Rigid Bodies and Application to Snooker Assignment

Rigid Bodies and Application to Snooker - Assignment ExampleIn this project I made a study of main laws and principles of laid remains dynamics from the practical and theoretical point of view as solutions to theoretical and practical exercises are provided. It required review and deeper study of vector abridgment and analytical geometry.Rigid ashes in mechanics is a system of material points, which doesnt change in time. So its an idealized system for which the distance in the midst of its particles remains constant in time under any motion. Phenomenological mechanics considers soused body to be a solid matter, in which particles are subjected to essential forces in the form of normal and tangent tensions. Such tensions are caused by external deformations. In case theyre re no deformations, there are no tensions inside rigid body. In many cases deformations are so small that can be neglected. So such model is an idealized rigid body, which is not able to deform and even thoug h internal tensions can take place because of external forces.Rigid body is a mechanical system with six-spot forms of freedom. In order to define the position of a rigid body its enough to know the position of at least 3 points A, B, C, which do not belong to one line. In order to prove that rigid body is described by six degrees of freedom we have to take point D. ... (XA-XB)2+(YA-YB)2+(ZA-ZB)2=AB2=const(XC-XB)2+(YC-YB)2+(ZC-ZB)2=CB2=const(XA-XC)2+(YA-YC)2+(ZA-ZC)2=AC2=constBecause the lengths of sides of triangle alphabet remain the same. So only six coordinates are left independent - rigid body has 6 degrees of freedom. If the body has fixed points the number of freedoms degrees reduces. If rigid body is fixed in one point - it has 3 degrees of freedom, if rigid body can only rotate around one axis is has one degree of freedom, if a body can slide across axis it has two degrees of freedom. In order to understand how x(t) and R(t) change over time we should remind the pursual formulasResultant is v= V+ w, R (using vector properties).Kinetic energy of a rigid body is tally kinetic energy of rotation plus total kinetic energy of motionT=.5 mv2 + 0.5Iw2Where I is moment of inactivity of a rigid body (mass analogue for rotational motion)Moment of inertia is defined as I=miRi2. Moment of inertia is additive so moment of inertia of a rigid body is organise from the sum of inertia moments of its parts. Any body, independently from rotational motion or rest has definite moment of inertia. Mass distribution in the limits of a body can be characterized by density p=m/VSo moment of inertia can be expressed as I=piRi2Vi, if density is constant I=pRi2ViIn limit it can be expressed in the by-line integralI=R2dm=pR2dVThe inertia tensor is a set of nine values (which can be written in the form of 3X3 matrix), which shows the dependence of shape and distribution of mass in the rigid body caused by its rotational motion. Its often explained as a scaling factor between angular momentum and angular velocity.1 Inertia tensor matrix has the following structure and its components are calculated