In this lecture, we will consider how to transfer from one orbit, or trajectory, to another. One of the assumptions that we shall make is that the velocity changes of the spacecraft, due to the propulsive effects, occur instantaneously. Although it obviously takes some time for the spacecraft to accelerate to the velocity of the new orbit, this assumption is reasonable when the burn time of the rocket is much smaller than the period of the orbit. In such cases, the Av required to do the maneuver is simply the difference between the

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

In this lecture, we consider the motion of a 3D rigid body. We shall see that in the general three dimensional case, the angular velocity of the body can change in magnitude as well as in direction, and, as a consequence, the motion is considerably more complicated than that in two dimensions. Rotation About a Fixed Point We consider first the simplified situation in which the 3D body moves in such a way that there is always a point, O, which is fixed. It is clear that, in this case, the path of any point in the rigid body which is at a

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 application of Newton's second law to a system of particles and derive some useful relationships expressing the conservation of angular momentum. Center of Mass Consider a system made up of n particles. A typical particle, i, has mass mi, and, at the instant considered, occupies the position Ti relative to a frame xyz. We can then define the center of mass, G, as the point

In this lecture, we will start from the general relative motion concepts introduced in lectures D11 and D12. and then apply them to describe the motion of 2D rigid bodies. We will think of a rigid body as a system of particles in which the distance between any two particles stays constant. The term 2-dimensional implies that particles move in parallel planes. This includes, for instance, a planar body moving within its plane

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 the previous lecture, we related the motion experienced by two observers in relative translational motion with respect to each other. In this lecture we will extend this relation to our third type of observer.That is, observers who accelerate and rotate with respect to each other. As a matter of illustration, let us consider a very simple situation, in which a particle at rest with respect

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 this lecture we will consider the equations that result from integrating Newtons second law, F=ma, in time. This will lead to the principle of linear impulse and momentum. This principle is very useful when solving problems in which we are interested in determining the global effect of a force acting on a particle over a time interval Linear momentum We consider the curvilinear motion of a particle of mass, m, under the influence of a force F. Assuming that