Flexibility - kau.edu.sa

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Flexibility
Definition
The ROM available in a jt. or group of jts.
The ability of a m. to relax & yield to a stretching force.
The freedom or ability to move part or parts of body in a wide R
of purposeful movs at the required speed.
The ability to move a jt. Through a normal ROM with undue
stress to musculo-tendinous unit.
Flexibility
 Extensibility of periarticular tissues to allow
normal motion of a joint. or a limb.

* Hypermobility
 The ROM in excess of the accepted
normal in most of joints. (may be normal)
* Laxity
 Excessive jt. instability & abnormal motion
of a given joint. (pathological)
Types of Flexibility

Ballistic Flexibility:
Normal ROM associated with bouncing &
rhythmic movements.

Dynamic Functional Flexibility:
Activities that need certain angle to certain joints.
& particular patterns
(usual activities).
Factors Affecting Flexibility
1.
Wrong habits: ↓ flexibility sitting for a long period (in school,
watching TV, computer work, sedentary games)
2.
Good habits: → ↑ flexibility.
3.
Genetic factors: females > males.
Hormonal factors: females > males.
Body build proportion & ↑ height → ↑ flexibility .
Body wt. ↑ → ↓ flexibility
Age: flexibility changes during a certain age
↑ 2-6y
↓ 6 - 12 y
↑ 18 - 22 y
↓ 35 - up
Skills & sport activities → ↑ flexibility.
4.
5.
6.
7.
8.
Contractile & Non-contractile Tissue

Contractile
1. Ms

Non-contractile
1. Skin
2. CT
3. Ligaments
4. Tendons
5. Capsules
6. Synovial Membrane
*N.B.

Soft tissues that can restrict joint motion are:
1. Skin
2. CT, tendons, ligaments
3. Ms
Each one has unique qualities affecting its extensibility (ability to elongate).

Factors affecting soft tissues lengthening:
1.
2.
3.
4.
5.
Velocity of stretch force
Intensity of stretch force
Duration of stretch force
Mechanical properties of contractile & non-contractile tissues
Neuro-physiological properties of contractile tissues
When soft tissue is stretched, either
elastic or plastic changes occur

Elasticity
Ability of soft tissue to return to its resting length
after passive stretching.

Plasticity
Tendency of soft tissue to assume new & greater
length after the stretch force has been removed.

Both contractile & non-contractile tissues have elastic & plastic
properties.
Degrees of Deformity

1st degree (grade I): can be corrected actively
(shortness).

2nd degree (grade II): can be corrected passively
(shortness).

3rd degree (grade III): can’t be corrected neither
actively nor passively (contractures).
Indications of Stretching
1.
Limited ROM due to adhesions & scar tissues formation leading to
shortening of Muscles, CT & Skin.
2.
Prevention of structural (neglected soft tissue → shortening →
structural limitations →skeletal deformities
(prophylactic intervention)
3.
When contractures interfere with every day functional activities or
nursing care.
3.
When there’s muscle weakness & opposing tissue tightness.
* Tight muscle must be elongated before weak m. can be effectively strengthened.
Goals of Stretching
1. To regain or re-establish normal joint. ROM & mobility of soft
tissue surrounding a jt..
2. To prevent irreversible contractures.
3.
To ↑ general flexibility of a part of the body prior to vigorous
strengthening excerises.
4. To prevent or minimize risk of musculo-tendinous injuries
related to specific physical activities & sports.
Precautions to Stretching
1.
Don’t passively stretch (force) a joint beyond its normal ROM
(Remember that normal ROM varies among normal individuals).
2.
Newly united # should be protected by stabilization between # site
& the joint where motion takes place.
3.
Extra caution in pts. with known or suspected osteoporosis due to:
a.
Disease
b. Prolonged bed rest
c. Aging
d. Prolonged use of steroids (cortisone)
4.
Avoid vigorous stretching of muscles & CT that have been
immobilized over a long period of time. CT (tendons & ligaments)
lose their tensile strength after prolonged immobilization.
Precautions to Stretching
a.
b.
c.
High intensity, short duration procedures →
traumas → soft tissue weakness
Stretching should be low intensity for long
duration stretch.
Strengthening excerises should be added to
stretching program at some point,
so patient will be able to develop appropriate
balance between flexibility & strength.
Contraindications to Stretching
1.
Acute infections.
9.
Joint laxity (hypermobility).
2.
Signs of inflammation.
10. Joint subluxation.
3.
Pain at the site to be stretched.
11. Joint dislocation.
4.
Osteomyelites
5.
Bone tumors.
6.
Advanced osteoporosis.
(acute
stage).
7.
Severe spasticity.
8.
Pain that isn’t yet evaluated by the
PT or physician.
12. Joint fusion.
13. Joint deformity.
14. Tissue adhesions (contractures).
15. Acute Traumas
Principles of Stretching
1.
Consider the best stretching type to ↑ ROM.
2.
Explain goals of stretching to the pt.
3.
Position the pt. in a comfortable relaxed pos. → allow the best plane
of motion to conduct stretching.
4.
Free the area from restrictive clothing, bandage, splints.
5.
6.
Apply superficial heating to the soft tissues to be stretched.
It will ↓ possibility of injury.
7.
Stretch direction is done opposite to tightness.
7.
The pt. must be completely relaxed throughout stretching ,employ
relaxation techniques before stretching.
8.
Stretching movs. or exs. are performed at least 3 times/week
to maintain flexibility. Progress to daily stretching routines.
9.
1-3 repetitions of each stretch must be done. The No. of
repetitions can be ↑ if the goal is to ↑ ROM.
10.
Degree of stretch can be ↑ or ↓ depending on time of stretch
& external force applied.
11.
Tension produced shouldn’t cause pain, the pt should only
feel a slight tension, which ↓ with stretching. (due to
adaptation)
12.
Stretch ↑ gradually, building to a max. as the tissues release.
13.
Stretch should be removed gradually to prevent rebouncing
or tightening of the ms.
Stretching depends on:
3.
Type of stretching force.
Intensity.
Duration
4.
Velocity.
1.
2.
Types:
A. Passive Stretching
PT applies an external force & controls stretch
1direction, 2speed, 3intensity & 4duration to the tight m..


Don’t confuse between:
1. Passive stretching → elongation of soft tissues beyond
full R of restriction.
2.
Passive ROM exs.
→ applied within the unrestricted (available) ROM.

Pt. must be completely relaxed during passive stretching.

Time of stretch force is usually 15 : 30 sec & repeated several
times during ex. session.
1. Manual Passive Stretching

Usually considered a short-duration stretch.

No specific No. of secs. are determined to be the most effective
duration.

In a study to stretch hip abd. of healthy subjects 15, 45 sec & 2
min at the same speed → no difference.

Intensity & duration of stretch depend on:

1.
Pt’s tolerance
2.
PT strength & endurance
Low intensity, long duration manual stretch is more
comfortable & tolerated by pt.
Maintained versus Ballistic Stretch
MANUAL PASSIVE
STRETCH
VERY
SLOW
MAINTAINED,
GENTLE
PROLONGED 15-30 SEC
OR MORE
Inhibit stretch reflex
&
↓ Muscle tension

HIGH INTENSITY
STRETCH
BOUNCING
QUICK
LENGTHENING
VERY SHORT
DURATION
Facilitate muscle contraction
&
↑ tension
2. Prolonged Mechanical Stretch

Low intensity external force (5-15 lb) (2: 6 kg) applied to shortened
tissues over a prolonged period by mech. equipments.

Stretch force is applied through positioning with weighted traction,
pulley system, dynamic splint & serial casts.

Prolonged stretch may be maintained for 20-30 min. or longer →
effective stretch & ↑ ROM
(low intensity mechanical stretch).

Comparing long-duration mech. stretch & manual passive stretch
( the latter is rather short & there’s transient & temporary achievement of ROM).
Examples
1.
Bohannon evaluated the effectiveness of an 8 min mechanical
hamstring stretch compared to a 20 min or longer using overhead
pulley system. The 8 min stretch resulted in a small ↑ in hamstring
flexibility, which was lost in 24 hrs.
It was suggested that 20 min or longer stretch is more effective to ↑
ROM & has a more permanent basis.
2.
Use of tilt table-wedge board standing for 30 min/daily → ↑
dorsiflexion in neurological pts.
3.
Dynasplint (dynamic splint) → prolonged low intensity stretch of
elbow, wrist, knee & ankle → ↑ ROM.
4.
Low intensity prolonged stretch of 5-12 lb applied for
1 hr/daily has been found to be more effective than manual passive
stretch over a 4-week period with pt’s with bilateral knee flexion
contractures. It is more comfortable
Permanent lengthening
(Plastic changes in contractile & non-contractile tissues)
has been reported with long duration stretch.

The term “Permanent lengthening” means
that the length is maintained after the
stretching force is removed.
3. Cyclic Mechanical Stretching

It is passive stretching using a mechanical devices as an
Autorange using a 20-sec high intensity (up to the
patient’s pain tolerance).
1.
Intensity of stretch,
Length of each cycle,
No. of stretch cycles/min Can be adjusted with the unit.
2.
3.

It is similar & may be used as a useful alternative to
manual passive stretching.
B. Active Inhibition





Is a tech. in which the patient reflexively relaxes muscles to be
elongated prior to stretching maneuver.
When the patyent is relaxed → minimal R to elongate the m.
It relaxes only the:
- Contractile structures,
- Not the CT.
The m. must be:
Normally innervated and Under voluntary control.
It Can’t be used with pts having:
1. Weakness,
2. Spasticity
3. Paralysis (From a neuromuscular disease).
It Can be used in:
Post-operative patient. (to avoid tightness).
Active Inhibition Techniques.
1.
2.
Contract-relax (hold-relax):
Pt performs an isometric contraction of tight muscle before it is
passively stretched (lengthened).
M. will relax as a result of ”autogenic inhibition”
(Golgi tendon organ may fire at ↓ tension).
Contract-relax-contract (hold-relax-contract).
a. 1st a contraction of tight muscle.
b. 2nd relaxation of tight muscle. (autogenic inhibition).
c. 3rd concentric contraction of opposite muscle (reciprocal
inhibition of tight m.).
In a study, the contract-relax-contract produced a greater ↑ in
ankle dorsiflexion > contract-relax tech. (in short calf m ).
* Both techs produce ↑ ROM > manual stretching.
3. Agonist Contraction

Agonist = muscle opposite to the tight muscle.

Antagonist = Tight muscle.

Patient dynamically contracts (shortens)
the muscle opposite to the tight muscle against
resistance → reciprocal inhibition of tight muscle.
C. Self-Stretching

Is a type of flexibility excerises. the patient carries
out by himself.

Pt. may passively stretch his tightness by:
1.
His/her own body weight. (& gravity)
2.
Active inhibition.
3.
Manual passive stretch (using sound limb)
Peripheral joint mobilization
Definition:
Peripheral joint mobilization (PJM) is the use of
skilled graded forces to mobilize joints:
to improve motion & normalize joint function.
Mobilization Techniques are used
to
1.
2.
3.
4.
5.
6.
Improve jt. nutrition.
Improve m. spasm & tension
Reduce pain.
Reverse jt. hypomobility.
Improve or restore motion.
Treat jt. dysfunction as stiffness.
The PT should recognize

When the mobilizing techniques
are indicated (at any ROM), or

If other stretching techniques are
more effective (stretching, stretching ex’s or CTM)
To regain the lost motion
A. Joint Mobilization


Is a type of passive mov performed by the PT at a speed slow enough
that the pt. can stop the mov.
The tech. may be applied with:
Oscillatory motion or
2. Sustained stretch
to

1.
↓Pain or
2.
↑mobility.
The tech. may use:
1.
2.
Physiologic or
Accessory movs.
1.
Mobilization Techniques

Physiologic
movements
1.
Movs. that the pt
can perform voluntarily as
flex, ext, abd, add & rot.

Accessory movements
1.
Movs within the jt &
surrounding tissues that are
necessary for normal ROM, can’t be
done by pt.
2. Described as (Osteokinematic).
i.e. mov of bones in relation to each
others
2. Described as
a. Component motions
b.
joint play.
Accessory motions

Accessory motions

B. Joint Play

Motions that
accompany active mov, but

Motions that occur in jts
& distensibility or give in of the jt
capsule,

Aren’t under voluntary control

Component mov is the often used term with
accessory mov.
As:

Can be demonstrated passively,

upward rot. of scapula & clavicle
occurring in
sh. flex.
Can’t be performed actively by the pt.


which allow bones to move.


Are necessary for normal functioning
through the ROM.
Include:
distraction, sliding, compression,
rolling, spinning of the jt surfaces.
Arthrokinematic
is used to describe these motions of bone
surfaces within the jt.
B. Manipulation

Is a passive mov using
accessory motions.
may be applied:
1. With a thrust , (cyropractic) or
2. Under anesthesia
physiologic or
It
B. Manipulation
THRUST

A sudden mov performed
with a high velocity.

Without anaesthesia.

Short amplitude motion,

Using:
1. Physiologic, or
Accessory movs.
Effects:
1. Snaps adhesions.
2. Stimulates jt receptors.

Medical procedure used to
breaking adhesions surrounding jt.

With anaesthesia.

Performed through:

1. Rapid thrust, or
Passive stretch.
Using:
restore full ROM by
can’t
be prevented by the pt.
Performed
At end of pathologic limit
(end of available ROM,
there’s restriction).

MANIPULATION UNDER ANAESTHESIA
when
2.
1. Physiologic, or
2. Accessory movs.
2.
Factors Affecting Jt. Motion
A. Jt. shape.
B. Types of Motion.
C. Other Accessory motions.
Factors Affecting Joint motion
A. Joint Shape
The type of motion is influenced
jt. shape.

by
Ovoid: one convex surface &
one concave (as A).

Sellar (Saddle): one surface is concave
in one
with the opposing surface
direction & convex in the other,
convex & concave
(as a horse back rider) (as B).
B. Types of Motion

When a bony lever
(bone)
moves about an axis of motion
surfaces on the opposing bone surface within the jt.
→ mov of bony
i.e.
1.
Movement of bony lever
- Called swing as (flex, ext, add, abd & rot.)
- Measured in degrees.
- Called ROM.
2. Motion of the bony surfaces
Within jt is a combination of rolling, sliding & spinning.
- These accessory motions allow greater angulation of
bone as it swings.
- This needs adequate capsule laxity or jt. play.
-
a. Roll

One bone rolls on another. The surfaces are incongruent.

New points on one surface meet new points on the opposing surface.

Results in angular motion.

Always rolling occur in the same direction of the mov.,
whether the surface is convex or concave.

There’s compression of the surface on the side to which the bone is
angulating & separation on the other side.

In normal functioning jts, pure rolling doesn’t occur alone, but in combination
with sliding & spinning.
b. Slide

One bone slides across another.

For pure slide, the surface must be congruent either flat or curved.

There’s no pure slide as jt surfaces aren’t completely congruent.

Same point on one surface comes into contact with new points on the
opposing surface.

Direction of slides depends on whether the surface is convex or
concave.

If the moving surface is convex
→ Sliding is in the opposite
direction of angular mov.

If the moving surface is concave
Sliding is in the same direction
as the angular mov.

This mechanical relationship is known as
convex-concave rule. It determines the
direction of the mobilization force, when jt
mobilization gliding techniques are used.
→
c. Combined roll-sliding in a joint

The more congruent the jt surface
→ the more sliding of one bony partner
on the other with movs.
(e.g. metatarsals & metacarpals)

The more incongruent the jt surface → the
more rolling movs.
(e.g. hip & sh)
For Joint Mobilization
Techniques


1.
2.
Sliding
Is used to
↓
Restore jt. play
Reverse jt.
Hypomobility


1.
Rolling
Is not used, as
↓
Jt. Compression
E.g. hip abd & add
sh abd & add

When PT passively moves the articulating surface
in
the direction in which the slide normally occurs
→ Translatory glide (glide)
↓
used to
1. Control pain
If applied gently
or
2. Stretch capsule
If with stretch force
d. Spin

Rotation of one bony segment about a stationary mechanical axis.

Spinning rarely occurs alone, but in combination with rolling &
sliding.

3 examples of spinning in the body jts:
1. Sh.: with flex/ext.
2. Hip: with flex/ext.
3. Radio-humeral: with sup./pron.
4. **Knee ext either by closed or open chain
C. Other Accessory Motions that
affect the Joint
e. Compression.
f. Traction (A&B).
e. Compression

Is the ↓ in jt space.

Occurs in LL & spinal jts with wt. bearing.

Some compression occurs as m. cont → provide stability to jt.

As one bone rolls on the other, some compression occurs on
the side to which the bone is angulating.

Normal intermittent compression loads
→ help in moving synovial fluid
so → maintain cartilage health.

Abnormal high compression loads
→ articular cartilage changes & deterioration.
(friction → erosion & OA)
f. Traction

Distraction or separation of jt surfaces.

For distraction to occur, the surfaces must be pulled apart. The mov
isn’t always the same.

Pulling on the long axis of one bone

Distraction of gleno-humeral jt requires a pull at a 90° to the glenoid fossa →

Distraction is used to:
shaft of humerus → glide jt surface) (long axis traction).
distraction & jt traction or jt separation.
1.
2.
Control or relieve pain: If applied gently.
Stretch the capsules: If applied with stretching force.
(as pulling the
Effects of Joint Motion
Skim. biologic activity by moving synovial fluid → brings nutrients to the
1.
avascular articular cartilage of jt surfaces & intra-articular fibrocartilage of the menisci.
Maintains extensibility & strength of:
articular & periarticular tissues (ligs, ms, tendons & capsules).
2.

3.
With immobilization
fibrofatty prolifiration
→ intra-articular adhesions & biochemical changes in [tendons, ligs & jt capsule]
→ m. contractures , jt stiffness & lig. weakening.
Provide proprioceptive feedback (awareness of pos. & mov. sense)
a. Static pos. & sense of speed of mov.
b. Change of speed of mov.
c. Sense of direction of mov.
d. Regulation of m. tone.
→
Indications of Joint Mobilization
Pain, m. guarding & spasm
1.
All can be treated with gentle jt play tech
neurophysiological & mechanical effects.
to stim.
a.
Neurophysiological effect:
b.
Mechanical effects:

Small-amplitude distraction or gliding mov → synovial fluid motion
→ bring
nutrients to the avascular portions of articular cartilage.
(↓ pain of
ischemia)
Gentle jt-play → maintain nutrient exchange → prevent painful effects of stasis when a jt is
painful or swollen & can’t move through a ROM.
(but not in acute or massive

oscillatory mov → stim mechanoreceptors
stimuli at spinal cord & brain stem levels.
swelling)
Small amplitude
→ ↓ transmission of nociceptive
Reversible Jt. Hypomobility
Can be treated with:
1. Progressive vigorous jt. play stretching techs.
→ elongate hypomobile structures.
2. Sustained or oscillatory stretch forces
→ mechanically distend shortened tissues.
Progressive Limitations
Diseases that progressively limit movement can be
treated by jt. play techs to:
1. Maintain available motion.
2. Retard progressive mechanical restrictions.
4. Functional Immobility (obligatory bedriddeness)
In this case the aim is to:
1. Maintain available jt. play.
2. Prevent degenerating.
3. Delay or reduce the effects of immobility.
N.B. Mobilization techniques

Can’t change the disease process of disorders as:
1.
2.

Rheumatoid arthritis.
Inflammation.
It only can:
Minimizes pain.
2. Maintain available jt. Play.
3. Reduces effects of mech. limitations.
1.
Contra-indications
Hypermobility: AS PJM →
1.
•
Potential necrosis of ligs. or capsules.
(Hypermobile pts may benefit from jt. play tech.
if kept within the limits of motion).
2. Joint Effusion

Never mobilize or stretch a swollen jt with:
1. mobilization or
passive stretching.
The capsule is already stretched by the extra fluid.
Extra fluid & m. response to pain (spasm) → limited motion.

Gentle oscillating motions that don’t stretch capsule or stress it →:
1. Improve fluid flow.
2.
3.

2.
↓ pain.
Maintains available jt play.
If pt response to jt play is ↑ pain or jt. Irritability:
The techs. were either:
1.
2.
Too vigorous, or
Don’t suit the current state of pathology.

3. Inflammation

Stretching in presence of inflammation →
1.
↑ Pain.
2.
↑ muscle guarding.
3.
Greater tissue damage.
4.
Spreading the inflam. process & infection.
Precautions


In most cases jt mobilization is safer than passive angular stretching.
But
Sometimes Jt mobilization can be used with extreme care in:
1. Malignancy -» spread & growth.
2. Excessive pain -» m. guarding, prevent mov.
3.
4.
5.
6.
Total jt. replacement
-» disl. or loosen int. fixation.
Bone disease (Osteoporosis, TB, Rickets).
Unhealed #
(site & stabilization)
-» re-fracture.
Hypomobility (in associated jts & m. weak)
-»↑ capsule laxity, lig. Weakness & jt disl.
Procedure for Applying Jt
Mobilization Techniques
A.
Evaluation
&
Assessment if
1. limited, or
2. painful mov.
1.
2.
Evaluate
Tissue limiting fun
State of pathology
Determine ttt direction
1. Relieving pain
2. Stretching a jt
3. Soft tissue limit
Determine Pain quality
1. Recovery Stage
2. Treatment Dose
3 Possibilities for pain
!!!! While moving or mobilizing & found!!!!
1.
Before tissue limitation
injury or active stage of a disease),
tech to relieve pain & maintain jt. play.
2.
(after acute
use pain inhibiting jt.
With tissue limitation
(as damaged tissue begins to heal) (subacute),
use gradual gentle stretching tech to tight tissue, but
exacerbate pain by injuring the tissues.
3.
After tissue limitations
stretching a tight capsule or a peri-articular tissue of ch. stiff jts),
use ↓ force of the stretching techniques.
Don’t
(as
If the capsule is limiting the
motion, there’s:
B. Grades of mov. (Dosage,
Amplitude)
2 systems of techniques:
1. Graded oscillation.
2. Sustained translatory jt. Play.
1. Graded Oscillation Tech.
Dosage
Grade I
Small amplitude rhythmic oscillations
at the beginning of ROM.
Grade II
Large amplitude rhythmic oscillations
within the ROM, but not reaching the limitation.
Grade III
Large amplitude rhythmic oscillations up to the limit
of available motion
& stressed into tissue resistance.
Small amplitude rhythmic
Grade IV
oscillations, up to the limit
of the available motion
& stressed into tissue resistance.
Small amplitude, high velocity, thrust tech..
Performed to break adhesions
Graded oscillation
technique
Techniques

Oscillations may be done using:
1- Physiologic (osteokinematic) motions.
2- Jt. play (arthrokinematic) techs.
Sustained Translatory jt. Play
Technique

Grade I (Loosen):
Small amplitude
3.
distraction, applied where no stress is placed on the capsule.
Equalizes cohesive force,
2. M. tension
3. Atmospheric pressure acting on the jt.

Grade II (Till tightness):
1.
2.
Distraction or glide applied to tightend tissues around jt.

Grade III:
Distraction or glide with large enough amplitude to place
a stretch on jt. capsule & surrounding peri-articular structures.
Sustained translatory
joint-play technique
Uses

Grade I: Used with gliding motions to relieve pain.

Grade II: Used to determine how sensitive the jt is.

Grade III:
to stretch jt. structures & ↑ jt play.

Technique:
Used
This grading system describes jt. play techniques
that separate or glide (slide) jt surfaces.
N.B.

The consistency between the dosage of the two grading systems is grade I
(no tension on the jt capsule or surroundings).

Grade III (sustained stretch) & IV (oscillation) are similar in dosage as they
are applied with a stretch force at the limit of the motion.

Using oscillating or sustained tech., depends on the pt’s response.

Pain management: use oscillating tech.

Loss of jt. play & ↓ function: use sustained stretch.

To maintain available R.:
Use either grade II! oscillating or II sustained technique.
C. Patient Position
1.
C. Patient Position The pt. & treated
extremity should be carefully
positioned & relaxed.
2. The jt. is positioned so that the
capsule has greatest laxity.
D. Stabilization
 Firm
& comfortable stabilization for
proximal parts by using:
Belt.
2. PT hand.
3. External assistance.
1.
E. Force
 Treatment
force
(gentle or strong) is applied as close to the opposing jt.
surfaces as possible.
 The larger the contact surface of PT hand
→ the more comfortable procedure will be.
e.g. use flat surface of the hand instead of forcing with the thumb.
F. Direction of Movement



Either parallel to, or perpendicular to ttt plane.
Jt. traction:
Perpendicular to ttt
plane.
The entire bone is moved → jt surfaces are separated.
Gliding tech.:
Parallel to the ttt plane.
This is determined by using the convex-concave rule.
- If the surface of moving bone is convex -→ opposite direction glide.
- If the surface of moving bone is concave → glide in the same direction.
The entire bone is moved, so there’s gliding of one jt. surface on the other.
No swing of the bone that causes rolling & compression of the jt. surfaces
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