D244BD RIGID BODY DYNAMICS KINETIC EWEGY In echure we derwed am kinenc a susem u dm T= Fere ts the velouty relanve to G. for a nald body we ca wate Uing the vechor nidontklyAxB=Ax

We have seen that the work done by a force F on a particle is given by dw =. dr. If the work done by F, when the particle moves from any position TI to any position T2, can be expressed as, W12=fdr=-(V(r2)-V(1)=V-v2, (1) then we say that the force is conservative. In the above expression, the scalar

In this lecture, we will derive expressions for the angular momentum and kinetic energy of a 3D rigid body. We shall see that this introduces the concept of the Inertia Tensor. Angular Momentum We start form the expression of the angular momentum of a system of particles about the center of mass

So far we have used Newton's second law= ma to establish the instantaneous relation between the sum of the forces acting on a particle and the acceleration of that particle. Once the acceleration is known,the velocity (or position) is obtained by integrating the expression of the acceleration (or velocity). There are two situations in which the cumulative effects of unbalanced forces acting on a particle are of interest to us. These involve:

In lecture D2 we introduced the position velocity and acceleration vectors and referred them to a fixed cartesian coordinate system. While it is clear that the choice of coordinate system does not affect the final answer, we shall see that, in practical problems, the choice of a specific system may simplify the calculations considerably. In previous lectures, all the vectors at all points in the trajectory were expressed in the

Inertial reference frames In the previous lecture, we derived an expression that related the accelerations observed using two reference frames, A and B, which are in relative motion with respect to each other. aA =aB+(aA/ B)'y'' 22 x (DA/ B) 'y'2'+ TA/B+ X TA/B). (1) Here, aA is the acceleration of particle A observed by one observer, and

In this lecture, we will particularize the conservation principles presented in the previous lecture to the case in which the system of particles considered is a 2D rigid body. Mass Moment of Inertia In the previous lecture, we established that the angular momentum of a system of particles relative to the center of mass, G, was

In this lecture, we will revisit the principle of work and energy introduced in lecture D7 for particle dynamics, and extend it to 2D rigid body dynamics. Kinetic Energy for a 2D Rigid Body We start by recalling the kinetic energy expression for a system of particles derived in lecture D17

In this lecture we will look at some applications of Newton's second law, expressed in the different coordinate systems that were introduced in lectures D3-D5. Recall that Newton's second law F=ma, (1) is a vector equation which is valid for inertial observers. In general, we will be interested in determining the motion of a particle given

In lecture D9, we saw the principle of impulse and momentum applied to particle motion. This principle was of particular importance when the applied forces were functions of time and when interactions between particles occurred over very short times, such as with impact forces. In this lecture, we extend these principles to two dimensional rigid body dynamics. Impulse and Momentum Equations Linear Momentum In lecture D18, we introduced the equations of motion for a two dimensional rigid body. The linear momen- tum for a system of particles is defined