2 = distance from axis of rotation squared
• Determining Mass Moment of Inertia:

  • I = mr^2
  • This inertia related to mass
  • Long bones = long axes
  • If increase mass increase resistance
• How does this relate to sport performance?

  • The ratio of muscular strength to segmental moments of inertia is an important contributor to performance capability in rotational sports
  • eg, Simone Biles: 4’8” or 1.47 m; 47 kg
  • Repositioning limbs in the air alters the angular acceleration
• How does this relate to sport performance?

  • Rotational athletes:
  • See change in stature, see shorter athletes (lower resistance); where long rotation is important
• How does this relate to sport performance?

  • Simone Biles was 1st woman to land triple double on floor
  • Launch velocity: 5.78 m/s, her hang time is 1.43 s
  • Only had 3-4 strides to get speed
  • Has to get significant height perform rotations in, in smaller height, if tucks can rotate quickly
  • Size benefits her + got to be strong to get speed & hold positions under forces
• How does this relate to sport performance?

  • Simone Biles was 1st woman to land Yurchenko double pike in competition
  • Very short air time. Speed on take off & many rotations in due to small size + also has to be strong
• Mass moment of inertia describes an object rotating around it’s CoM:

  • I = mr^2
• A simpler example to start…

  • I = m1 * r1^2 + m2 * r2^2
  • I - (5)(0.75)2 + (5)(0.75)2
  • I = 2.813 + 2.813
  • I = 5.63 kgm^2
  • I =∑mr^2
• A simpler example to start…

  • I =∑mr^2
  • Spin barbell around CoM; middle will be CoM
  • Mass moment of inertia related to __
  • Mass x resistance
  • Mass moments of inertia units are kg/m
• What happens if the object rotates about an axis that is not at the CoM?

  • I = Icm +mk^2 – Radius of Gyration
  • I = (m1 * r1^2 + m2 * r2^2) + mk^2
  • Local = (m1 * r1^2 + m2 * r2^2)
  • Remote = mk^2
  • The total inertia (Itot) = ICM + mk^2
  • This is the parallel axes theorem
  • Resistance as pinned to joint (2 axes of rotation)
  • Only works when axes are parallel
  • Good for human movement eg hip pinned (& knee free)
• What happens if the object rotates about an axis that is not at the CoM?

  • Happens a lot in human movement, as joints & limbs rotate around mass can have 2 axes of rotation
  • Rotate around 1 end (around that axis)
  • Now have momentum of inertia = moment of inertia around CoM + remote term (distance from CoM)
  • Resistance to rotate + mass of object & distance of new axis from distance of rotation (+ mk^2)
• What happens if the object rotates about an axis that is not at the CoM?

  • Remote is total mass + distance of new/2nd axis from CoM/1st axis
  • So total around remote axis eg hip joint
• We can think of our limbs in the same manner…

  • The total inertia (Itot) = ICM + mk^2
  • During running, the thigh not only rotates about the hip axis (left, white arrow), which is also called the remove axis, but also about its own local axis (right, grey arrow)
  • The total moment of inertia is the sum of the moments of inertia about both the remote & local axes
• We can think of our limbs in the same manner…

  • The total inertia (Itot) = ICM + mk^2
  • Athletes with high rotational component & short time (sprinters)
  • Muscle hypertrophy closest to axis of rotation which makes it easier to rotate + more force/strength for rotation
  • Muscle mass closer to proximal end
  • While cyclist closet to distal end as don’t need to rotate limb
• Consider the mass effect on a swinging limb:

  • The moment of inertia of a swinging leg would thus be…
  • I = ∑mr2 + mk^2
  • Ithigh = ICM + mk^2
  • Ishank = ICM + mk^2
  • Ifoot = ICM + mk^2
  • Therefore…
  • (Itot) = Ithigh + Ishank + Ifoot
  • Consider: What happens to the I of the swinging limb from A to B?
  • Which is harder to rotate?
  • How might this affect the running mechanics?
  • Have turn for each segment put all together = what going on in limb
• Which skill do you think would be physically harder to perform:

  • Layout or a tuck
  • Layout harder to perform than a tuck
  • In order to control that rotation takes a lot of effort to get acceleration required