Biomech - Angular Kinetics ch 7

Inertia

resistance of a body to changing motion

Angular inertia

property of an object to resist changes in its angular motion

not dependent on mass

Moment of inertia

Quantifies how an object’s mass is distributed relative to a specific axis of rotation

mass farther from axis creates more angular resistance

mass closer to the axis created less angular resistance

Moment of inertia equation

$$I_{a}=\sum m_{i}r_{i}^2$$

$I_{a}$ = inertia about axis a thru center of gravity (cg)

m = mass of particle i (kg)

r_i = radius from particle i to axis of rotation a (m)

Radius of gyration

simpler approach to consider mass distribution

describes how far from axis of rotation mass has to be concentrated to create enough resistance to change the motion back to original shape

Radius of gyration equation

$$I_{a}=mk_{a}^2$$

I_a = moment of inertia about axis a through cg

m = mass of object (kg)

k_a = radius of gyration about axis a through cg (m)

Distribution of mass

Distribution has a greater effect

Double the mass = 2 x I

Double the radius of gyration = 4 x I

Moments of inertia in the Human body

3 principal axes are

Anteroposterior (cartwheel)

Transverse (somersault)

Longitudinal (twist)

Manipulating inertia of the human body

altering relative angel at joint or joints changes I by changing the mass distribution

tucking versus laying position of divers & gymnasts

Tangential velocity (V_T) & Inertia (I) greater with ____

Longer radius

When choosing length of swinging implement what is the tradeoff?

Ease of swing versus effectiveness of velocity achieved

Angular Momentum

Quantifies the angular motion of an object, describing both:

• current state of motion

• resistance to changing motion

Angular Momentum equation

$$H_{a}=I_{a}w_{a}$$

H_a= Angular momentum about axis a (kg x m²/s)

I_a = moment of inertia about axis a (kg x m²/s)

w_a = angular velocity about axis a (rad/s)

Angular momentum Vector quantity

magnitude & direction

• magnitude: Iw

• Direction: same as w

H_a will change with change in I or w.

m is constant in human movement

Angular momentum of human body

Frequently, many limbs rotating at different w’s

• H_a is the sum of angular momentum of individual segments

Angular momentum of human Body equation

$$H_{a}=\sum\left(I_{\frac{i}{cg}}w_{i}\right)$$

H_a = angular momentum

I_i/cg = Moment of inertia

w_i = angular velocity

Newton’s first law: Angular interpretation

Angular momentum of an object remains constant unless a net external torque acts on it

• if I’s are constant, w is constant

for rigid projectile, gravity only external force

• weight acts thru cg

• weight has no moment arm, no external torque on projectile

Angular interpretation mathematically

H remains constant unless a net external torque acts on the body

$$H_{i}=I_{i}w_{i}=I_{f}w_{f}=H_{f}$$

Angular interpretation & Human body

human body is a system of rigid links with modifiable I

H is constant

• I increase, proportional decrease in w

• I decrease, proportional increase in w

Faster spin, reduce I (tuck arms)

Slower spin, increase I (arms out)

Control w of Limbs or Trunk

Repositioning one or more segments of body must be countered by the repositioning of one or more segments in the opposite direction to have constant H_cg

• repositioning means changing w

Newton’s Second Law: Angular Interpretation

If net external torque acts on body, body will accelerate in the direction of the torque. Acceleration is proportional to the torque and inverse to its inertia

Newton’s Second law: Angular interpretation Equation

$$\sum T_{a}=I_{a}\alpha_{a}$$

$\sum T_{a}$ = net external torque

I_a =moment of inertia

$\alpha_{a}$ = angular acceleration

Both I and $\alpha$ can change or be changed