- Distinguish angular motion from rectilinear and curvilinear motion.
- Discuss the relationships among angular kinematic variables.
- Correctly associate angular kinematic quantities with their units of measure.
- Explain the relationships between angular and linear displacement, angular and linear velocity, and angular and linear acceleration.
- Solve quantitative problems involving angular kinematic quantities and the relationships between angular and linear kinematic quantities.

Why is a driver longer than a 9-iron? Why do batters slide their hands up the handle of the bat to lay down a bunt but not to drive the ball? How does the angular motion of the discus or hammer during the windup relate to the linear motion of the implement after release?

These questions relate to angular motion, or rotational motion around an axis. The axis of rotation is a line, real or imaginary, oriented perpendicular to the plane in which the rotation occurs, like the axle for the wheels of a cart. In this chapter, we discuss angular motion, which, like linear motion, is a basic component of general motion.

Understanding angular motion is particularly important for the student of human movement, because most volitional human movement involves rotation of one or more body segments around the joints at which they articulate. Translation of the body as a whole during gait occurs by virtue of rotational motions taking place at the hip, knee, and ankle around imaginary mediolateral axes of rotation. During the performance of jumping jacks, both the arms and the legs rotate around imaginary anteroposterior axes passing through the shoulder and hip joints. The angular motion of sport implements such as golf clubs, baseball bats, and hockey sticks, as well as household and garden tools, is also often of interest.

As discussed in Chapter 2, clinicians, coaches, and teachers of physical activities routinely analyze human movement based on visual observation. What is actually observed in such situations is the angularkinematics of human movement. Based on observation of the timing and range of motion (ROM) of joint actions, the experienced analyst can make inferences about the coordination of muscle activity producing the joint actions and the forces resulting from those joint actions.

Much of the reported description of the developmental stages of motor skills is based on analysis of angularkinematics. For example, three developmental stages for kicking among children age 2–6 have been identified (2). In stage 1, the child kicks using a small ROM of hip flexion, with no coordinated motion apparent at any other joint. In stage 2, knee extension is coordinated with hip flexion, and the arms are abducted at the shoulders to promote balance. Stage 3 is characterized by increased hip flexion and knee extension, and elbow flexion is present in addition to shoulder abduction to improve balance. The knowledgeable analyst can obtain a great deal of information about the relative developmental and skill levels of the performer through careful observation of angular kinematics.

As reviewed in Appendix A, an angle is composed of two sides that intersect at a vertex. Quantitativekinematic analysis can be achieved by projecting filmed images of the human body onto a piece of paper, with joint centers then marked with dots and the dots connected with lines representing the longitudinal axes of the body segments (Figure 11-1). A protractor can be used to make hand measurements of angles of interest from this representation, with the joint centers forming the vertices of the angles between adjacent body segments. (The procedure for measuring angles with a protractor is reviewed in Appendix A.) Videos and films of human movement can also be analyzed using this same basic procedure to evaluate the angles present at the joints of the human body and the angular orientations of the body segments. The angle assessments are usually done with computer software from stick figure representations of the human body constructed in computer memory.

Assessing the angle at a joint involves measuring the angle of
one body segment relative to the other body segment articulating
at the joint. The **relative angle** at
the knee is the angle formed between the longitudinal axis of the
thigh and the longitudinal axis of the lower leg (Figure 11-2).
When joint ROM is quantified, it is the relative joint angle that
is measured.

The convention used for measuring relative joint angles is that in anatomical reference position, all joint angles are at 0˚. As discussed in Chapter 5, joint motion is then measured directionally. For example, when the extended arm is elevated 30˚ in front of the body in the sagittal plane, the arm is in 30˚ of flexion at the shoulder. When the leg is abducted at the hip, the ROM in abduction is likewise measured from 0˚ in anatomical reference position.

Other angles of interest are often the orientations of the body
segments themselves. As discussed in Chapter 9, when the
trunk is in flexion, the angle of inclination of the trunk directly
affects the amount of force that must be generated by the trunk
extensor muscles to support the trunk in the position assumed. The
angle of inclination of a body segment, referred to as its **absolute angle**, is measured with respect
to an absolute reference line, usually either horizontal or vertical. Figure
11-3 shows quantification of segment angles with respect to
the right horizontal.

Goniometers are commonly used by clinicians for direct measurement of relative joint angles on a live human subject. A goniometer is essentially a protractor with two long arms attached. One arm is fixed so that it extends from the protractor at an angle of 0˚. The other arm extends from the center of the protractor and is free to rotate. The center of the protractor is aligned over the joint center, and the two arms are aligned over the longitudinal axes of the two body segments that connect at the joint. The angle at the joint is then read at the intersection of the freely rotating arm and the protractor scale. The accuracy of the reading depends on the accuracy of the positioning of the goniometer. Knowledge of the underlying joint anatomy is essential for proper location of the joint center of rotation. Placing marks on the skin to identify the location of the center of rotation at the joint and the longitudinal axes of the body segments before aligning the goniometer is sometimes helpful, particularly if repeated measurements are being taken at the same joint.

Other instruments available for quantifying angles relative to
the human body are the electrogoniometer and various inclinometers.
The electrogoniometer (referred to as an *elgon*)
was developed by Peter Karpovich in the late 1950s. An elgon is
simply a goniometer with an electrical potentiometer at its vertex.
When the arms of the elgon are attached with tape or velcro straps
over a joint center, changes in the relative angle at the joint
cause proportional changes in the electrical current emitted by
the elgon. Inclinometers are other devices used for direct assessment
of human body segment angles. These are usually gravitationally
based instruments that identify the absolute angle of orientation
of a body segment. Because accurate positioning of inclinometers
is critical to the accuracy of the readings obtained, the measurement
validity and reliability of these devices is controversial, particularly
for measuring spinal orientation (11).

Quantification of joint angles is complicated by the fact that
joint motion is often accompanied by displacement of one bone with
respect to the articulating bone at the joint. This phenomenon is caused
by normal asymmetries in the shapes of the articulating bone surfaces.
One example is the tibiofemoral joint, at which medial rotation
and anterior displacement of the femur on the tibial plateau accompany
flexion (Figure 11-4). As a result, the location of the
exact center of rotation at the joint changes slightly when joint angle
changes. The center of rotation at a given joint angle, or at a
given instant in time during a dynamic movement, is called the **instant center**. The exact location
of the instant center for a given joint may be determined through
measurements taken from roentgenograms (X rays), which are usually
taken at 10˚ intervals throughout the ROM at the joint
(12). The instant center at the tibiofemoral joint of the
knee shifts during angular movement at the knee due to the ellipsoid
shapes of the femoral condyles (13).

The interrelationships among angularkinematic quantities are similar to those discussed in Chapter 10 for linear kinematic quantities. Although the units of measure associated with the angular kinematic quantities are different from those used with their linear counterparts, the relationships among angular units also parallel those present among linear units.

Consider a pendulum swinging back and forth from a point of support. The pendulum is rotating around an axis passing through its point of support perpendicular to the plane of motion. If the pendulum swings through an arc of 60˚, it has swung through an angular distance of 60˚. If the pendulum then swings back through 60˚ to its original position, it has traveled an angular distance totaling 120˚ (60˚ + 60˚). Angular distance is measured as the sum of all angular changes undergone by a rotating body.

The same procedure may be used for quantifying the angular distances through which the segments of the human body move. If the angle at the elbow joint changes from 90˚ to 160˚ during the flexion phase of a forearm curl exercise, the angular distance covered is 70˚. If the extension phase of the curl returns the elbow to its original position of 90˚, an additional 70˚ have been covered, resulting in a total angular distance of 140˚ for the complete curl. If 10 curls are performed, the angular distance transcribed at the elbow is 1400˚ (10 × 140˚).

Just as with its linear counterpart, **angular
displacement** is assessed as the difference in the initial and
final positions of the moving body. If the angle at the knee of
the support leg changes from 5˚ to 12˚ during
the initial support phase of a running stride, the angular distance
and the angular displacement at the knee are 7˚. If extension
occurs at the knee, returning the joint to its original 5˚ position,
angular distance totals 14˚ (7˚ + 7˚),
but angular displacement is 0˚, because the final position
of the joint is the same as its original position. The relationship
between angular distance and angular displacement is represented
in Figure 11-5.

Like linear displacement, angular displacement is defined by both magnitude and direction. Since rotation observed from a side view occurs in either a clockwise or a counterclockwise direction, the direction of angular displacement may be indicated using these terms. The counterclockwise direction is conventionally designated as positive (+), and the clockwise direction as negative (−) (Figure 11-6). With the human body, it is also appropriate to indicate the direction of angular displacement with joint-related terminology such as flexion or abduction.

Three units of measure are commonly used to represent angular distance and angular displacement. The most familiar of these units is the degree. A complete circle of rotation transcribes an arc of 360˚, an arc of 180˚ subtends a straight line, and 90˚ forms a right angle between perpendicular lines (Figure 11-7).

Another unit of angular measure sometimes used in biomechanical
analyses is the **radian**. A line connecting
the center of a circle to any point on the circumference of the
circle is a radius. A radian is defined as the size of the angle
subtended at the center of a circle by an arc equal in length to
the radius of the circle (Figure 11-8). One complete circle
is an arc of 2π radians, or 360˚. Because
360˚ divided by 2π is 57.3˚,
one radian is equivalent to 57.3˚. Because a radian is
much larger than a degree, it is a more convenient unit for the
representation of extremely large angular distances or displacements. Radians
are often quantified in multiples of pi (π).

The third unit sometimes used to quantify angular distance or displacement is the revolution. One revolution transcribes an arc equal to a circle. Dives and some gymnastic skills are often described by the number of revolutions the human body undergoes during their execution. The one-and-a-half forward somersault dive is a descriptive example. Figure 11-9 illustrates the way in which degrees, radians, and revolutions compare as units of angular measure.

Angular speed is a scalar quantity and is defined as the angular distance covered divided by the time interval over which the motion occurred:

The lowercase Greek letter sigma (σ) represents angular speed, the lowercase Greek letter phi (φ) represents angular distance, and t represents time.

**Angular velocity** is calculated
as the change in angular position, or the angular displacement,
that occurs during a given period of time:

The lowercase Greek letter omega (ω) represents angular velocity, the capital Greek letter theta (Θ) represents angular displacement, and t represents the time elapsed during the velocity assessment. Another way to express change in angular position is angular position2 − angular position1, in which angular position1 represents the body’s position at one point in time and angular position2 represents the body’s position at a later point:

Because angular velocity is based on angular displacement, it must include an identification of the direction (clockwise or counterclockwise, negative or positive) in which the angular displacement on which it is based occurred.

Units of angular speed and angular displacement are units of angular distance or angular displacement divided by units of time. The unit of time most commonly used is the second. Units of angular speed and angular velocity are degrees per second (deg/s), radians per second (rad/s), revolutions per second (rev/s), and revolutions per minute (rpm).

Moving the body segments at a high rate of angular velocity is a characteristic of skilled performance in many sports. Angular velocities at the joints of the throwing arm in Major League Baseball pitchers have been reported to reach 2320 deg/s in elbow extension and 7240 deg/s in internal rotation (5). Interestingly, these values are also high in the throwing arms of youth pitchers, with 2230 deg/s in elbow extension and 6900 deg/s in internal rotation documented (5). Comparison of different types of pitches thrown by collegiate baseball pitchers showed internal rotation values of 7550 deg/s for fastballs, 6680 deg/s for change-ups, 7120 deg/s for curveballs, and 7920 deg/s for sliders (4). Figures 11-10, 11-11, 11-12, and 11-13 display the patterns of joint angle and joint angular velocity at the elbow and shoulder during underhand and overhand throws executed by collegiate softball players. A study of world-class male and female tennis players has documented a sequential rotation of segmental rotations. Analysis of the cocked, preparatory position showed the elbow flexed to an average of 104˚ and the upper arm rotated to about 172˚ of external rotation at the shoulder. Proceeding from this position, there was a rapid sequence of segmental rotations, with averages of trunk tilt of 280 deg/s, upper torso rotation of 870 deg/s, pelvis rotation of 440 deg/s, elbow extension of 1510 deg/s, wrist flexion of 1950 deg/s, and shoulder internal rotation of 2420 deg/s for males and 1370 deg/s, for females (6). Angular velocity of the racket during serves executed by professional male tennis players has been found to range from 1900 to 2200 deg/s (33.2 to 38.4 rad/s) just before ball impact (3).

As discussed in Chapter 10, when the human body becomes a projectile during the execution of a jump, the height of the jump determines the amount of time the body is in the air. When figure skaters perform a triple or quadruple axel, as compared to a axel or double axel, this means that either jump height or rotational velocity of the body must be greater. Measurements of these two variables indicate that it is the skater’s angular velocity that increases, with skilled skaters rotating their bodies in excess of 5 rev/s while airborne during the triple axel (10). Higher rotational velocities of the body with increasing skill difficulty have also been documented in gymnastics, with representative values of 6.80 rad/s for the handspring, 7.77 rad/s for the handspring incorporating a somersault and one-half twist, and 10.2 rad/s for the backward somersault layout with two twists (1).

**Angular acceleration** is the rate
of change in angular velocity, or the change in angular velocity
occurring over a given time. The conventional symbol for angular
acceleration is the lowercase Greek letter alpha (α):

The calculation formula for angular acceleration is therefore the following:

In this formula, ω1 represents angular velocity at an initial point in time, ω2 represents angular velocity at a second or final point in time, and t1 and t2 are the times at which velocity was assessed. Use of this formula is illustrated in Sample Problem 11.1.

Just as with linear acceleration, angular acceleration may be positive, negative, or zero. When angular acceleration is zero, angular velocity is constant. Just as with linear acceleration, positive angular acceleration may indicate either increasing angular velocity in the positive direction or decreasing angular velocity in the negative direction. Similarly, a negative value of angular acceleration may represent either decreasing angular velocity in the positive direction or increasing angular velocity in the negative direction.

Units of angular acceleration are units of angular velocity divided by units of time. Common examples are degrees per second squared (deg/s2), radians per second squared (rad/s2), and revolutions per second squared (rev/s2). Units of angular and linearkinematic quantities are compared in Table 11-1.

Displacement | Velocity | Acceleration | |
---|---|---|---|

Linear | meters | meters/second | meters/second2 |

Angular | radians | radians/second | radians/second2 |

A golf club is swung with an averageangular acceleration of 1.5 rad/s2. What is the angular velocity of the club when it strikes the ball at the end of a 0.8 s swing? (Provide an answer in both radian and degree-based units.)

The formula to be used is the equation relating angular acceleration, angular velocity, and time:

Substituting in the known quantities yields the following:

It may also be deduced that the angular velocity of the club at the beginning of the swing was zero:

In degree-based units:

Because representing angular quantities using symbols such as
curved arrows would be impractical, angular quantities are represented
with conventional straight vectors, using what is called the **right hand rule**. According to this
rule, when the fingers of the right hand are curled in the direction
of an angular motion, the vector used to represent the motion is
oriented perpendicular to the plane of rotation, in the direction
the extended thumb points in (Figure 11-14). The magnitude
of the quantity may be indicated through proportionality to the
vector’s length.

Angular speed, velocity, and acceleration may be calculated as instantaneous or average values, depending on the length of the time interval selected. The instantaneous angular velocity of a baseball bat at the instant of contact with a ball is typically of greater interest than the average angular velocity of the swing, because the former directly affects the resultant velocity of the ball.

The greater the radius is between a given point on a rotating body and the axis of rotation, the greater is the linear distance undergone by that point during an angular motion (Figure 11-15). This observation is expressed in the form of a simple equation:

The curvilinear distance traveled by the point of interest s
is the product of r, the point’s **radius
of rotation**, and φ, the angular distance through
which the rotating body moves, which is quantified in radians.

For this relationship to be valid, two conditions must be met: (a) The linear distance and the radius of rotation must be quantified in the same units of length, and (b) angular distance must be expressed in radians. Although units of measure are normally balanced on opposite sides of an equal sign when a valid relationship is expressed, this is not the case here. When the radius of rotation (expressed in meters) is multiplied by angular displacement in radians, the result is linear displacement in meters. Radians disappear on the right side of the equation in this case because, as may be observed from the definition of the radian, the radian serves as a conversion factor between linear and angular measurements.

The same type of relationship exists between the angular velocity of a rotating body and the linear velocity of a point on that body at a given instant in time. The relationship is expressed as the following:

The linear (tangential) velocity of the point of interest is v, r is the radius of rotation for that point, and ω is the angular velocity of the rotating body. For the equation to be valid, angular velocity must be expressed in radian-based units (typically rad/s), and velocity must be expressed in the units of the radius of rotation divided by the appropriate units of time. Radians are again used as a linear-angular conversion factor, and are not balanced on opposite sides of the equals sign:

The use of radian-based units for conversions between linear and angular velocities is shown in Sample Problem 11.2.

During several sport activities, an immediate performance goal is to direct an object such as a ball, shuttlecock, or hockey puck accurately, while imparting a relatively large amount of velocity to it with a bat, club, racket, or stick. In baseball batting, the initiation of the swing and the angular velocity of the swing must be timed precisely to make contact with the ball and direct it into fair territory. A 40 m/s pitch reaches the batter 0.41 s after leaving the pitcher’s hand. It has been estimated that a difference of 0.001 s in the time of initiation of the swing can determine whether the ball is directed to center field or down the foul line, and that a swing initiated 0.003 s too early or too late will result in no contact with the ball (7). Similarly, there is a very small window of time during which gymnasts on the high bar can release from the bar to execute a skillful dismount. For high-bar finalists in the 2000 Olympic Games in Sydney, the release window was an average of 0.055 s (8).

With all other factors held constant, the greater the radius of rotation at which a swinging implement hits a ball, the greater the linear velocity imparted to the ball. In golf, longer clubs are selected for longer shots, and shorter clubs are selected for shorter shots. However, the magnitude of the angular velocity figures as heavily as the length of the radius of rotation in determining the linear velocity of a point on a swinging implement. Little Leaguers often select long bats, which increase the potential radius of rotation if a ball is contacted, but are also too heavy for the young players to swing as quickly as shorter, lighter bats. The relationship between the radius of rotation of the contact point between a striking implement and a ball and the subsequent velocity of the ball is shown in Figure 11-16.

It is important to recognize that the linear velocity of a ball
struck by a bat, racket, or club is *not* identical
to the linear velocity of the contact point on the swinging implement.
Other factors, such as the directness of the hit and the elasticity
of the impact, also influence ball velocity.

Two baseballs are consecutively hit by a bat. The first ball is hit 20 cm from the bat’s axis of rotation, and the second ball is hit 40 cm from the bat’s axis of rotation. If the angular velocity of the bat was 30 rad/s at the instant that both balls were contacted, what was the linear velocity of the bat at the two contact points?

The formula to be used is the equation relating linear and angular velocities:

For ball 1:

For ball 2:

The acceleration of a body in angular motion may be resolved into two perpendicular linear acceleration components. These components are directed along and perpendicular to the path of angular motion at any point in time (Figure 11-17).

The component directed along the path of angular motion takes
its name from the term *tangent.* A tangent
is a line that touches, but does not cross, a curve at a single
point. The tangential component, known as **tangential
acceleration**, represents the change in linear speed for a body
traveling on a curved path. The formula for tangential acceleration
is the following:

Tangential acceleration is at, v1 is the tangential linear velocity of the moving body at an initial time, v2 is the tangential linear velocity of the moving body at a second time, and t is the time interval over which the velocities are assessed.

When a ball is thrown, the ball follows a curved path as it is accelerated by the muscles of the shoulder, elbow, and wrist. The tangential component of ball acceleration represents the rate of change in the linear speed of the ball. Because the speed of projection greatly affects a projectile’s range, tangential velocity should be maximum just before ball release if the objective is to throw the ball fast or far. Once ball release occurs, tangential acceleration is zero, because the thrower is no longer applying a force.

The relationship between tangential acceleration and angular acceleration is expressed as follows:

Linear acceleration is at, r is the radius of rotation, and α is angular acceleration. The units of linear acceleration and the radius of rotation must be compatible, and angular acceleration must be expressed in radian-based units for the relationship to be accurate.

Although the linear speed of an object traveling along a curved
path may not change, its direction of motion is constantly changing.
The second component of angular acceleration represents the rate of
change in direction of a body in angular motion. This component
is called **radial acceleration**, and
it is always directed toward the center of curvature. Radial acceleration
may be quantified by using the following formula:

Radial acceleration is ar, v is the tangential linear velocity of the moving body, and r is the radius of rotation. An increase in linear velocity or a decrease in the radius of curvature increases radial acceleration. Thus, the smaller the radius of curvature (the tighter the curve) is, the more difficult it is for a cyclist to negotiate the curve at a high velocity (see Chapter 14).

During execution of a ball throw, the ball follows a curved path because the thrower’s arm and hand restrain it. This restraining force causes radial acceleration toward the center of curvature throughout the motion. When the thrower releases the ball, radial acceleration no longer exists, and the implement follows the path of the tangent to the curve at that instant. The timing of release is therefore critical: If release occurs too soon or too late, the ball will be directed to the left or the right rather than straight ahead. Sample Problem 11.3 demonstrates the effects of the tangential and radial components of acceleration.

Both tangential and radial components of motion can contribute to the resultantlinear velocity of a projectile at release. For example, during somersault dismounts from the high bar in gymnastics routines, although the primary contribution to linear velocity of the body’s center of gravity is generally from tangential acceleration, the radial component can contribute up to 50% of the resultant velocity (9). The size of the contribution from the radial component, and whether the contribution is positive or negative, varies with the performer’s technique.

A windmill-style softball pitcher executes a pitch in 0.65 s. If her pitching arm is 0.7 m long, what are the magnitudes of the tangential and radial accelerations on the ball just before ball release, when tangential ball speed is 20 m/s? What is the magnitude of the total acceleration on the ball at this point?

To solve for tangential acceleration, use the following formula:

Substitute in what is known and assume that v1 = 0:

To solve for radial acceleration, use the following formula:

Substitute in what is known:

To solve for total acceleration, perform vector composition of tangential and radial acceleration. Since tangential and radial acceleration are oriented perpendicular to each other, the Pythagorean theorem can be used to calculate the magnitude of total acceleration.

An understanding of angular motion is an important part of the study of biomechanics, because most volitional motion of the human body involves the rotation of bones around imaginary axes of rotation passing through the joint centers at which the bones articulate. The angular kinematic quantities—angular displacement, angular velocity, and angular acceleration—possess the same interrelationships as their linear counterparts, with angular displacement representing change in angular position, angular velocity defined as the rate of change in angular position, and angular acceleration indicating the rate of change in angular velocity during a given time. Depending on the selection of the time interval, either average or instantaneous values of angular velocity and angular acceleration may be quantified.

Angularkinematic variables may be quantified for the relative angle formed by the longitudinal axes of two body segments articulating at a joint, or for the absolute angular orientation of a single body segment with respect to a fixed reference line. Different instruments are available for direct measurement of angles on a human subject.

1. The relative angle at the knee changes from 08 to 858 during the knee flexion phase of a squat exercise. If 10 complete squats are performed, what is the total angular distance and the total angular displacement undergone at the knee? (Provide answers in both degrees and radians.) (Answer: φ = 1700˚, 29.7 rad; Θ = 0)

2. Identify the angular displacement, the angular velocity, and the angular acceleration of the second hand on a clock over the time interval in which it moves from the number 12 to the number 6. Provide answers in both degree- and radian-based units. (Answer: Θ = −180˚, − π rad; ω = −6 deg/s, − π/30 rad/s; α = 0)

3. How many revolutions are completed by a top spinning with a constant angular velocity of 3 π rad/s during a 20 s time interval? (Answer: 30 rev)

4. A kicker’s extended leg is swung for 0.4 s in a counterclockwise direction while accelerating at 200 deg/s2. What is the angular velocity of the leg at the instant of contact with the ball? (Answer: 80 deg/s, 1.4 rad/s)

5. The angular velocity of a runner’s thigh changes from 3 rad/s to 2.7 rad/s during a 0.5 s time period. What has been the averageangular acceleration of the thigh? (Answer: −0.6 rad/s2, −34.4 deg/s2)

6. Identify three movements during which the instantaneous angular velocity at a particular time is the quantity of interest. Explain your choices.

7. Fill in the missing corresponding values of angular measure in the table below.

8. Measure and record the following angles for the drawing shown below:

a. The relative angle at the shoulder

b. The relative angle at the elbow

c. The absolute angle of the upper arm

d. The absolute angle of the forearm

Use the right horizontal as your reference for the absolute angles.

9. Calculate the following quantities for the diagram shown below:

a. The angular velocity at the hip over each time interval

b. The angular velocity at the knee over each time interval

Would it provide meaningful information to calculate the averageangular velocities at the hip and knee for the movement shown? Provide a rationale for your answer.

10. A tennis racket swung with an angular velocity of 12 rad/s strikes a motionless ball at a distance of 0.5 m from the axis of rotation. What is the linear velocity of the racket at the point of contact with the ball? (Answer: 6 m/s)

1. A 1.2 m golf club is swung in a planar motion by a right-handed golfer with an arm length of 0.76 m. If the initial velocity of the golf ball is 35 m/s, what was the angular velocity of the left shoulder at the point of ball contact? (Assume that the left arm and the club form a straight line, and that the initial velocity of the ball is the same as the linear velocity of the club head at impact.) (Answer: 17.86 rad/s)

2. David is fighting Goliath. If David’s 0.75 m sling is accelerated for 1.5 s at 20 rad/s2, what will be the initial velocity of the projected stone? (Answer: 22.5 m/s)

3. A baseball is struck by a bat 46 cm from the axis of rotation when the angular velocity of the bat is 70 rad/s. If the ball is hit at a height of 1.2 m at a 45˚ angle, will the ball clear a 1.2 m fence 110 m away? (Assume that the initial linear velocity of the ball is the same as the linear velocity of the bat at the point at which it is struck.) (Answer: No, the ball will fall through a height of 1.2 m at a distance of 105.7 m.)

4. A polo player’s arm and stick form a 2.5 m rigid segment. If the arm and stick are swung with an angular speed of 1.0 rad/s as the player’s horse gallops at 5 m/s, what is the resultant velocity of a motionless ball that is struck head-on? (Assume that ball velocity is the same as the linear velocity of the end of the stick.) (Answer: 7.5 m/s)

5. Explain how the velocity of the ball in Problem 4 would differ if the stick were swung at a 30˚ angle to the direction of motion of the horse.

6. List three movements for which a relative angle at a particular joint is important and three movements for which the absolute angle of a body segment is important. Explain your choices.

7. A majorette in the Rose Bowl Parade tosses a baton into the air with an initial angular velocity of 2.5 rev/s. If the baton undergoes a constant acceleration while airborne of −0.2 rev/s2 and its angular velocity is 0.8 rev/s when the majorette catches it, how many revolutions does it make in the air? (Answer: 14 rev)

8. A cyclist enters a curve of 30 m radius at a speed of 12 m/s. As the brakes are applied, speed is decreased at a constant rate of 0.5 m/s2. What are the magnitudes of the cyclist’s radial and tangential accelerations when his speed is 10 m/s? (Answer: ar = 3.33 m/s2; at = −0.5 m/s2)

9. A hammer is being accelerated at 15 rad/s2. Given a radius of rotation of 1.7 m, what are the magnitudes of the radial and tangential components of acceleration when tangential hammer speed is 25 m/s? (Answer: ar = 367.6 m/s2; at = 25.5 m/s2)

10. A speed skater increases her speed from 10 m/s to 12.5 m/s over a period of 3 s while coming out of a curve of 20 m radius. What are the magnitudes of her radial, tangential, and total accelerations as she leaves the curve? (Remember that ar and at are the vector components of total acceleration.) (Answer: ar = 7.81 m/s2; at = 0.83 m/s2; a = 7.85 m/s2)

• Absolute angles should consistently be measured in the same direction from a single reference—either horizontal or vertical.

• The counterclockwise direction is regarded as positive, and the clockwise direction is regarded as negative.

• Pi (π) is a mathematical constant equal to approximately 3.14, which is the ratio of the circumference to the diameter of a circle.

*Current research in sports biomechanics,*Basel, 1987, Karger.

*Biomechanics of sport,*Boca Raton, FL, 1989, CRC Press.

*Biomechanics in sports XI,*Amherst, MA, 1993, International Society of Biomechanics in Sports.

*Biomechanics of the musculoskeletal system*(3rd ed), Philadelphia, 2001, Lippincott Williams & Wilkins.

Bruggemann G-P: Biomechanics in gymnastics. In Van Gheluwe B
and Atha J: *Current research in sports biomechanics,* Basel,
1987, Karger.

Fleisig GS, Jameson EG, Dillman CJ, and Andrews JR: Biomechanics
of overhead sports. In Garrett WE Jr. and Kirkendall DT: *Exercise and sport science,* Philadelphia,
2000, Lippincott Williams & Wilkins.

- www.exploratorium.edu/baseball/index.html
*Explains scientific concepts related to baseball pitching and hitting.*

- www.exploratorium.edu/cycling/index.html
*Explains scientific concepts related to bicycle wheels and gear ratios.*

- www.exploratorium.edu/hockey/index.html
*Explains scientific concepts related to hockey, including how to translate rotational motion of the arms into linear motion of the puck.*