Angular Momentum of One Particle
Angular momentum of one particle is a measure of its rotational motion. It is a vector quantity defined as the cross product of the particle's position vector and its linear momentum. The magnitude of the angular momentum is proportional to the particle's mass, velocity, and the distance from the axis of rotation.
4 Key excerpts on "Angular Momentum of One Particle"
eBook - ePubIn Search of Divine Reality
Science as a Source of Inspiration
- Lothar Schäfer(Author)
- 1997(Publication Date)
- University of Arkansas Press(Publisher)
...More precisely, angular momentum is quantized. In addition, common elementary particles, like electrons, protons, or neutrons, were unexpectedly found to carry an intrinsic spin angular momentum, as though they were spheres spinning about an axis like a top. This intrinsic momentum is also quantized, being allowed a single, fixed value for each type of particle. Furthermore, when its direction is probed with respect to an axis in the laboratory, only a limited number of orientations with respect to that axis is found to exist. We say that orientation is quantized and often refer to it as space quantization. For a classical particle, the amount of spin can be measured in terms of speed, but for the point-like quantum particles speed of spin has no meaning and magnitude of spin can only be measured in terms of angular momentum. If we use the magnitude of the spin of photons as a unit, assigning it the value of 1, then most particles either have 0, ½, or 1 unit of angular momentum. Electrons and protons are spin- ½ particles. Invariance of spin means that the amount cannot be changed; that is, it cannot be accelerated nor stopped. Spin is an intrinsic property that elementary particles own, regardless of observation. SPACE QUANTIZATION Space quantization is an amazing phenomenon. We can take an electron as a specific example. Each electron has intrinsic angular momentum with fixed magnitude (√ 3)(h/4 π), where h is Planck’s constant. In addition, when the components of this spin are measured along a direction in space, only one of two possible values can be found, either +h/4 π or -h/4 π, showing that only two orientations are allowed for this vector property with respect to a given axis, one up and one down, respectively...
eBook - ePub- Ronald J. Anderson(Author)
- 2020(Publication Date)
- Wiley(Publisher)
...This is because angular momentum is defined as the moment of linear momentum. Equation 2.35 defines angular momentum about the reference point due to the mass particle as, (2.46) where the linear momentum is defined to be, resulting in, (2.47) We assume that the absolute velocity of the reference point,, and the angular velocity of the rigid body,, are known. We can then write an expression for the velocity of particle (noting that the position vector has no rate of change of magnitude because both and are in the same rigid body and cannot move apart) as, (2.48) Referring to Figure 2.3, we can write expressions for the position of with respect to and the absolute velocity of point in the reference frame with unit vectors. Let the distance in the direction from point to point be. Similarly, we define and to be the distances in the and directions respectively. The position vector is then, Let the absolute velocity of have scalar components,, and. The velocity of is then, Further, let the angular velocity of the coordinate system be, The cross product can then be written as, and the velocity of particle is, (2.49) Equation 2.49 can be substituted into Equation 2.47 to give, after performing another cross product and gathering some terms, an expression for the angular momentum about due to particle as follows: (2.50) To get the total angular momentum vector about point, we write Equation 2.50 for every particle and add the resulting equations together to get, (2.51) or, (2.52) where the terms used in Equation 2.52 are defined as follows.. This is the total mass of the rigid body.. This summation is one component of that used to locate the center of mass of the body with respect to the reference point. is the ‐component of.. is the ‐component of.. is the ‐component of.. This term is a function of the spatial distribution of mass particles around the ‐axis (i.e. direction) passing through point. The term is always positive because of the sum of squares term...
- Paul Grimshaw, Neil Fowler, Adrian Lees, Adrian Burden(Authors)
- 2007(Publication Date)
- Routledge(Publisher)
...Angular acceleration is defined as the rate of change of angular velocity and is calculated by angular velocity (final – initial) divided by the time taken. Clockwise and anti-clockwise rotation Clockwise rotation is movement in the same direction as the hands of a clock (i.e., clockwise) when you look at it from the front. Clockwise rotation is given a negative symbol (−ve) for representation. Anti-clockwise rotation is the opposite movement to clockwise rotation and it is given a positive symbol (+ve) for representation. Absolute and relative angles An absolute angle is the angle measured from the right horizontal (a fixed line) to the distal aspect of the segment or body of interest. A relative joint angle is the included angle between two lines that often represent segments of the body (i.e., the relative knee joint angle between the upper leg (thigh) and the lower leg (shank)). In a relative angle both elements (lines) that make up the angle can be moving. Included angle and vertex An included angle is the angle that is contained between two lines that meet or cross (intersect) at a point. Often these lines are used to represent segments of the human body. The vertex is the intersection point of two lines. In human movement the vertex is used to represent the joint of interest in the human body (i.e., the knee joint) Angular motion Angular motion is rotatory movement about an imaginary or real axis of rotation and where all parts on a body (and the term body need not necessarily be a human body) or segment move through the same angle. Angular kinematics describes quantities of angular motion using such terms as angular displacement, angular velocity and angular acceleration. Fig. A3.1 identifies two examples of angular motion in more detail. Angular distance or displacement (scalar or vector quantity) is usually expressed in the units of degrees (where a complete circle is 360 degrees)...
eBook - ePubBiomechanics of Human Motion
Applications in the Martial Arts, Second Edition
- Emeric Arus, Ph.D.(Authors)
- 2017(Publication Date)
- CRC Press(Publisher)
...More precisely, the angular speed tells us the change in angle per unit of time, which is measured in radians/s. Angular speed can also be measured in degrees, for example, 360 o /s. Even if the term angular speed is equivalent to rotational velocity, there is a difference, which is the rotation (revolution) per minute, for example, 60 rpm. Angular speed represents the magnitude of angular velocity. The angular velocity, whose sign is also a Greek letter omega (ω), is a vector quantity; that is why in most physics books or other books, the authors use the terms angular velocity and angular speed interchangeably. The magnitude of angular velocity can be described in two terms. One is the average angular velocity, which represents the angular displacement of an object between two points of an angle such as θ 1 and θ 2 at the time intervals t 1 and t 2, respectively. Then, the equation for average angular velocity is = (θ 2 -θ 1 / t 2 - t 1)= (∆θ/∆ t); the other term for the magnitude is the instantaneous angular velocity, which is the limit of the magnitude ratio as Δ t approaches 0. The formula is: ω = lim as Δ t → 0 Δθ/Δ t = dθ/d t, both and ω being measured in m/s. To find out the average angular acceleration and the instantaneous angular acceleration, we proceed in an analogous fashion to linear velocities and accelerations. The average angular acceleration () of a rotating body in the interval from t 1 to t 2 can be defined using the following formula, where ω represents the instantaneous angular velocities. The formula for average angular acceleration =ω 2 -ω 1 / t 2 - t 1 = ∆ω/∆ t,and the instantaneous angular acceleration α = lim as Δ t → 0 Δω/Δ t = dω/d t, both and α being measured in rad/s 2. Speaking about angular motion, we should mention two different and important accelerations. One of these is the so-called radial component of the linear acceleration that is named radial acceleration, also called centripetal acceleration...
