++
As discussed in Chapter 3, the rotary effect created
by an applied force is known as torque,
or moment of force. Torque, which may
be thought of as rotary force, is the
angular equivalent of linear force. Algebraically, torque is the
product of force and the force’s moment
arm, or the perpendicular distance from the force’s
line of action to the axis of rotation:
++
++
Thus, both the magnitude of a force and the length of its moment
arm equally affect the amount of torque generated (Figure 13-1).
Moment arm is also sometimes referred to as force
arm or lever arm.
++
++
As may be observed in Figure 13-2, the moment arm is
the shortest distance between the force’s line of action
and the axis of rotation. A force directed through an axis of rotation
produces no torque, because the force’s moment arm is zero.
++
++
Within the human body, the moment arm for a muscle with respect
to a joint center is the perpendicular distance between the muscle’s
line of action and the joint center (Figure 13-3). As a
joint moves through a range of motion, there are changes in the
moment arms of the muscles crossing the joint. For any given muscle,
the moment arm is largest when the angle of pull on the bone is
closest to 90˚. At the elbow, as the angle of pull moves
away from 90˚ in either direction, the moment arm for the
elbow flexors is progressively diminished. Since torque is the product
of moment arm and muscle force, changes in moment arm directly affect
the joint torque that a muscle generates. For a muscle to generate
a constant joint torque during an exercise, it must produce more force
as its moment arm decreases.
++
++
In the sport of rowing, where adjacent crew members traditionally
row on opposite sides of the hull, the moment arm between the force
applied by the oar and the stern of the boat is a factor affecting
performance (Figure 13-4). With the traditional arrangement,
the rowers on one side of the boat are positioned farther from the
stern than their counterparts on the other side, thus causing a
net torque and a resulting lateral oscillation about the stern during
rowing (28). The Italian rig eliminates this problem by
positioning rowers so that no net torque is produced, assuming that
the force produced by each rower with each stroke is nearly the
same (Figure 13-4). Italian and German rowers have similarly
developed alternative positionings for the eight-member crew (Figure
13-5).
++
++
++
Another example of the significance of moment arm length is provided
by a dancer’s foot placement during preparation for execution
of a total body rotation around the vertical axis. When a dancer
initiates a turn, the torque producing the turn is provided by equal
and oppositely directed forces exerted by the feet against the floor.
A pair of equal and opposite forces is known as a force couple. Because the forces in a couple
are positioned on opposite sides of the axis of rotation, they produce
torque in the same direction. The torque generated by a couple is
therefore the sum of the products of each force and its moment arm.
Turning from fifth position, with a small distance between the feet,
requires greater force production by a dancer than turning at the
same rate from fourth position, in which the moment arms of the
forces in the couple are longer (Figure 13-6). Significantly
more force is required when the torque is generated by a single
support foot, for which the moment arm is reduced to the distance
between the metatarsals and the calcaneus (15).
++
++
Torque is a vector quantity, and is therefore characterized by
both magnitude and direction. The magnitude of the torque created
by a given force is equal to Fd⊥, and the direction of
a torque may be described as clockwise or counterclockwise. As discussed
in Chapter 11, the counterclockwise direction is conventionally
referred to as the positive (+) direction, and
the clockwise direction is regarded as negative (−).
The magnitudes of two or more torques acting at a given axis of
rotation can be added using the rules of vector composition (see Sample
Problem 13.1).
++
Two children sit on opposite sides of a playground seesaw. If
Joey, weighing 200 N, is 1.5 m from the seesaw’s axis of
rotation, and Susie, weighing 190 N, is 1.6 m from the axis of rotation,
which end of the seesaw will drop?
++
++
++
The seesaw will rotate in the direction of the resultanttorque
at its axis of rotation. To find the resultant torque, the torques
created by both children are summed according to the rules of vector composition.
The torque produced by Susie’s body weight is in a counterclockwise
(positive) direction, and the torque produced by Joey’s
body weight is in a clockwise (negative) direction.
++
++
+++
Resultant Joint
Torques
++
The concept of torque is important in the study of human movement,
because torque produces movement of the body segments. As discussed
in Chapter 6, when a muscle crossing a joint develops tension,
it produces a force pulling on the bone to which it attaches, thereby
creating torque at the joint the muscle crosses.
++
Much human movement involves simultaneous tension development
in agonist and antagonist muscle groups. The tension in the antagonists
controls the velocity of the movement and enhances the stability
of the joint at which the movement is occurring. Since antagonist
tension development creates torque in the direction opposite that
of the torque produced by the agonist, the resulting movement at
the joint is a function of the net torque. When net muscle torque
and joint movement occur in the same direction, the torque is termed concentric, and muscle torque in the
direction opposite joint motion is considered to be eccentric. Although these terms are
generally useful descriptors in analysis of muscular function, their
application is complicated when two-joint or multijoint muscles
are considered, since there may be concentric torque at one joint
and eccentric torque at a second joint crossed by the same muscle.
++
Because directly measuring the forces produced by muscles during
the execution of most movement skills is not practical, measurements
or estimates of resultant joint torques (joint moments) are often
studied to investigate the patterns of muscle contributions. A number
of factors, including the weight of body segments, the motion of
the body segments, and the action of external forces, may contribute
to net joint torques. Young infants generate irregular patterns
of joint torques, possibly because of their inexperience in predicting
the magnitude and direction of external forces (14). Among
adults, however, joint torque profiles are typically matched to
the requirements of the task at hand and provide at least general
estimates of muscle group contribution levels.
++
Interestingly, however, with advanced age there is commonly a
redistribution of lower extremity joint torques during walking gait,
with the elderly using the hip extensors more and the knee extensors
and plantar flexors less than young adults walking at the same pace
(5). Because hip extension torque has been shown to be
significantly related to walking speed and stride length among the
elderly, researchers have suggested that strengthening the hip extensors may
improve gait characteristics in this group (1).
++
To better understand muscle function during running, a number
of investigators have studied resultant joint torques at the hip,
knee, and ankle throughout the running stride. Figure 13-7 displays
representative resultant joint torques and angular velocities for
the hip, knee, and ankle during a running stride, as calculated
from film and force platform data. In Figure 13-7, when
the resultant joint torque curve and the angular velocity curves
are on the same side of the zero line, the torque is concentric;
the torque is eccentric when the reverse is true. As may be observed
from Figure 13-7, both concentric and eccentric torques
are present at the lower-extremity joints during running.
++
++
During both running and walking, individuals with anterior cruciate
ligament (ACL) injury use greater extensor torques at the hip and
ankle and lower extensor torques at the knee as compared to uninjured
people (6). Research indicates that as compared to normal
individuals, those with ACL injuries also display greater co-contraction
of the knee flexor muscles during knee extension, which contributes
to stabilization of the knee (19).
++
Lower-extremity joint torques during cycling at a given power
are affected by pedaling rate, seat height, length of the pedal
crank arm, and distance from the pedal spindle to the ankle joint.
Average hip and knee torques during cycling under cruising conditions
have been reported to be minimum at approximately 105 rotations
per minute (24). Figure 13-8 shows the changes
in average resultanttorque at the hip, knee, and ankle joints with
changes in pedaling rate at a constant power.
++
++
It is widely assumed that the muscular force (and subsequently,
joint torque) requirements of resistance exercise increase as the
amount of resistance increases. However, this is true only as long as
movement kinematics remain constant. It has been shown, for example,
that during the squat exercise, use of a wide stance as compared
to a narrow stance produces greater torques at both the hip and
the knee (7).
++
Another factor influencing joint torques during exercise is movement
speed. When other factors remain constant, increased movement speed
is associated with increased resultant joint torques during exercises
such as the squat (23). However, increased movement speed
during weight training is generally undesirable, because increased
speed increases not only the muscle tension required but also the
likelihood of incorrect technique and subsequent injury. Acceleration
of the load early in the performance of a resistance exercise also
generates momentum, which means that the involved muscles need not
work so hard throughout the range of motion as would otherwise be
the case. For these reasons, it is both safer and more effective
to perform exercises at slow, controlled movement speeds.
++
++
When muscles develop tension, pulling on bones to support or
move the resistance created by the weight of the body segment(s)
and possibly the weight of an added load, the muscle and bone are functioning
mechanically as a lever. A lever is
a rigid bar that rotates about an axis, or fulcrum.
Force applied to the lever moves a resistance. In the human body,
the bone acts as the rigid bar; the joint is the axis, or fulcrum;
and the muscles apply force. The three relative arrangements of
the applied force, resistance, and axis of rotation for a lever
are shown in Figure 13-9.
++
++
With a first-class lever, the applied
force and resistance are located on opposite sides of the axis.
The playground seesaw is an example of a first-class lever, as are
a number of commonly used tools, including scissors, pliers, and
crowbars (Figure 13-10). Within the human body, the simultaneous
action of agonist and antagonist muscle groups on opposite sides
of a joint axis is analogous to the functioning of a first-class
lever, with the agonists providing the applied force and the antagonists
supplying a resistance force. With a first-class lever, the applied
force and resistance may be at equal distances from the axis, or
one may be farther away from the axis than the other.
++
++
In a second-class lever, the applied
force and the resistance are on the same side of the axis, with
the resistance closer to the axis. A wheelbarrow, a lug nut wrench,
and a nutcracker are examples of second-class levers, although there
are no completely analogous examples in the human body (Figure
13-10).
++
With a third-class lever, the force
and the resistance are on the same side of the axis, but the applied
force is closer to the axis. A canoe paddle and a shovel can serve
as third-class levers (Figure 13-10). Most muscle-bone
leversystems of the human body are also of the third class for
concentric contractions, with the muscle supplying the applied force
and attaching to the bone at a short distance from the joint center
compared to the distance at which the resistance supplied by the
weight of the body segment or that of a more distal body segment
acts (Figure 13-11). As shown in Figure 13-12,
however, during eccentric contractions, it is the muscle that supplies
the resistance against the applied external force. During eccentric
contractions, muscle and bone function as a second-class lever.
++
++
++
A leversystem can serve one of two purposes (Figure 13-13).
Whenever the moment arm of the applied force is greater than the
moment arm of the resistance, the magnitude of the applied force
needed to move a given resistance is less than the magnitude of the
resistance. Whenever the resistance arm is longer than the force
arm, the resistance may be moved through a relatively large distance.
The mechanical effectiveness of a lever for moving a resistance
may be expressed quantitatively as its mechanical
advantage, which is the ratio of the moment arm of the force
to the moment arm of the resistance:
++
++
++
Whenever the moment arm of the force is longer than the moment
arm of the resistance, the mechanical advantage ratio reduces to
a number that is greater than one, and the magnitude of the applied
force required to move the resistance is less than the magnitude
of the resistance. The ability to move a resistance with a force
that is smaller than the resistance offers a clear advantage when a
heavy load must be moved. As shown in Figure 13-10, a wheelbarrow
combines second-class leverage with rolling friction to facilitate
transporting a load. When removing a lug nut from an automobile
wheel, it is helpful to use as long an extension as is practical
on the wrench to increase mechanical advantage.
++
Alternatively, when the mechanical advantage ratio is less than
one, a force that is larger than the resistance must be applied
to cause motion of the lever. Although this arrangement is less
effective in the sense that more force is required, a small movement
of the lever at the point of force application moves the resistance
through a larger range of motion (see Figure 13-13).
++
During wheelchair propulsion, mechanical advantage is the ratio
of handrim radius to wheel radius. Since handrim radius is always
smaller than wheel radius, the mechanical advantage for wheelchair propulsion
is always less than one. This is advantageous, because movements
of the handrims translate to larger movements of the wheels, and
the resistance, which is the force of rolling friction, is relatively
low. For a given applied force on the pushrim, wheelchair velocity
is proportional to mechanical advantage. Researchers have found
that a mechanical advantage of 0.43 is more mechanically efficient
than mechanical advantages ranging up to 0.87 for wheelchair propulsion,
because at lower mechanical advantage (and lower velocity), the
wheelchair occupant is able to apply force more directly in line
with the path of handrim rotation (31). During wheelchair
propulsion, mechanical advantage has been shown to have a significant
effect on oxygen uptake, energy cost, mechanical efficiency, and
stroke frequency (29).
++
++
Skilled athletes in many sports intentionally maximize the length
of the effective moment arm for force application to maximize the
effect of the torque produced by muscles about a joint. During execution
of the serve in tennis, expert players not only strike the ball
with the arm fully extended but also vigorously rotate the body
in the transverse plane, making the spine the axis of rotation and
maximizing the length of the anatomical lever delivering the force.
The same strategy is employed by accomplished baseball pitchers.
As discussed in Chapter 11, the longer the radius of rotation,
the greater the linear velocity of the racket head or hand delivering
the pitch, and the greater the resultant velocity of the struck
or thrown ball.
++
In the human body, most muscle–bone leversystems are
of the third class, and therefore have a mechanical advantage of
less than one. Although this arrangement promotes range of motion
and angular speed of the body segments, the muscle forces generated
must be in excess of the resistance force or forces if positive
mechanical work is to be done.
++
The angle at which a muscle pulls on a bone also affects the
mechanical effectiveness of the muscle–bone leversystem.
The force of muscular tension is resolved into two force components—one perpendicular
to the attached bone and one parallel to the bone (Figure 13-14).
As discussed in Chapter 6, only the component of muscle
force acting perpendicular to the bone—the rotary component—actually
causes the bone to rotate about the joint center. The component
of muscle force directed parallel to the bone pulls the bone either
away from the joint center (a dislocating component) or toward the
joint center (a stabilizing component), depending on whether the
angle between the bone and the attached muscle is less than or greater
than 90˚. The angle of maximum mechanical advantage for
any muscle is the angle at which the most rotary force can be produced.
At a joint such as the elbow, the relative angle present at the
joint is close to the angles of attachment of the elbow flexors.
The maximum mechanical advantages for the brachialis, biceps, and
brachioradialis occur between angles at the elbow of approximately
75˚ and 90˚ (Figure 13-15).
++
++
++
As joint angle and mechanical advantage change, muscle length
also changes. Alterations in the lengths of the elbow flexors associated
with changes in angle at the elbow are shown in Figure 13-16.
These changes affect the amount of tension a muscle can generate,
as discussed in Chapter 6. The angle at the elbow at which
maximum flexion torque is produced is approximately 80˚,
with torque capability progressively diminishing as the angle at
the elbow changes in either direction (30).
++
++
The varying mechanical effectiveness of muscle groups for producing
joint rotation with changes in joint angle is the underlying basis
for the design of modern variable-resistance strength-training devices.
These machines are designed to match the changing torque-generating
capability of a muscle group throughout the range of motion at a
joint. Machines manufactured by Universal (the Centurion) and Nautilus
are examples. Although these machines offer more relative resistance through
the extremes of joint range of motion than free weights, the resistance
patterns incorporated are not an exact match for average human strength
curves (8).
++
Isokinetic machines represent another approach to matching torque-generating
capability with resistance. These devices are generally designed
so that an individual applies force to a lever arm that rotates
at a constant angular velocity. If the joint center is aligned with
the center of rotation of the lever arm, the body segment rotates
with the same (constant) angular velocity of the lever arm. If volitional
torque production by the involved muscle group is maximum throughout
the range of motion, a maximum matched resistance theoretically
is achieved. However, when force is initially applied to the lever
arm of isokinetic machines, acceleration occurs, and the angular
velocity of the arm fluctuates until the set rotational speed is
reached (8). Because optimal use of isokinetic resistance machines
requires that the user be focused on exerting maximal effort throughout
the range of motion, some individuals prefer other modes of resistance
training.
++
+++
Equations of
Static Equilibrium
++
Equilibrium is a state characterized by balancedforces and torques
(no net forces and torques). In keeping with Newton’s first
law, a body in equilibrium is either motionless or moving with a
constant velocity. Whenever a body is completely motionless, it
is in static equilibrium. Three conditions
must be met for a body to be in a state of static equilibrium:
+
1. The sum of all vertical forces (or force components)
acting on the body must be zero.
2. The sum of all horizontal forces (or force components)
acting on the body must be zero.
3. The sum of all torques must be zero:
++
++
The capital Greek letter sigma (Σ) means the sum of, Fv represents
vertical forces, Fh represents horizontal forces, and T
is torque. Whenever an object is in a static state, it may be inferred
that all three conditions are in effect, since the violation of
any one of the three conditions would result in motion of the body.
The conditions of static equilibrium are valuable tools for solving
problems relating to human movement (see Sample Problems 13.2, 13.3,
and 13.4).
++
How much force must be produced by the biceps brachii, attaching
at 90˚ to the radius at 3 cm from the center of rotation
at the elbow joint, to support a weight of 70 N held in the hand
at a distance of 30 cm from the elbow joint? (Neglect the weight
of the forearm and hand, and neglect any action of other muscles.)
++
++
++
Since the situation described is static, the sum of the torques
acting at the elbow must be equal to zero:
++
++
Two individuals apply force to opposite sides of a frictionless
swinging door. If A applies a 30 N force at a 40˚ angle
45 cm from the door’s hinge and B applies force at a 90˚ angle
38 cm from the door’s hinge, what amount of force is applied
by B if the door remains in a static position?
++
++
++
The equations of static equilibrium are used to solve for FB.
The solution may be found by summing the torques created at the
hinge by both forces:
++
++
The quadriceps tendon attaches to the tibia at a 30˚ angle
4 cm from the joint center at the knee. When an 80 N weight is attached
to the ankle 28 cm from the knee joint, how much force is required
of the quadriceps to maintain the leg in a horizontal position?
What is the magnitude and direction of the reaction force exerted
by the femur on the tibia? (Neglect the weight of the leg and the
action of other muscles.)
++
++
++
The equations of static equilibrium can be used to solve for
the unknown quantities:
++
++
The equations of static equilibrium can be used to solve for
the vertical and horizontal components of the reaction force exerted
by the femur on the tibia. Summation of vertical forces yields the
following:
++
++
Summation of horizontal forces yields the following:
++
++
The Pythagorean theorem can now be used to find the magnitude
of the resultant reaction force:
++
++
The tangent relationship can be used to find the angle of orientation
of the resultant reaction force:
++
++
+++
Equations of
Dynamic Equilibrium
++
Bodies in motion are considered to be in a state of dynamic equilibrium, with all acting
forces resulting in equal and oppositely directed inertial forces.
This general concept was first identified by the French mathematician
D’Alembert, and is known as D’Alembert’s
principle. Modified versions of the equations of static equilibrium,
which incorporate factors known as inertia vectors, describe the conditions of dynamic equilibrium. The
equations of dynamic equilibrium may be stated as follows:
++
++
The sums of the horizontal and vertical forces acting on a body
are ΣFx and ΣFy;
māx and māy are
the products of the body’s mass and the horizontal and
vertical accelerations of the body’s center of mass; ΣTG is
the sum of torques about the body’s center of mass, and
is the product of the body’s moment of inertia about the
center of mass and the body’s angular acceleration (see Sample
Problem 13.5). (The concept of moment of inertia is discussed
in Chapter 14.)
++
A familiar example of the effect of D’Alembert’s
principle is the change in vertical force experienced when riding
in an elevator. As the elevator accelerates upward, an inertial
force in the opposite direction is created, and body weight as measured
on a scale in the elevator increases. As the elevator accelerates
downward, an upwardly directed inertial force decreases body weight
as measured on a scale in the elevator. Although body weight remains
constant, the vertical inertial force changes the magnitude of the
reaction force measured on the scale.
++
A 580 N skydiver in free fall is accelerating at −8.8
m/s2 rather than −9.81 m/s2 because
of the force of air resistance. How much drag force is acting on
the skydiver?
++
++
++
Since the skydiver is considered to be in dynamic equilibrium,
D’Alembert’s principle may be used. All identified
forces acting are vertical forces, so the equation of dynamic equilibrium
summing the vertical forces to zero is used:
++
++
Given that ΣFy = −580
N + Fd, substitute the known information
into the equation:
++