PEP 294: Lecture Notes
XIII-XIV. Angular Kinetics
Torque (pp. 396-405)
- Related problems in Ch. 13: IP1, 2, 3, 4, 5, AP1, 3, 4 & 5
1. Torque
 | Torque (moment of force): |
- rotary force that causes angular motion
- proportional to angular acceleration
- angular equivalent of force (cause of angular motion)
- torque = (force)(moment arm)
- vector quantity (magnitude + direction):
counterclockwise (+) or clockwise ( -)
- standard unit: Nm (not J)
| Moment arm: |
- shortest (perpendicular) distance between the line of action of the force and the axis of rotation
- change in
=> change in T
2. Muscle Torque
| Muscle torque: |
- torque produced by a muscle
- cause of the joint motion
-
:
(a) perpendicular distance between the joint center and muscle's line of action
(b) varies as joint angle changes
(c) maximum when angle of pull is 90°
| Resultant joint torque: |
- sum of the torques produced by the muscles around a joint
- agonist torque vs. antagonist torque
- Example: Seesaw case (p. 402)
TJoey (clockwise) = -(200)(1.5) = -300 Nm
TSusie (counterclockwise) = (190)(1.6) = 304 Nm
Tnet = TSusie + TJoey = 304 - 300 = 4 Nm
Net T is positive: counterclockwise motion (Susie's end falls.)
| Joint motion vs. torque: |
- concentric torque: produced through concentric activation of muscle (same direction)
- eccentric torque: produced through eccentric activation of muscle (opposite direction)
Levers (pp. 405-415)
- Related problems in Ch. 13: IP6, 7, 8, AP7, 9 & 10
1. Lever System
| Lever: simple machine consisting of a rigid barlike body that rotates about an axis |
| Elements: |
- body
- fulcrum: point of support & rotation
- resistance (R)
- force (F)
| Torques acting in a lever: |
- resistance torque = (resistance)(resistance arm)
- force torque = (force)(force arm)
| Equilibrium in a lever: |
- net torque = 0:
force torque - resistance torque = 0 => force torque = resistance torque
(R)(RA) = (F)(FA)
- net force = 0:
force on fulcrum + resistance + force = 0
Ffulcrum + R + F = 0
- Example -- biceps brachii (p. 413): FA = 3 cm, R = 70 N, RA = 30 cm => F
(F)(FA) = (R)(RA)
(F)(0.03) = (70)(0.3)
F = 2.1 / 0.03 = 700 N
| Mechanical advantage: |
- mechanical advantage: force gain
MA = resistance / force = force arm / resistance arm
MA = R / F = FA / RA
- mechanical advantage of muscle:
(a) change in angle of pull => FA & MA changes
(b) for a given F, FA is max when angle of pull is 90°
(c) optimal joint angle (max MA) varies from one muscle to another
2. Types of Lever
| First class lever: |
- fulcrum between R and F
- R & F in the same direction
- MA depends on the location of fulcrum
- example: atlanto-occipital joint
| Second class lever: |
- FA > RA => F < R
- Ffulcrum & F in the same direction
- advantage in force: mechanical advantage > 1
- most of the lever used in daily life
- example: elbow in push-up
| Third class level: |
- RA > FA => F > R
- Ffulcrum & R in the same direction
- advantage in ROM and speed: mechanical advantage < 1
- most of the levers in human body ( > 95 % )
- example: elbow in lifting
Center of Gravity (pp. 415-424)
1. Center of Gravity
| Center of gravity (CG): |
- a point around which the weight of a body is balanced in all directions
- center of gravity = center of mass (CM)
| CG of objects: |
- within or outside the object
- posture vs. CG
2. Computation of CG
| Reaction board method: |
- reaction board
- use of the equilibrium equation of lever system
| Segmental method: |
- Lab #9
- human body CG: affected by the posture
- steps:
(a) quantification of the posture -- digitizing
(b) computation of the segmental CGs
(c) computation of the body CG -- segmental masses & segmental CGs
Moment of Inertia and Angular Momentum (pp. 436-453)
1. Moment of inertia
| Moment of inertia (MOI): |
- inertia of the object in angular motion
(a) tendency to maintain the current state of angular motion
(b) difficulty in changing state of angular motion
- function of mass distribution about the axis of rotation
I = mr2
I = S(mr2)
"The more closely mass is distributed to the axis of rotation, the easier it is to initiate or stop angular motion."
- minimum when the axis of rotation passes through the CM
- Standard unit: kg m2
| Human body vs. MOI: |
- MOI: function of posture
- MOI: function of axis of rotation
| Torque: |
- torque = (MOI)(angular acceleration)
T = I a
- vector quantity
2. Angular Momentum
| Angular momentum: |
- quantity of angular motion
- angular momentum = (MOI)(angular velocity)
H = I w
- vector quantity
| Conservation of angular momentum: |
- T = 0 => H = I w = constant
"The total angular momentum of a given system remains constant in the absence of external torques."
- For constant H: I is inversely proportional to w.
example: diving
| Transfer of angular momentum: |
- H of one body part may transfer to another.
- H of each body part may change while the total H remains constant.
Comparison: Linear vs. Angular Kinetics
| Linear | Angular | |
| Inertia | mass (m) | moment of inertia (I) |
| Cause of motion | force (F = ma) | torque (T = Ia) |
| Momentum | M = mv | H = Iw |
