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 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

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

Non-Inertial Reference Frame Gravitational attraction The Law of Universal Attraction was already introduced in lecture D1. The law postulates that the force of attraction between any two particles, of masses M and m, respectively, has a magnitude, F, given by F= (1) where r is the distance between the two particles, and G is the universal constant of gravitation. The value of G is empirically determined to be

We will start by studying the motion of a particle. We think of particle as a body which has mass, but has negligible dimensions. Treating bodies as particles is, of course, an idealization which involves an approximation. This approximation may be perfectly acceptable in some situations and not adequate in some other cases. For instance, if we want to study the motion of planets it is common to consider each planet as a particle

is a vector equation that relates the magnitude and direction of the force vector, to the magnitude and direction of the acceleration vector. In the previous lecture we derived expressions for the acceleration vector expressed in cartesian coordinates. This expressions can now be used in Newton's second law, to produce the equations of motion expressed in cartesian coordinates

In this lecture we will look at some other common systems of coordinates. We will present polar coordinates in two dimensions and cylindrical and spherical coordinates in three dimensions. We shall see that these systems are particularly useful for certain classes of problems Like in the case of intrinsic coordinates presented in the previous lecture, the reference frame changes from point to point. However, for the coordinate systems to be presented below, the reference frame depends only on the position of the particle. This is in contrast with the intrinsic coordinates, where the reference frame is a function of the position, as well as the path

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

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 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