Topic 08 – Angular Kinematics and Kinetics

Topic 08 – Angular Kinematics and Kinetics

Lesson Learning Outcomes

• Define relative and absolute angular position, and distinguish between the two
• Define angular displacement
• Define average angular velocity and instantaneous angular velocity
• Define average angular acceleration and instantaneous angular acceleration
• Name the units of measurement for angular position, displacement, velocity, and acceleration
• Explain the relationship between average linear speed and average angular velocity
• Explain the relationship between instantaneous linear velocity and instantaneous angular velocity
• Define tangential acceleration and explain its relationship to angular acceleration
• Define centripetal acceleration and explain its relationship to angular velocity and tangential velocity
• Define moment of inertia
• Explain how the human body’s moment of inertia may be manipulated
• Explain Newton’s first, second and third law of motion as it applies to angular motion
• Define angular impulse and angular momentum
• Explain the relationship between angular impulse and angular momentum

Guiding Questions

• Why is it important to understand angular motions on human movements?
• How are angular motions measured and described?
• How does angular kinetics affect human movement and the techniques used in sports?

Definition of Angle

An angle is formed by the intersection of two lines, two planes, or a line and a plane.

Angular Position

Angular position refers to the orientation of a line with another line or plane. If the other line or plane is fixed and immovable relative to the earth, the angular position is an absolute angular position. If the other line or plane is capable of moving, the angular position is a relative angular position.

Mathematically, an angle measured in radius units (we’ll call these radians) is

Angular Displacement

Angular displacement is the change in absolute angular position experienced by a rotating line. Angular displacement is thus the angle formed between the final position and the initial position of a rotating line.

If the initial position of the arm is 5° from the vertical and its final position is 170° from the vertical, the angular displacement is +165°.

Angular and Linear Displacement

In the example, the hand moves 10 times as far as the insertion of the biceps tendon. The linear distance (arc length) traveled by a point on a rotating object is directly proportional to the angular displacement of the object and the radius, the distance that point is from the axis of rotation of the object. If the angular displacement is measured in radians, the linear distance traveled (the arc length) is equal to the product of the angular displacement and the radius. This is true only if the angular displacement is measured in radians. This relationship is expressed

mathematically in the equation below:

Angular Velocity

Its units of measurement are radians per second (rad/s), degrees per second (°/s), revolutions

per minute (rpm), and so on. Angular velocity is abbreviated with the Greek letter omega (ω). Angular velocity is a vector quantity, just like linear velocity, so it has direction associated with it. The direction of an angular velocity is determined using the right-hand thumb rule, as with angular displacement.

Average angular velocity is computed as the change in angular position (angular displacement) divided by time. Mathematically,

Instantaneous angular velocity is an indicator of how fast something is spinning at a specific instant in time.

The average angular velocity of a batter’s swing may determine whether or not she contacts the ball, but it is the bat’s instantaneous velocity at ball contact that determines how fast and how far the ball will go.

Relationship Between Angular and Linear Velocity

The relationship between angular displacement and linear distance traveled provides the answer. Consider a swinging golf club. All points on the club undergo the same angular displacement, and thus the same average angular velocity, because they all take the same time to undergo that displacement. But a point on the club closer to the club head (and farther from the axis of rotation) moves through a longer arc length than a point farther from the club head (and closer to the axis of rotation). The two points travel their respective arc lengths in the same time. The point farther from the axis of rotation must have a faster linear speed because it moves a longer distance but in the same time. Mathematically, this relationship can be derived from the relationship between angular displacement and linear distance traveled (arc length).

Dividing both sides by the time it takes to rotate through the displacement gives us

The average linear speed of a point on a rotating object is equal to the average angular velocity of the object times the radius (the distance from the point on the object to the axis of rotation of the object). At an instant in time, this relationship becomes

The instantaneous linear velocity of a point on a rotating object is equal to the instantaneous angular velocity of the rotating object times the radius. The direction of this instantaneous linear velocity is perpendicular to the radius and tangent to the circular path of the point. The

instantaneous linear velocities for the two points on the golf club are shown below.

Angular Acceleration

Angular acceleration is the rate of change of angular velocity. Its units of measurement are radians per second per second (rad/s/s or rad/s2), degrees per second per second (°/s/s or °/s2), or some unit of angular velocity per unit of time. Angular acceleration is abbreviated with the Greek letter α (alpha). Just like linear acceleration, angular acceleration is a vector quantity; it has direction and size. The right-hand thumb rule is used to describe directions of angular acceleration vectors.

Average angular acceleration is computed as the change in angular velocity divided by time. Mathematically,

Angular acceleration occurs when something spins faster and faster or slower and slower, or when the spinning object’s axis of spin changes direction.

Angular and Linear Acceleration

When the angular velocity of a spinning object increases, the linear velocity of a point on the object increases as well. The angular and linear accelerations of a point on a spinning object are related.

Tangential Acceleration

The component of linear acceleration tangent to the circular path of a point on a rotating object is called the tangential acceleration of that point. Remember that a line is tangent to a circle if the line intersects the circle at just one point. A line from this point to the center of the circle—a radial line—is perpendicular to the tangent line. Tangential acceleration is related to the angular

acceleration of the object in the following way:

A point on a rotating object undergoes a linear acceleration tangent to its rotational path and equal to the angular acceleration of the object times the radius.

Centripetal Acceleration

The linear acceleration directed toward the center of the circle (axis of rotation) is called centripetal acceleration (or radial acceleration), and the force causing it is called centripetal force. It is directly proportional to the square of the tangential linear velocity and the square of the angular velocity. If angular velocity is held constant, centripetal acceleration is directly proportional to the radius of rotation. If the tangential linear velocity is held constant, centripetal acceleration is inversely proportional to the radius of rotation.

Mathematically, centripetal acceleration can be defined using two different equations:

Angular Inertia

The property of an object to resist changes in its angular motion is angular inertia or rotary inertia.

The quantity that describes angular inertia is called moment of inertia. Moment of inertia is abbreviated with the letter I. Theoretically, an object may be considered to be composed of many particles of mass. The moment of inertia of such an object about an axis through its center of gravity can be defined mathematically as follows:

Each particle provides some resistance to change in angular motion. This resistance is equal to the mass of the particle times the square of the radius from the particle to the axis of rotation. The sum of all the particles’ resistances to rotation is the total moment of inertia of the object. The units of measurement for moment of inertia are units of mass times units of length squared, or kg·m2 in SI units. Moment of inertia may also be represented mathematically as

An object may have more than one moment of inertia because an object may rotate about more than one axis of rotation.

MANIPULATING THE MOMENTS OF INERTIA OF THE HUMAN BODY

A rigid object has many different moments of inertia because it may have many axes of rotation. But for any one axis of rotation, only one moment of inertia is associated with that axis. The human body is not a rigid object, though, because humans can move their limbs relative to each other. These movements may change the distribution of mass about an axis of rotation, thus changing the moment of inertia about that axis. A human’s moment of inertia about any axis is variable. There is more than one value for the moment of inertia about an axis. This means that humans can manipulate their moments of inertia.

ANGULAR MOMENTUM OF A RIGID BODY

Angular momentum is the angular analog of linear momentum, so it is the product of the angular analog of mass (moment of inertia) times the angular analog of linear velocity (angular velocity). Mathematically, then, the angular momentum of a rigid body is

Angular momentum is abbreviated with the letter H. The units for angular momentum are kilogram meters squared per second (kg·m2/s). Angular momentum is a vector quantity, just like linear momentum, so it has size and direction. The direction of angular momentum is the same as the direction of the angular velocity that defines it. The right-hand thumb rule is used to determine direction.

Angular momentum depends on two variables: moment of inertia and angular velocity. For rigid objects, changes in angular momentum also depend on changes in only one variable—angular velocity—because the moment of inertia of a rigid object does not change. For nonrigid objects, however, changes in angular momentum may result from changes in angular velocity or changes in moment of inertia, or both, because angular velocity and moment of inertia are both variable.

ANGULAR MOMENTUM OF THE HUMAN BODY

Mathematically, the angular momentum about an axis through the center of gravity of a multisegmented object such as the human body is defined below:

In other words, the sum of the angular momenta of all the body segments gives an approximation of the angular momentum of the entire body.

Angular Interpretation of Newton’s 1^st Law

The angular momentum of an object remains constant unless a net external torque is exerted on it. For a rigid object whose moments of inertia are constant, this law implies that the angular velocity remains constant. Its rate of rotation and its axis of rotation do not change unless an external torque acts to change it.

The angular momentum of the human body is constant unless external torques act on it. Mathematically, this can be represented as

Because the body’s moment of inertia is variable and can be changed by altering limb positions, the body’s angular velocity also changes to accommodate the changes in the moment of inertia. In this case, Newton’s first law does not require that the angular velocity be constant, but rather that the product of the moment of inertia times the angular velocity be constant if no external torques act. For this to occur, any increases in moment of inertia created by the person moving limbs farther from the axis of rotation cause decreases in angular velocity (vice versa) to keep angular momentum constant.

Angular Interpretation of Newton’s 2^nd Law

The change in angular momentum of an object is proportional to the net external torque exerted on it, and this change is in the direction of the net external torque. For a rigid object with constant moments of inertia, we can state this law more simply by substituting torque for force, angular acceleration for acceleration, and moment of inertia for mass in the linear version of this law. Mathematically, for a rigid object with constant moments of inertia, this law is stated as

If the external torques acting on an object do not sum to zero, the object will experience an angular acceleration in the direction of the net torque. Its angular velocity will speed up or slow down, or its axis of rotation will change direction. If an object’s angular velocity or axis of rotation changes, a net external torque must be acting on the object to cause the angular acceleration.

For a nonrigid object with a variable moment of inertia, the equation above does not apply. In this case, the net external torque equals the rate of change of momentum. Mathematically, this can be expressed for average net torque as follows:

A net external torque acting on a nonrigid object with variable moments of inertia will cause a large and quick change or a small and slow change in angular momentum if the net torque is large or small respectively, provided the torques act for equal time intervals.

The change in angular momentum may be seen as

1. A speeding up or slowing down of the object’s angular velocity
2. A change in the direction of the axis of rotation
3. A change in the moment of inertia.

Angular Impulse and Angular Momentum

Angular impulse is the change in angular momentum. The angular analog of the impulse–momentum relationship may be derived from equations below.

In many sport skills, the athlete must cause a change in the angular momentum of the entire body or an implement or individual body part. The angular impulse–momentum relationship shown in equation above indicates how this is accomplished. A larger external torque acting over a longer duration will create a larger change in angular momentum.

Angular Interpretation of Newton’s 3^rd Law

For every torque exerted by one object on another, the other object exerts an equal torque back on the first object but in the opposite direction.

Example: Knee Extension

Muscles produce torques about joints by creating forces on the limbs on either side of the joint. The vastus group of the quadriceps femoris are knee extensor muscles. When these muscles contract, they create a torque on the lower leg that causes it to rotate (or tend to rotate) in one direction, and they create an equal but opposite torque on the thigh that causes it to rotate (or tend to rotate) in the opposite direction. These two opposite rotations produce extension at the knee joint.

SUMMARY

• Angular kinematics is concerned with the description of angular motion. Angles describe the orientation of two lines.
• Absolute angular position refers to the orientation of an object relative to a fixed reference line or plane, such as horizontal or vertical.
• The definitions of angular displacement, angular velocity, and angular acceleration are similar to those for their linear counterparts.
• Tangential linear velocity and acceleration of a point on a rotating object are directly proportional to the radius.
• Centripetal acceleration (also called radial acceleration) of an object rotating in a circular path is the component of linear acceleration directed toward the axis of rotation.
• The basics of angular kinetics, the causes of angular motion, are explained by angular interpretations of Newton’s laws of motion.
• Angular inertia, called moment of inertia, is an object’s resistance to change in its angular motion.
• Angular momentum is the product of moment of inertia and angular velocity.
• The angular interpretation of Newton’s first law says that objects do not change their angular momentum unless a net external torque acts on them.
• The angular interpretation of Newton’s second law explains what happens if a net external torque does act on an object.
• An angular interpretation of Newton’s third law explains that torques act in pairs.