Angular Momentum
Angular momentum is a measure of the rotational motion of an object. It is a vector quantity defined as the cross product of the position vector and the linear momentum of an object. In simpler terms, it describes how an object is spinning or rotating and is conserved in the absence of external torques.
7 Key excerpts on "Angular Momentum"
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 - ePub- W. Bolton(Author)
- 2015(Publication Date)
- Routledge(Publisher)
...Chapter 5 Angular motion 5.1 Introduction This chapter is concerned with describing angular motion, deriving and using the equations for such motion and relating linear motion of points on the circumference of rotating objects with their angular motion. The term torque is introduced. 5.1.1 Basic terms The following are basic terms used to describe angular motion. Angular displacement The angular displacement is the angle swept out by the rotation and is measured in radians. Thus, in Figure 5.1, the radial line rotates through an angular displacement of θ in moving from OA to OB. One complete rotation through 360° is an angular displacement of 2 π rad; one quarter of a revolution is 90° or π /2 rad. As 2 π rad 5 360°, then 1 rad 5 360°/2 π or about 57°. Figure 5.1 Angular motion 2 Angular velocity Angular velocity ω is the rate at which angular displacement occurs, the unit being rad/s. 3 Average angular velocity The average angular velocity over some time interval is the change in angular displacement during that time divided by the time...
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...
eBook - ePub- John Bird(Author)
- 2019(Publication Date)
- Routledge(Publisher)
...Chapter 23 Linear and angular motion Why it is important to understand: Linear and angular motion This chapter commences by defining linear and angular velocity and also linear and angular acceleration. It then derives the well-known relationships, under uniform acceleration, for displacement, velocity and acceleration, in terms of time and other parameters. The chapter then uses elementary vector analysis, similar to that used for forces in chapter 20, to determine relative velocities. This chapter deals with the basics of kinematics. A study of linear and angular motion is important for the design of moving vehicles. At the end of this chapter, you should be able to: appreciate that 2π radians corresponds to 360° define linear and angular velocity perform calculations on linear and angular velocity using v = ωτ and ω = 2πn define linear and angular acceleration perform calculations on linear and angular acceleration using v 2 = v 1 + at, ω 2 = ω 1 + at and a = τα select appropriate equations of motion when performing simple calculations appreciate the difference between scalar and vector quantities use vectors to determine relative velocities, by drawing and by calculation 23.1 Introduction This chapter commences by defining linear and angular velocity and also linear and angular acceleration. It then derives the well-known relationships, under uniform acceleration, for displacement, velocity and acceleration, in terms of time and other parameters. The chapter then uses elementary vector analysis, similar to that used for forces in chapter 20, to determine relative velocities. This chapter deals with the basics of kinematics. 23.2 The radian The unit of angular displacement is the radian, where one radian is the angle subtended at the centre of a circle by an arc equal in length to the radius, as shown in Figure 23.1. The relationship between angle in radians θ, arc length s and radius of a circle τ is: s = r θ (1) Science and Mathematics for Engineering...
eBook - ePub- William Bolton(Author)
- 2012(Publication Date)
- Routledge(Publisher)
...4 Linear and angular motion 4.1 Introduction This chapter is concerned with the behaviour of dynamic mechanical systems when there is uniform acceleration. The terms and basic equations associated with linear motion with uniform acceleration and angular motion with uniform angular acceleration, Newton’s laws of motion, moment of inertia and the effects of friction are revised and applied to the solution of mechanical system problems. The terms scalar quantity and vector quantity are used in this chapter, so as a point of revision: Scalar quantities are those that only need to have their size to be given in order for their effects to be determined, e.g. mass. Vector quantities are those that need to have both their size and direction to be given in order for their effects to be determined, e.g. force where we need to know the direction as well as the size to determine its effect. 4.2 Linear motion The following are basic terms used in the description of linear motion, i.e. motion that occurs in a straight line path rather than rotation which we will consider later in this chapter: 1 Distance and displacement The term distance tends to be used for distances measured along the path of an object, whatever form the path takes; the term displacement, however, tends to be used for the distance travelled in a particular straight line direction (Figure 4.1). For example, if an object moves in a circular path the distance travelled is the circumference of the path whereas the displacement might be zero if it ends up at the same point it started from...
eBook - ePub- A. D. Johnson(Author)
- 2017(Publication Date)
- CRC Press(Publisher)
...Angular motion equivalent parameters need to be inserted as follows: Linear motion Angular. motion v = s t (2.1) ω = θ t (2.6) v 2 = v 1 + a t (2.2) ω 2 = ω 1 + α t (2.10) s = (v 1 + v 2) t 2 (2.3) θ = (ω 1 + ω 2) t 2 (2.11) s = v 1 t + 1 2 a t 2 (2.4) θ = ω 1 t + 1 2 α t 2 (2.12) v 2 2 =[--=PLGO-SEPARATOR=. --]v 1 2 + 2 a s (2.5) ω 2 2 = ω 1 2 + 2 α θ (2.13) Example 2.15 A wheel, initially at rest, is subjected to a constant angular acceleration of 2.5 rad/s 2 for 60s. Find: the angular velocity attained; the number of revolutions made in that time. Solution The angular velocity can be found using equation (2.10), but α = 2.5rad/s 2, f = 60s, so ω 2 = ω 1 + α t = 0 + (2.5 × 60) = 150 rad/s The angular displacement can be found using equation (2.12) : θ = ω 1 t + 1 2 α t 2 so that θ = 2.5 × 60 2 2 = 4500 rad To convert the angular displacement to revolutions it is recognized that one revolution represents 2π rad: n = 4500 2 π = 716 rev Example 2.16 A wheel initially has an angular velocity of 50 rad/s. When brakes are applied the wheel comes to rest in 25 s. Find the average retardation. Solution The angular retardation can be found using equation (2.10) : ω 2 = ω 1 + α t but ω 1 = 50 rad/s, ω 2 = 0, t = 25 s, so that 0 = 50 + (α × 25) giving α = − 50 25 = − 2rad/s 2 Example 2.17 A drum starts from rest and attains a rotation of 210 rev/min in 6.2 s with uniform acceleration. A brake is then applied which brings the drum to rest in a further 5.5 s with uniform retardation. Find the total number of revolutions made by the drum. Solution The total number of revolutions can be found by using equation (2.11) : θ = (ω 1 + ω 2) t 2 Fig. 2.17 Consider, initially, area A under the velocity–time graph (Figure 2.17)...
