CH1_single particle

# CH1_single particle - CHAPTER 1 NEWTONIAN MECHANICS FOR...

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Unformatted text preview: CHAPTER 1 NEWTONIAN MECHANICS FOR SINGLE PARTICLE 1.1 Frame of Reference Before we can discuss the physical laws, we have to talk about the concepts of the reference frame. Indeed, before we make any description of the motion of a single particle, firstly we have to specify the reference frame because the motion of a particle is relative. Or, simply speaking we asked the question where we are as we make the description? In order to illustrate that the motion of a particle is relative, we try to describe a particle moving in the three dimensional space from two different reference frame, in which one is moving relative to the other. Consider that we are sitting in the reference frame S and there is any reference frame S' which is moving with respect to S. The position vector of the origin of S' with respect to S is . The position vectors of the particle P observed from the reference frames S and S' are denoted by and respectively. From the figure, it is easy to say that: (1.1) If the frames & the observed object P have relative motions, we can differentiate the equation to obtain the relation of the various velocity vectors, (1.2) or P x y 0 z Frame S Frame S’ x’ y’ z’ 0’ Differentiate once more with respect to time can yield the relation between the acceleration vectors: (1.3) Or conclusively, we have the general transformation formulae for the position vectors, the velocity vectors and the acceleration vectors observed from different reference frames: (1.4) For a very important special case, if the relative velocities between the two reference frames are constant, i.e. v S'S = const., then we have: (1.5) These frames are called inertial reference frames , in which the accelerations are the same in all frames and the transform is known as the Galilean transform , which is to be used in the transformation of position vectors, velocity vectors and acceleration vectors between inertial frames. 1.2 Newton’s Laws of Motion for single particle 1.2.1 Newton’s 1st Law: the law of inertia Newton's 1st law states that: A free particle always moves with constant velocity without acceleration. Or we can say: If the velocity of a particle is changed, there must exist an “ interaction” from another body. We usually call this interaction a ‘ force’ Newton's first law is indeed obtained from observations or a law concluded from previous experience. Inertia Mass Consider that we are performing a collision experiment with two particles (not necessarily identical particles) on a two dimensional plane, for example a smooth table. If the initial velocity vectors of the two particles were and respectively and after the collision, their velocity were found to be and respectively. After performing numerous trials with different initial velocities and final velocity being measured, it was found that: (1) always in opposite direction of (2) constant We can repeat the experiment by changing different particles and we found that different particles have different degree of resistance to change its magnitude of the velocity after...
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CH1_single particle - CHAPTER 1 NEWTONIAN MECHANICS FOR...

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