Mechanics

10 CHAPTER 1. GENERAL PRINCIPLES 1.2.4 Units of Measurement (SI System) Table 1.1 — Systems of Units Name Length Time Mass Force — meter second kilogram newton — m s kg N=  kg · m s 2  Table 1.2 — Preﬁxes — Exponential Form Preﬁx SI Symbol Multiple 1 000 000 000 10 9 giga G 1 000 000 10 6 mega M 1 000 10 3 kilo k Submultiple 0 . 001 10 - 3 milli m 0 . 000 001 10 - 6 micro μ 0 . 000 000 001 10 - 9 nano n Chapter 2 Kinematics of a Particle 2.1 Introduction Suppose we want to analyse a rigid body moving through space. There are six diﬀerent vector values that are necessary to accurately describe the movement of that object. ~s ( t )= x ( t ) ˆ ı + y ( t ) ˆ  + z ( t ) ˆ k + θ |{z} YAW + ψ |{z} PITCH + γ |{z} ROLL The vectors x , y and z describe the location of the particle. By analysing the location of the particle with respect to time, we can determine the velocity of the object. Average speed is a scalar value concerned with distance, where as average velocity is a vector concerned with direction and displacement. Average Speed = Total Distance Total Time (2.1) Average Velocity = Total Displacement Total Time (2.2) We begin with the simplest model of a moving body, a particle. That is, we focus on 11 12 CHAPTER 2. KINEMATICS OF A PARTICLE those objects whose dimensions, size and shape can be ignored for the purposes of the analysis. Consequently, the body’s rotation is neglected and all the mass is assumed to be concentrated at the mass centre. 2.2 Rectilinear Kinematics — Continuous Motion Consider the following assumptions of rectilinear kinematics. • Particle moves along a straight line (rectilinear path) • The kinematics of the particle are characterized by specifying, at any given instant, the particles – Position – Velocity – Acceleration 2.2.1 Position Position is always a vector (magnitude and direction) but here, since the direction is always along the coordinate axis, we need only specify the coordinate s as positive or negative. 2.2.2 Average Speed and Average Velocity Average speed and average velocity are not the same thing — and they are calculated diﬀerently! It’s not just because velocity is a vector and speed is scalar (magnitude of velocity), but also because or the following key distinction. • Average Velocity (Vector) deals with displacement (Vector). • Average Speed (Scalar) deals with distance (Scalar) The average speed is not the magnitude of the average velocity. 2.2. RECTILINEAR KINEMATICS — CONTINUOUS MOTION 13 Example (2.1) Suppose each part of the trip took 2 seconds. So the total time taken to travel 7 . 5m is 10sec. The direction of travel, or the position of the ﬁnal resting place is not important in the calculation of average speed. Using Equation (2.1) Average Speed = Total Distance Travelled Total Time Taken The Average Speed is 7 . 5 10 =0 . 75m/s Of the other hand, the average velocity is calculated using Equation (2.2) as follows. Using Equation (2.1) Average Velocity = Total Displacement frim Origin Total Time Taken Thus the Average Velocity is 5m to the right 10sec =0 . 5m/s to the right Velocity does not depend on how you got to the ﬁnish — only how far you ended up from your original position and the total time taken to get there. Displacement Displacement of the particle ≡ change in the particles position Again, displacement is vector. (diﬀerent from ‘total distance travelled’ which is always a positive scalar)

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