Physics

Classical Angular Momentum

Classical angular momentum refers to the rotational equivalent of linear momentum in classical mechanics. It is a measure of the rotational motion of an object and is defined as the product of the object's moment of inertia and its angular velocity. In classical physics, angular momentum is conserved in the absence of external torques, making it a fundamental quantity in the description of rotational motion.

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  • Instant Notes in Sport and Exercise Biomechanics
    eBook - ePub

    Instant Notes in Sport and Exercise Biomechanics

    Second Edition

    • Paul Grimshaw, Michael Cole, Adrian Burden, Neil Fowler(Authors)
    • 2019(Publication Date)
    • Garland Science
      (Publisher)
    2 /s.
    Angular momentum = moment of inertia × angular velocity (kg.m2 /s)
    The angular momentum of an object about a particular axis will remain constant unless the object is acted on by an unbalanced eccentric force (such as another athlete, a ball or an implement) or a couple (a pair of equal and opposite parallel forces).
    The importance of angular momentum within sport and exercise can be seen by considering how a soccer player learns to kick a ball effectively, how a golfer transfers angular movement of a club to the linear movement of the golf ball or indeed how a sprinter manages to move the limb quickly through the air in order to make the next contact with the ground that is needed to push off and move forward with speed.
    The inertia of an object is referred to as the resistance offered by the stationary object to move linearly and it is directly proportional to its mass. The moment of inertia , however, is defined as the reluctance of an object to begin rotating or to change its state of rotation about an axis. Moment of inertia is related to the mass of the object (body or body part) and the location (distribution) of this mass from the axis of rotation. Without specific reference to a particular axis of rotation the moment of inertia value has little meaning.
    Figure C6.1 shows the moment of inertia values in some selected athletic situations during sport. It is important to reiterate that moment of inertia is specific to the axis of rotation about which the body is moving (e.g. either the centre of gravity (transverse) axis of the body as in diving or the high bar (transverse axis) in gymnastics). The greater the spread of mass from the rotation centre (axis) the greater will be the moment of inertia. In Figure C6.1
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  • Biomechanics of Human Motion
    eBook - ePub

    Biomechanics of Human Motion

    Applications in the Martial Arts, Second Edition

    • Emeric Arus, Ph.D.(Authors)
    • 2017(Publication Date)
    • CRC Press
      (Publisher)
    2
  • Circular cone with the axis through the centers of x, y, and z axes and meets the thin edge point of the cone. The cone longitudinal axis is through the x line/axis. I = 3/10 m ⋅ r 2
  • Solid sphere about any diameter = 2r . I = 2/5 m ⋅ r 2 .
    • Figure 9.4 demonstrates the parallel axis theorem. Looking at Figure 9.4 , the black round spots represent CoM or CoG. Two other small circles (they are not black) also represent the moment of inertia of body parts about transverse axes through their CoM. The 0.28, 0.40, and 0.50 m represent the distances between the CoM of different body segments.
    Figure 9.3 shows in percentage the location of the mass centers of body segments.

    9.7 ANGULAR MOMENTUM

    Recall that linear momentum is a vector quantity that has a magnitude and direction. A body or an object that has a motion has a momentum that measures the velocity and the quantity of that mass or body. To put into simpler words, the momentum is a measure of the force needed to start or stop a motion. Therefore, linear momentum (p ) = m ⋅ v (N ⋅ s) or kg ⋅ m/s.
    In a similar way, the angular or rotary momentum H or (L ) = moment of inertia times angular velocity. L = I ⋅ ω or kg ⋅ m2 /s. The momentum of a diver who dives in a straight line is the equation p = m ⋅ v ; when the diver starts to rotate he has the equation L = I ⋅ ω.
    In contact sports such as karate, boxing, and so on, the linear momentum is closely related to the linear impulse (see Section 8.5 ); in the similar way in rotary motion, the rotary impulse (which will be described later) is also related to angular momentum. When a body rotates normally, it will not stop until a force intervenes in its rotation route?
    Let us take an example of a discus thrown. The discus will spin in the direction liberated by the thrower. The discus will rotate around its center (CoG) and also advances forward. The angular momentum L = I ⋅ ω (kg ⋅ m2 /s). For any rotating object or body that is free from net torques, such as throwing a discus where the body rotates about a central axis, we can write the angular momentum L = I o ωo = constant, where I o and ωo
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  • Magnetic Resonance Imaging
    eBook - ePub

    Magnetic Resonance Imaging

    Physical Principles and Sequence Design

    • Robert W. Brown, Y.-C. Norman Cheng, E. Mark Haacke, Michael R. Thompson, Ramesh Venkatesan(Authors)
    • 2014(Publication Date)
    • Wiley-Blackwell
      (Publisher)
    Sec. 2.3 .

    2.2.1 Torque and Angular Momentum

    Nonzero total torque on a system implies that the system’s total angular momentum must change according to
    (2.14 )
    This equation, discussed in most introductory mechanics textbooks, can be derived, as a problem, for a single point mass.
    Problem 2.3
    Consider a point mass m moving at velocity (t) with position (t) defined by some origin. Its angular momentum relative to that origin is therefore = with . Derive (2.14 ) by taking the time derivative of this angular momentum. Note (2.2 ) and (2.3 ).
    The generality of (2.14 ) follows by considering a system as a limit of many point particles. The total angular momentum is the corresponding limit of
    (2.15)
    with respect to some origin.

    2.2.2 Angular Momentum of the Proton

    We next formulate the connection between the proton intrinsic angular momentum (or what is often referred to as its ‘spin’) and its moment. The connections for other nuclear particles are also of interest.
    The proton spin can be thought of as leading to a circulating electric current, and, hence, an associated magnetic moment. The direct relationship between the magnetic moment and the spin angular momentum vector is found from experiment,
    (2.16 )
    The proportionality constant in (2.16 ) is called the gyromagnetic (or magnetogyric) ratio and depends on the particle or nucleus. For the proton, it is found to be6
    (2.17 )
    or, what may be referred to as ‘gamma-bar,’
    (2.18)
    where T is the tesla unit of magnetic field and is equal to 10,000 gauss (G). Of all the numbers in MR, is probably the one most often used in back-of-the-envelope calculations. From (2.16
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