# Dynamics Of Particles And Of Rigid Bodies In Plane Motion Pdf

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- Dynamics of Rigid Bodies
- Engineering Systems in Motion: Dynamics of Particles and Bodies in 2D Motion
- Engineering Systems in Motion: Dynamics of Particles and Bodies in 2D Motion

This course is an introduction to the study of bodies in motion as applied to engineering systems and structures. We will study the dynamics of particle motion and bodies in rigid planar 2D motion. This will consist of both the kinematics and kinetics of motion. Kinematics deals with the geometrical aspects of motion describing position, velocity, and acceleration, all as a function of time.

## Dynamics of Rigid Bodies

In physics , a rigid body also known as a rigid object [2] is a solid body in which deformation is zero or so small it can be neglected. The distance between any two given points on a rigid body remains constant in time regardless of external forces or moments exerted on it.

A rigid body is usually considered as a continuous distribution of mass. In the study of special relativity , a perfectly rigid body does not exist; and objects can only be assumed to be rigid if they are not moving near the speed of light. In quantum mechanics , a rigid body is usually thought of as a collection of point masses. For instance, molecules consisting of the point masses: electrons and nuclei are often seen as rigid bodies see classification of molecules as rigid rotors.

The position of a rigid body is the position of all the particles of which it is composed. To simplify the description of this position, we exploit the property that the body is rigid, namely that all its particles maintain the same distance relative to each other. If the body is rigid, it is sufficient to describe the position of at least three non- collinear particles. This makes it possible to reconstruct the position of all the other particles, provided that their time-invariant position relative to the three selected particles is known.

However, typically a different, mathematically more convenient, but equivalent approach is used. The position of the whole body is represented by:.

Thus, the position of a rigid body has two components: linear and angular , respectively. The linear position can be represented by a vector with its tail at an arbitrary reference point in space the origin of a chosen coordinate system and its tip at an arbitrary point of interest on the rigid body, typically coinciding with its center of mass or centroid.

This reference point may define the origin of a coordinate system fixed to the body. There are several ways to numerically describe the orientation of a rigid body, including a set of three Euler angles , a quaternion , or a direction cosine matrix also referred to as a rotation matrix.

All these methods actually define the orientation of a basis set or coordinate system which has a fixed orientation relative to the body i. In general, when a rigid body moves, both its position and orientation vary with time. In the kinematic sense, these changes are referred to as translation and rotation , respectively. Indeed, the position of a rigid body can be viewed as a hypothetic translation and rotation roto-translation of the body starting from a hypothetic reference position not necessarily coinciding with a position actually taken by the body during its motion.

Velocity also called linear velocity and angular velocity are measured with respect to a frame of reference. The linear velocity of a rigid body is a vector quantity, equal to the time rate of change of its linear position.

Thus, it is the velocity of a reference point fixed to the body. During purely translational motion motion with no rotation , all points on a rigid body move with the same velocity. However, when motion involves rotation, the instantaneous velocity of any two points on the body will generally not be the same. Two points of a rotating body will have the same instantaneous velocity only if they happen to lie on an axis parallel to the instantaneous axis of rotation.

Angular velocity is a vector quantity that describes the angular speed at which the orientation of the rigid body is changing and the instantaneous axis about which it is rotating the existence of this instantaneous axis is guaranteed by the Euler's rotation theorem. All points on a rigid body experience the same angular velocity at all times. During purely rotational motion, all points on the body change position except for those lying on the instantaneous axis of rotation.

The relationship between orientation and angular velocity is not directly analogous to the relationship between position and velocity. Angular velocity is not the time rate of change of orientation, because there is no such concept as an orientation vector that can be differentiated to obtain the angular velocity.

The angular velocity of a rigid body B in a reference frame N is equal to the sum of the angular velocity of a rigid body D in N and the angular velocity of B with respect to D: [5].

In this case, rigid bodies and reference frames are indistinguishable and completely interchangeable. The velocity of point P in reference frame N is defined as the time derivative in N of the position vector from O to P: [6]. The result is independent of the selection of O so long as O is fixed in N. The acceleration of point P in reference frame N is defined as the time derivative in N of its velocity: [6]. By differentiating the equation for the Velocity of two points fixed on a rigid body in N with respect to time, the acceleration in reference frame N of a point Q fixed on a rigid body B can be expressed as.

The acceleration in reference frame N of the point R moving in body B while B is moving in frame N is given by. If C is the origin of a local coordinate system L , attached to the body,. In 2D, the angular velocity is a scalar, and matrix A t simply represents a rotation in the xy -plane by an angle which is the integral of the angular velocity over time.

Vehicles , walking people, etc. This integer is the winding number with respect to the origin of the velocity. Compare the amount of rotation associated with the vertices of a polygon. Any point that is rigidly connected to the body can be used as reference point origin of coordinate system L to describe the linear motion of the body the linear position, velocity and acceleration vectors depend on the choice.

Two rigid bodies are said to be different not copies if there is no proper rotation from one to the other. A rigid body is called chiral if its mirror image is different in that sense, i. In the opposite case an object is called achiral: the mirror image is a copy, not a different object. Such an object may have a symmetry plane, but not necessarily: there may also be a plane of reflection with respect to which the image of the object is a rotated version.

For a rigid rectangular transparent sheet, inversion symmetry corresponds to having on one side an image without rotational symmetry and on the other side an image such that what shines through is the image at the top side, upside down.

We can distinguish two cases:. A sheet with a through and through image is achiral. We can distinguish again two cases:.

The configuration space of a rigid body with one point fixed i. From Wikipedia, the free encyclopedia. Physical object which does not deform when forces or moments are exerted on it. Second law of motion. History Timeline Textbooks. Newton's laws of motion. Analytical mechanics Lagrangian mechanics Hamiltonian mechanics Routhian mechanics Hamilton—Jacobi equation Appell's equation of motion Koopman—von Neumann mechanics. Core topics. Circular motion Rotating reference frame Centripetal force Centrifugal force reactive Coriolis force Pendulum Tangential speed Rotational speed.

Main article: Rigid body dynamics. Introduction to Statics and Dynamics. Oxford University Press. In mathematics , however, linear has a different meaning. In both contexts, the word "linear" is related to the word "line". In mathematics, a line is often defined as a straight curve.

For those who adopt this definition, a curve can be straight, and curved lines are not supposed to exist. In kinematics , the term line is used as a synonym of the term trajectory , or path namely, it has the same non-restricted meaning as that given, in mathematics, to the word curve.

In short, both straight and curved lines are supposed to exist. In topology and meteorology , the term "line" has the same meaning; namely, a contour line is a curve. Dynamics Online. Rigid body at Wikipedia's sister projects. Authority control GND : Categories : Rigid bodies Rigid bodies mechanics Rotational symmetry. Hidden categories: Articles with short description Short description is different from Wikidata Pages using Sister project links with hidden wikidata Commons category link is on Wikidata Wikipedia articles with GND identifiers.

Namespaces Article Talk. Views Read Edit View history. Help Learn to edit Community portal Recent changes Upload file. Download as PDF Printable version. Formulations Newton's laws of motion Analytical mechanics Lagrangian mechanics Hamiltonian mechanics Routhian mechanics Hamilton—Jacobi equation Appell's equation of motion Koopman—von Neumann mechanics.

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## Engineering Systems in Motion: Dynamics of Particles and Bodies in 2D Motion

The study of the motion of a rigid body on a plane RBP motion is usually one of the most challenging topics that students face in introductory physics courses. In this paper, we discuss a couple of problems which are typically used in basic physics courses, in order to highlight some aspects related to RBP motion which are not usually well understood by physics students. The first problem is a pendulum composed of a rod and disk. The angular frequency of the pendulum is calculated in two situations: disk fixed to the rod and disk free to spin. A detailed explanation of the change in the angular frequency from one case to another is given. The second problem is a ladder which slides touching a frictionless surface.

A force of N is applied to the centre of a circular disc, of mass 10 kg and radius 1 m, resting on a floor as shown in the figure. A partical of unit mass is moving on a plane. The following figure shows the velocity-time plot for a particle traveling along a staright line. Two disks A and B with identical mass m and radius R are initialy at rest. They roll down from the top of identical inclined planes without slipping. Disk A has all of its mass concentrated at the rim, while Disk B has its mass uniformly distributed. A point mass M is released from rest and slides down a spherical bowl of radius R from a height H as shown in the figure below.

Kinematic equations relate the variables of motion to one another. In Part 3, vectors are used to solve the problem. Kinematics studies how the position of an object changes with time. This is the community to discuss both parts of Engineering Mechanics: Statics and Dynamics. Therefore, it is necessary to solve many problems independently.

Dynamics & Vibrations, NAV. 3. Introduction. 5. Plane Kinematics. Particle. Rigid Body. Size. Small & Not importantBig & Important. Motion.

## Engineering Systems in Motion: Dynamics of Particles and Bodies in 2D Motion

Plane Motion: When all parts of the body move in a parallel planes then a rigid body said to perform plane motion. Straight Line Motion: It defines the three equations with the relationship between velocity, acceleration, time and distance travelled by the body. In straight line motion, acceleration is constant. Distance travelled in n th second:.

Затем она, наверное, вмонтирует алгоритм в защищенный чип, и через пять лет все компьютеры будут выпускаться с предустановленным чипом Цифровой крепости. Никакой коммерческий производитель и мечтать не мог о создании шифровального чипа, потому что нормальные алгоритмы такого рода со временем устаревают. Но Цифровая крепость никогда не устареет: благодаря функции меняющегося открытого текста она выдержит людскую атаку и не выдаст ключа.

Да. Я решила ее издать. - Издать? - Он с сомнением покачал головой.

Пожилой человек вдруг поднялся и куда-то побежал, видимо, вызвать скорую. Танкадо явно терял последние силы, но по-прежнему совал кольцо прямо в лицо тучному господину. Тот протянул руку, взял Танкадо за запястье, поддерживая остававшуюся на весу руку умирающего. Танкадо посмотрел вверх, на свои пальцы, на кольцо, а затем, умоляюще, - на тучного господина.

Кадр казался неестественно вытянутым по вертикали и неустойчивым, как бывает при дрожащем объективе, - это было результатом удаления кадров, процесса, сокращающего видеозапись вдвое и экономящего время. Объектив, скользнув по огромной площади, показал полукруглый вход в севильский парк Аюнтамьенто. На переднем плане возникли деревья.

Танкадо, задыхаясь и не в силах произнести ни звука, в последней отчаянной надежде посмотрел на тучного господина. Пожилой человек вдруг поднялся и куда-то побежал, видимо, вызвать скорую. Танкадо явно терял последние силы, но по-прежнему совал кольцо прямо в лицо тучному господину. Тот протянул руку, взял Танкадо за запястье, поддерживая остававшуюся на весу руку умирающего. Танкадо посмотрел вверх, на свои пальцы, на кольцо, а затем, умоляюще, - на тучного господина.