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 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 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

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

Lecture D32: Damped Free Vibration Spring-Dashpot-Mass System k Spring Force Fs =-kx, k>0 Dashpot Fd =-cx, c>0 Newton's Second Law (mx =EF) mx +cx+kx (Define)Natural Frequency wn=k/m,and

Lecture D33: Forced Vibration Fosinwt m Spring Force Fs =-kx, k>0 Dashpot Fd =-ci, c>0 Forcing Fext Fo sin wt Newton's Second Law (mix =CF) mx+cx+kx= Fo sin wt =k/m,=c/(2mwn)

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

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

When the only force acting on a particle is always directed to- wards a fixed point, the motion is called central force motion. This type of motion is particularly relevant when studying the orbital movement of planets and satellites. The laws which gov- ern this motion were first postulated by Kepler and deduced from observation. In this lecture, we will see that these laws are a con- sequence of Newton's second law. An understanding of central

In this lecture, we consider the problem of a body in which the mass of the body changes during the motion, that is, m is a function of t, i.e. m(t). Although there are many cases for which this particular model is applicable, one of obvious importance to us are rockets. We shall see that a significant fraction of the mass of a rocket is the fuel, which is expelled during flight at a high velocity and thus, provides the propulsive force for the rocket