- 1. Define the terms
*mechanics*and*biomechanics,*and differentiate between them. - 2. Define the terms
*kinematics, kinetics, statics,*and*dynamics,*and state how each relates to the structure of biomechanics study. - 3. Convert the units of measurement employed in the study of biomechanics from the U.S. system to the metric system, and vice versa.
- 4. Describe the nature of scalar and vector quantities, and identify such quantities as one or the other.
- 5. Demonstrate the use of the graphic method for the combination and resolution of two-dimensional vectors.
- 6. Demonstrate the use of the trigonometric method for the combination and resolution of two-dimensional vectors.
- 7. Identify the scalar and vector quantities represented in individual motor skills and describe the vector quantities using vector diagrams.

In the context of this text, kinesiology is an area of study concerned with the musculoskeletal analysis of human motion and the study of mechanical principles and laws as they relate to the study of human motion. Students of human motion capable of accurately analyzing the musculoskeletal actions occurring in the execution of a movement are well on their way toward knowing what is happening during that movement. Those who have taken the further step of acquiring a working knowledge of how human motion is governed by physical laws and principles have added an additional dimension to their understanding of how and why the motion occurs as it does. Together, all this information provides a scientific foundation upon which to make appropriate decisions concerning the safest, most effective, and most efficient execution of any movement pattern. It is only through such study that definitive answers may be found concerning the “best” way for an individual to perform a skill and the reasons the method selected is indeed the best. In sport, for example, the record-breaking “form” of one athlete may or may not be appropriate for another of different body build and size. In fact, although they break records, top performers’ techniques may include actions that, if eliminated, would result in even greater performance. Unless subjected to scientific scrutiny, discrimination between the success factors and deterrent factors may be confused or not even identified.

Where *forces* and *motion* are
concerned, the area of scientific study that provides accurate answers
to what is happening, why it is happening, and to what extent it
is happening is called *mechanics.* It
is that branch of physics concerned with the effect that **forces** have on bodies and the motion
pro-duced by those forces. The study of mechanics is engaged in by
those people whose occupations or professions require an understanding
of force, matter, space, and time. Engineers, to a large extent,
are involved with the application of mechanics. Navigation, astronomy,
space, and communications experts all study mechanics. The same
is true of individuals concerned with the study of human motion
and the forces causing it. The laws and principles used to explain
the motion of planets or the strength of buildings and bridges apply equally
to humans. *All* motion, including motions
of the human body and its parts, is the result of the application
of forces and is subject to the laws and principles that govern
force and motion.

When the study of mechanics is limited to living structures,
especially the human body, it is called *biomechanics.***Biomechanics** is an interdisciplinary
science based on many of the fundamental disciplines found in the
physical and life sciences. Generally, biomechanics is considered
to be that aspect of the science concerned with the basic laws governing
the effect forces have on the state of rest or motion of animals
or humans, whereas the applied areas of biomechanics deal with solving
practical problems. Anatomists, orthopedists, space engineers, industrial
engineers, biomedical engineers, physical therapists, physical educators,
dancers, and coaches all have an interest in biomechanics and in applying
its principles to the improvement of human movement. Professional
applications may differ, but the same basic laws of biomechanics
provide a common foundation for all. In this context the part of
biomechanics that applies to sport, dance, and physical education
may be considered to be that part of kinesiology that involves the
mechanics of human motion.

The study of biomechanics is divided into two areas, statics
and dynamics. **Statics** covers situations
in which all forces acting on a body are balanced, and the body
is in equilibrium. A body in a static situation would exhibit either
no motion or a constant, unchanging motion. With a knowledge of
the principles of statics, one may have a better understanding of levers
and a greater ability to solve problems such as locating the body’s
center of gravity or center of buoyancy.

The branch of biomechanics dealing with bodies subject to *unbalance* is called **dynamics.** The principles of dynamics
explain circumstances in which an excess of force in one direction
or a turning force causes an object to change speed or direction.
In this situation an object might move from a static position to
one of motion or might undergo some change in the motion state. Principles
of work, energy, and accelerated motion are included in the study
of dynamics.

The terms *kinematics* and *kinetics* are also part of the vocabulary
of the study of mechanics. **Kinematics** has
been referred to as the geometry of motion. It describes the motion
of bodies in terms of time, displacement, velocity, and acceleration.
The motion occurring may be in a straight line (linear kinematics)
or rotating about a fixed point (angular kinematics). Kinematics
is concerned only with the analytical and mathematical descriptions
of all kinds of motion and *not* with
the forces that cause the motion. The branch of mechanics that considers
the *forces* that produce or change motion
is called **kinetics.** The most complex
of biomechanics studies, it is the area least developed in physical
education and physical therapy, and of most challenge to research
today. Linear kinetics is concerned with the causes of linear motion,
and angular kinetics deals with the forces that cause angular motion.

In the study of human motion, as in the study of any science, careful measurement and the use of mathematics are essential for the classification of facts and the systematizing of knowledge. Mathematics is the language of science. It enables us to express relationships quantitatively rather than merely descriptively. It provides objective evidence of the superiority of one technique over another and thus forms the basis for developing effective measures for improving both the safety and the effectiveness of performance. Furthermore, it makes possible continuing advancement of knowledge through research. Had it not been for the use of mathematics, the contributions of great scientists like Archimedes, Galileo, and Newton would not have been possible.

In the biomechanical aspects of kinesiology, as in all mechanics, the depth of understanding of the principles and laws that apply to it is greatly increased through experimental and mathematical evidence. Hence, it is to the student’s advantage to become conversant with appropriate mathematical concepts and techniques. The mathematics needed for the quantitative treatment of the simple mechanics discussed in this text is not difficult. It consists of elementary algebra and right triangle trigonometry. A review of these essential mathematical concepts is presented in Appendix D.

The units of measurement employed in the study of biomechanics are expressed in terms of space, time, and mass. Currently in the United States, there are two systems of unit measurement for these quantities, the U.S. system and the metric system. Although the metric system is currently used in research and literature, a comparison of equivalent values is helpful. Table 10.1 presents some common units used in biomechanics study and their U.S. and metric equivalents.

Unit | Metric System | U.S. | Equivalents |
---|---|---|---|

Length | centimeter (cm) | inch (in) | 1 in = 2.54 cm |

meter (m) = 100 cm | foot (ft) | 1 cm = 0.3937 in | |

kilometer (km) = 1000 m | mile (mi) = 5280 ft | 1 ft = 0.305 m | |

1 m = 3.28 ft | |||

1 mi = 1.609 km | |||

1 km = 0.621 mi | |||

Area | square meter (100 cm2) | square foot (144 in2) | 1 in2 = 6.45 cm2 |

1 cm2 = 0.155 in2 | |||

Volume | cubic cm (cm3) | cubic in (in3) | 1 qt = 0.946 liter |

liter (1000 cm3) | quart (57.75 in3) | 1 liter = 1.06 qt | |

1 in3 = 16.39 cm3 | |||

1 cm3 = 0.06 in3 | |||

Mass | kilogram (kg) | slug (32 lb) | 1 kg = 0.068 slug |

1 slug = 14.6 kg | |||

Force (weight) | newton (0.102 kg) | pound (lb) | 1 lb = 0.454 kg |

1 kg = 2.21 lb | |||

1 N = 0.225 lb | |||

Time | second | second |

In the metric or decimal system, all units differ in size by a multiple of 10. In ascending order, linear units are millimeters, centimeters, meters, and kilometers. In the U.S. system the basic unit of length is the foot. Other possible units are inches, yards, and miles.

In the metric system, square centimeters or square meters are used for area, and cubic centimeters, liters, or cubic meters are used for volume. In the U.S. system, area units are square inches or feet, and cubic inches, cubic feet, quarts, or gallons denote volume.

**Mass** is the quantity of matter
a body contains. The **weight** of a body
depends on its quantity of matter and the strength of the gravitational
attraction acting on it. The measure of gravitational *force* is called weight. The mass of
an object will not change even if taken to the moon, but its weight will.
The kilogram, equal to the mass of a liter of water, is the unit
of mass in the metric system. The unit of force (weight) is the
newton (N), and for most of the United States a mass of 1 kilogram
weighs approximately 9.80 newtons. In the U.S. system the pound
is the basic unit of force (weight). The mass unit is the slug (from
the English word for sluggish). A mass of 1 slug weighs approximately
32 pounds for the gravitational pull present at the latitude and
longitude of most of the United States.

The basic unit of time for both systems of measurement is the second.

Quantities that are used in the description of motion may be classified as either scalar or vector in nature.

**Scalar** quantities are single quantities.
They possess only size or amount. This size or amount is referred to
as magnitude and completely describes the scalar quantity. The units
of measure described in the previous section are primarily scalar
quantities because they are described only by magnitude. Examples
of scalar quantities would be such things as a speed of 8 kilometers
per hour, a temperature of 70 degrees, an area of 2 square kilometers,
a mass of 10 kilograms, or a height of 2 meters.

There are also double quantities that cannot be described by
magnitude alone. These double quantities are called **vector** quantities. A vector quantity
is described by both magnitude and direction. Examples of vector quantities
would be a velocity of 8 kilometers per hour in a northwest direction,
10 newtons of force applied at a 30-degree angle, a displacement
of 100 meters from the starting point. The importance of clearly
designating vector quantities can be seen if the direction component
of the double quantity is altered. For instance, if two people on
opposite sides of a door push with equal magnitudes (amounts) of
force, the door will not move. If, on the other hand, they both
push on the same side of the door, thus changing the *direction* of one of the forces, the
result will be very different. The nature of the movement of the
door depends on both the *amount* and *direction* of the force. Force, therefore,
is a vector quantity. If the individual who ran 8 kilometers runs
8 more kilometers, the total distance run will be 16 kilometers.
However, if the runner goes 8 kilometers in one direction, reverses,
and runs back to the starting point, the change in position, or **displacement,** is zero. The runner
is 0 kilometers from the starting point. Displacement, then, is
also a vector quantity possessing both magnitude and direction.
Numerous quantities in biomechanics are vector quantities. In addition
to force, displacement, and velocity already mentioned, some other examples
are momentum, acceleration, friction, work, and power. Vector quantities
exist whenever *direction* and *amount* are inherent characteristics
of the quantities.

A vector is represented by an arrow whose length is proportional to the magnitude of the vector. The direction in which the arrow points indicates the direction of the vector quantity. Figure 10.1 shows examples of arrows indicating the vector quantities of force, displacement, and velocity.

Vector quantities are equal if magnitude and direction are the same for each vector. Although all of the following vectors are of the same length (magnitude), only two are equal vector quantities. They are the two that also have the same direction (d and f).

Vectors may be combined by addition, subtraction, or multiplication.
They are added by joining the head of one with the tail of the next
while accounting for magnitude and direction. The combination results
in a new vector called the **resultant.** The
resultant vector is represented by the distance between the first
tail and the last head. Figure 10.2 shows examples of vectors that
have been combined by addition. Note that the head of the resultant ** R** meets
the

*head*of the last component vector. These drawings also show that very different component vectors may produce the same resultant.

The subtraction of vectors is done by changing the sign of one vector (multiply by −1) and then adding as before (Figure 10.3a). The multiplication of a vector by a number changes its magnitude only, not its direction (Figure 10.3b).

As just explained, the combination of two or more vectors results
in a new vector. Conversely, any vector may be broken down or resolved
into two component vectors acting at right angles to each other.
The vector in Figure 10.1c represents the velocity with which the
shot was put. Should one wish to know how much of that velocity
was in a horizontal direction and how much in a vertical direction,
the vector must be resolved into horizontal and vertical components.
In Figure 10.4, ** A** and

**are the vertical and horizontal**

*B**components*of resultant

**The vector addition of these components once again would result in the resultant vector**

*R.***The arrows over**

*R.***and**

*A, B,***indicate that they are vector quantities.**

*R*In describing motion it is helpful to have a frame of reference
within which one can locate a position in space or change in position.
Frame of reference describes the orientation of objects in a given
space. When discussing movement of the whole body, the term *global reference frame* is often used.
The global reference frame in this case could be described in terms
of the cardinal planes as presented in Chapter 2, or it could be
broadened to include the position of the body in relation to some
external frame of reference such as a room, a playing field, or
an external observer. A *local reference
frame* is more finite and may be used to describe individual
joint motions in terms of their individual axes. As an example,
flexion of the shoulder joint might be described globally as being
in the sagittal plane, while locally the motion might be described
as it relates to the plane of the scapula. In mechanics, frame of
reference is used to describe motion. Consider someone riding down
a river on a boat. The person tosses a ball straight up and then
catches it as it lands. The person on the boat (local frame of reference)
will see the ball as traveling only in a vertical direction. However, someone
standing on the shore observing (global reference) will see the
ball travel horizontally as well as vertically. It is possible to
locate an object in three dimensions, but to simplify understanding,
the description that follows is limited to motion in two dimensions,
that is, one plane.

The position of a point *P* can be
located using either *rectangular* coordinates
or *polar* coordinates. In the two-dimensional
rectangular coordinate system, the plane is divided into four quadrants
by two perpendicular intersecting lines. The horizontal line is
the *x* axis and the vertical line is
the *y* axis. Values along either axis
are measured from the point of intersection of the two axes where *x* and *y* are
both equal to zero (0,0). The location of point *P* is
represented by two numbers, the first equal to the number of *x* units and the second to the number
of *y* units away from the intersection
or origin. In Figure 10.5 the rectangular coordinates for point *P* are (*x, y*)
in a and (13, 5) in b. In this latter example, point *P* is at the head of vector ** R** and
the location of the vector tail is at the origin. This vector’s
location in space is established. The tail is at (0, 0), and the
head is at (13, 5).

###### Figure 10.5

A point in space may be located in two dimensions using
an *x* and *y* axis
as a frame of reference. In (a) point *P* is *x* units from the *y* axis
and *y* units from the *x* axis. In (b) point *P* is 13*x* units
from the *y* axis and 5*y* units from the *x* axis.
In (c) point *P* is *r* units
from the *x, y* intersection and θ degrees
from the *x* axis.

Point *P* may also be described using
polar coordinates. These consist of the distance (*r*) of point *P* from
the origin and the angle (θ) that the line *r* makes with the *x* axis.
The polar coordinates for point *P* in
Figure 10.5c are (*r,* θ).
If *P* is the vector head, *r* equals the vector’s magnitude,
and θ is equal to its direction. In polar terms
the vector’s description is (*r,* θ).

In these coordinate systems, degrees are customarily measured
in a counterclockwise direction. Also, by convention, *x* values to the right of the *y* axis are positive (+)
and those to the left are negative (−). The *y* values above the *x* axis
are positive and those below are negative. Point *A* in
Figure 10.6 has (*x, y*) coordinates
of (4.3, 2.5) and (*r,* θ)
coordinates of (5, 30°). The (*x,
y*) coordinates for point *B* are
(−1.5, −3) and the polar coordinates
are (3.4, 240°).

Within this frame of reference, quantities encountered in the
study of biomechanics may be portrayed and handled graphically.
Consider a jumper who takes off with a velocity of 5.5 meters per second
(m/sec) at an angle of 18 degrees. Because the takeoff
velocity has both magnitude and direction, it is a vector quantity
and may be *resolved* into its components.
By selecting a linear unit of measurement to represent a unit of
velocity, such as 1 cm to represent 1 meter per second of velocity,
and by constructing a line of the appropriate length at an angle
of 18 degrees to the *x* axis, one can
determine the horizontal and vertical components of velocity for
that jump. This is done by constructing a right triangle in which
the hypotenuse is the vector representing the total velocity of
the jump, the vertical side is the vertical velocity, and the horizontal
side is the horizontal velocity (Figure 10.7). The horizontal and
vertical velocity components are determined by carefully measuring
the lengths of the respective sides of the triangle and converting
those amounts to velocity values by using the conversion factor.
In this illustration, the conversion factor selected was 1 cm = 1
m/sec. The values obtained from measuring the lengths of
the sides and applying the conversion factor were 5.2 m/sec
of horizontal velocity component and 1.7 m/sec of vertical
velocity.

The *combination* of vectors to determine
the resultantvector may also be obtained graphically by the construction of
a parallelogram, the sides of which are linear representations of
the two vectors. The first step is to mark a point *P* on a piece of paper. This represents
the point at which the two vectors are applied. From this point
two vector lines are drawn to scale, with the correct angle between
them—that is, the same angle that actually exists between
the two vectors. By using these two lines as two sides, a parallelogram
is constructed with the addition of two more lines to form the other
two sides (Figure 10.8). The diagonal is then drawn from the point
of application *P* to the opposite corner.
This diagonal represents both in magnitude and direction the composite effect
of the two separate vectors. It is the resultant vector *R.*

When one wishes to determine the resultant of *three* or
more vectors acting at one point, a similar procedure is followed.
First, the resultant of two of the vectors is found. A second parallelogram
is then constructed using the third vector as one side and the resultant
of the first two vectors as the second side. The resultant vector
of this second parallelogram is the resultant of all three vectors
(Figure 10.9).

###### Figure 10.9

Parallelogram method used for determining the composite effect of three or more forces applied to the same point. R1 is resultant of combined forces A and B. R2 is resultant of combined forces R1 and C. Thus A + B + C = R2. For example, the three rasti exert forces through the patellar tendon.

Another graphic method combines the vectors by adding them head
to tail. Suppose muscle *J* has a force
of 1000 newtons and is pulling on bone E–F at an angle
of 10 degrees, and muscle *K* has a force
of 800 newtons and is pulling at an angle of 40 degrees (Figure
10.10a). The composite effect of these two muscles may be described
in terms of the amount of force and the direction or angle of pull
of that composite force. Again a linear unit of measure is selected
to represent a unit of force, and the vectors are placed in reference
to the *x, y* axes so that the tail of
the force vector for muscle *K* is added
to the head of the force vector for muscle *J* (Figure
10.10b). The scale used in this example is 1 cm = 400
N. The resultant vector was drawn connecting the tail of *J* with the head of *K.* Its
length represents the resultant force of the two muscles. The force
in newtons was obtained by multiplying the vector length by the
conversion factor, 1 cm = 400 N. The direction
of the resultant force is indicated by the angle between the *x* axis and the resultant (measured
with a protractor). Careful use of a protractor and ruler produced an *R* of 4.4 cm, which is equivalent to
1760 newtons pulling in a direction of 23.5 degrees. Like the parallelogram
technique, this method is not limited to two vectors. Any number
of vectors may be combined in this fashion.

###### Figure 10.10

Combination of vectors using graphic method. Vector *J* is added to vector *K* taking into account magnitude and
direction. Resultant *R* is drawn by
connecting the tail of *J* to the head
of *K.* When measured with a ruler, *R* equals 7 cm, which according to the
scale is equivalent to 1750 N. The resultant angle θ,
when measured with a protractor, is 23.5°.

Although the graphic method has value for portraying the situation, it does have serious drawbacks when it comes to calculating results. Accuracy is difficult to control in the drawing and measuring process, and the procedure is slow and unwieldy. A more accurate and efficient approach makes use of trigonometric relationships for both combining and resolving vectors.

Any vector may be resolved into horizontal and vertical components
if the trigonometric relationships of a right triangle are employed.
Let us use the previous example of a jumper whose velocity at takeoff
was 9.6 m/sec in the direction of 18 degrees with the horizontal.
To find the horizontal velocity (*V*x)
and vertical velocity (*V*y)
at takeoff a right triangle is constructed. With the takeoff velocity
(*R*) as the hypote-nuse of the triangle,
the vertical and horizontal components of velocity become the vertical
and horizontal sides of the triangle (Figure 10.11). To obtain the
values of *V*x and *V*y the sine and cosine functions
are used as shown in Figure 10.11. The horizontal velocity of the jump *V*x turns out to be 9.1 m/sec
and the vertical velocity *V*y is
3.0m/sec.

The combination of vectors is also possible with the use of right triangle trigonometric relationships. If two vectors are applied at right angles to each other, the solution should appear reasonably obvious because it is the reverse of the example just explained. If a volleyball is served with a vertical velocity of 15 m/sec and a horizontal velocity of 26 m/sec, the velocity of the serve and the angle of release may be determined as shown in Figure 10.12. The resultant velocity—that is, the velocity of the serve—is 30 m/sec and the angle of projection is 30 degrees.

If more than two vectors are involved or if they are not at right
angles to each other as shown in previous examples, the resultant
may be obtained by determining the *x* and *y* components for each individual vector
and then summing these individual components to obtain the *x* and *y* components
of the resultant. Once the *x* and *y* components are known, the magnitude
and direction of *R* may be obtained.
Let us consider the problem treated graphically in Figure 10.10,
in which we were interested in determining the composite force
of the two muscles and the resultant direction or angle of pull.
To solve this problem trigonometrically, the horizontal and vertical
components for each muscle must first be determined, as in Figure
10.12.

Next, to obtain the *y* component
of the resultant effect of the *two* muscles,
the *y* values for muscles *J* and *K* are
summed (∑*y* = 173.6 + 514.2; ∑*y* = 687.8 N). Similarly,
the *x* component for *R* is
the sum of the *x* values for *J* and *K* (∑*x* = 984.8 + 612.8; ∑*x* = 1597.6 N).

As we have seen before, a knowledge of the horizontal (*x*) and vertical (*y*)
components makes it possible to determine the resultantvector.
A triangle is formed and the unknown parts are found. Figure 10.13
presents the solution once the *x* and *y* values for muscles *J* and *K* have
each been summed. The summed *F*y and *F*x values form the two sides
of the triangle and the unknown resultant force is the hypotenuse. Using
the tangent relationship between the two sides, the resultant angle
of pull of the two muscles was calculated to be 23.3 degrees. With
this additional information, the composite or resultant force of
1739 newtons was found using the sin θ = opposite/hypotenuse
relationship.

In the explanation of coordinate systems it was shown that values
of *x* and *y* may
be negative and that values of θ may exceed 90
degrees (see Figure 10.6). An example of a problem with these additional
factors is that of the hiker who plots the stated course and then
must determine the resultantdisplacement at the completion of the
course:

- A hiker walks the following course: 2000 meters at 30 degrees, 1000 meters at 100 degrees, and 500 meters at 225 degrees. What is her resultant displacement? The solution is presented in Figure 10.14.

The ability to handle variables of motion and force as vector
or scalar quantities should improve one’s understanding
of motion and the forces causing it. The effect that a muscle’s
angle of pull has on the force available for moving a limb is better
understood when it is subjected to vector analysis. Similar study
of the direction and force of projectiles improves one’s
understanding of the effect of gravity, angle of release, and force
of release upon the flight of a projectile. The effect of several
muscles exerting their combined forces on a single bone is also
clarified when treated quantitatively as the combination of vector
quantities to obtain a *resultant.* Indeed,
the effect that a change in any variable may produce becomes much
more apparent. Without the use of a vector relationship, it would
be difficult if not impossible to describe motion and forces in
meaningful, quantitative terms.

*Fundamentals of biomechanics: Equilibrium, motion, and deformation.*2nd ed. New York: Springer.

- 1. Define the following key terms:
- Statics
- Dynamics
- Kinematics
- Kinetics
- Scalar
- Vector
- Component vector

- 2. Express the following units in metric terms:
- a. A force of 25 pounds
- b. A mass of 5 slugs
- c. A distance of 11 inches
- d. A velocity of 20 feet per second
- e. A volume of 3 quarts

- See Appendix H, Chapter 10.
- 3. Determine the distance between each set of points (scale: 1 unit = 10 cm).
- See Appendix H, Chapter 10.
- 4. Find the
*x*and*y*component for each of the following vectors:- a. 45 m/sec at 25°
- b. 85 N at 135°
- c. 118 kg at 310°
- d. 25 m/sec2 at 210°

- See Appendix H, Chapter 10.
- 5. A basketball official runs 20 meters along the sideline in one direction, reverses, and runs 8 meters. What is the distance run? What is the displacement? Draw a vector diagram.
- See Appendix H, Chapter 10.
- 6. The muscular force of a muscle is 650 N and the muscle is pulling on the bone at an angle of 15 degrees. What are the vertical and horizontal components of this force?
- See Appendix H, Chapter 10.
- 7. At the moment of release, a baseball has a horizontal velocity component of 25 meters per second and a vertical velocity component of 14 meters per second. At what angle was it released, and what was its initial velocity in the direction of the throw in m/sec? In ft/sec?
- See Appendix H, Chapter 10.
- 8. A child is being pulled in a sled by a person holding a rope that has an angle of 20 degrees with the horizontal. The total force being used to move the sled at a constant forward speed is 110 N. How much of the force is horizontal? Vertical?
- See Appendix H, Chapter 10.
- 9. An orienteer runs the following course: 1000 meters at 45°,
1500 meters at 120°, 500 meters at 190°.
- a. Draw the course to scale accurately.
- b. Determine the resultantdisplacement graphically.
- c. Determine the resultantdisplacement trigonometrically.
- d. Explain any differences you have between your graphic and trigonometric results.
- e. Express the orienteer’s position at the end of the course in terms of rectangular coordinates; polar coordinates.

- See Appendix H, Chapter 10.
- 10. A football lineman charges an opponent with a force of 175 pounds in the direction of 310 degrees. The opponent charges back with a force of 185 pounds in the direction of 90 degrees. What is the resultant force and in what direction will it act?
- See Appendix H, Chapter 10.
- 11. Referring to Figures 2.3 and 7.21, make a tracing of the
femur and adductor longus muscle. Draw a straight line to represent
the mechanical axis of the femur and another to represent the muscle’s
line of pull.
- a. Using a protractor, determine the angle of pull of the muscle (angle formed by muscle’s line of pull and mechanical axis of bone).
- b. Assuming a total muscle force of 900 N, calculate the force components.

- See Appendix H, Chapter 10.
- 12. Muscle
*A*has a force of 450 N and is pulling on a bone at an angle of 15 degrees. Muscle*B*has a force of 600 N and is pulling on the same bone at the same spot but at an angle of 30 degrees. Muscle*C*has a force of 325 N and is pulling at the same spot with an angle of pull of 10 degrees. What is the composite effect of these muscles in terms of amount of force and direction? - See Appendix H, Chapter 10.
- 13. Name as many vector and scalar quantities you can think of that are part of the games of football, tennis, or golf.
- 14. Choose one of the following sports:
- Volleyball
- Soccer
- Ice hockey
- a. For that sport make a list of the individual motor skills represented and classify each skill according to the classification model presented in Chapter 1, System for Classification of Motor Skills.
- b. Name the underlying mechanics objective for each skill identified (Chapter 1, Mechanical Principles).
- c. Identify the scalar and vector quantities that are a part
of each skill identified. For each skill, draw and label a vector
diagram for
*one*of the vector quantities named.