Course: PGPathshala-Biophysics Paper 1:Foundations of Bio-Physics Module 22: Kinematics, kinetics, dynamics, inertia Kinematics is a branch of classical mechanics in physics in which one learns about the properties of motion such as position, velocity, acceleration, etc. of bodies or particles without considering the causes or the driving forces behindthem. Since in kinematics, we are interested in study of pure motion only, we generally ignore the dimensions, shapes or sizes of objects under study and hence treat them like point objects. On the other hand, kinetics deals with physics of the states of the system, causes of motion or the driving forces and reactions of the system. In a particular state, namely, the equilibrium state of the system under study, the systems can either be at rest or moving with a time-dependent velocity. The kinetics of the prior, i.e., of a system at rest is studied in statics. Similarly, the kinetics of the later, i.e., of a body moving with a velocity is called dynamics. Introduction Classical mechanics describes the area of physics most familiar to us that of the motion of macroscopic objects, from football to planets and car race to falling from space. NASCAR engineers are great fanatics of this branch of physics because they have to determine performance of cars before races. Geologists use kinematics, a subarea of classical mechanics, to study ‘tectonic-plate’ motion. Even medical researchers require this physics to map the blood flow through a patient when diagnosing partially closed artery. And so on. These are just a few of the examples which are more than enough to tell us about the importance of the science of motion. In this module, we will discuss the basic components of kinematics such as position, velocity and acceleration and the rules governing them. Following this, we will learn about fundamental rules of motion which govern the state of the system. Learning of the concepts has been attempted through appropriate examples at various stages in the discussion. Objectives In this chapter, we aim to learn the following subjects: Motion Components of kinematics Equations of motion Relative motion Kinetics Laws of motion Learning through given illustrations 1 1. What is motion? The basic question to be asked is that ‘what exactly is motion?’ and ‘how do we measure it?’ Well, in the simplest words, ‘motion can be defined as the change in position of an object with reference to a fixed point or object with which we attach our coordinate system and measure quantities relative to it. In the broadest sense, motion is divided into two categories, the translational motion, and the rotational motion 2. Components of motion Now, before proceeding further, let’s recall some basic definitions and terms related to motion. The physical quantities can be ‘scalar’ or ‘vector’. Scalar quantities are those which have only magnitude, for example, energy, length, mass, speed, temperature and time are all scalar quantities. On the other hand, vector quantities need both the magnitude and direction to represent them. So to simplify things we use special language of vectors. In one dimensional motion, a particle can move in two directions only, i.e., along positive and negative axis. However, while studying two and three dimensional motion, this convention becomes less effective. Here vectors come in handy. Some of the commonly used physical quantities that are vector quantities are displacement, velocity, acceleration, force, momentum, etc. To deal with such quantities, we use vector algebra. 2.1 Position To define the position of any object, we consider a reference coordinate system, in most of the cases it’s the Cartesian co-ordinates xyz-system in three dimensions. Obviously, for one dimensional motion, the position can be defined either using x-axis or y-axis or z-axis depending on the choice of the person. Similarly, for motion in a two dimensional plane, the choice can be either xy-plane or yz-plan or xz-plane. A vector is denoted by some alphabet with an arrow over it, e.g., A . But for keeping the notations simpler in this text, we will use bold letters to represent the vector quantities, i.e., the same vector A will be written as A. The direction of the vector is usually denoted by some unit vector (or with respect to some reference axis). Here, unit vector is a vector which has magnitude one but points in a particular direction. For instance, unit vectors iˆ , ˆj and kˆ point along the direction of positive x-axis, y-axis and z-axis, respectively of the xyz-reference frame. Illustration 1: Using a given xyz Cartesian coordinate system, the position of two objects on a plane can be represented as (x1,y1) = (4,11) = 4 + 11 ˆj and (x2,y2) = (11,4) = 11 + 4 as shown here in the figure below. 2 Figure 1: Illustration for representing the positions of objects by points in a Cartesian coordinate system. 2.2 Distance and displacement One should not confuse distance with displacement as the two quantities have different meaning. Distance is a scalar quantity and it is the total length of the path traversed by a moving body or object. Whereas displacementis a vector quantity and defined as the shortest distance between the initial and final position of a body. The difference between the two quantities will become clear in the following illustrations. Illustration 2: Suppose an object moves from point P to point Q, then the displacement can be represented by a vector pointing from P to Q. The displacement vector PQ (as well as other displacement vectors) represent only the overall effect of the motion. For instance, all the three paths connecting P and Q, as shown in the figure below correspond to the same displacement vector, i.e., vector PQ. Illustration 3: A particle starts from point A and stops at point B, both of which are diametrically opposite to each other while covering the semi-circular part. If radius of the circular path is 5m then find: 3 (a) Distance covered by the particle, and (b) Displacement of the particle. Solution: (a) Distance covered = ½ × (circumference of the circle) = π × radius of the circle = π × 5 = 15.7 m (b) Displacement of the particle = distance between points A and B = length of circles diameter = 2 × radius = 10 m. Note that a vector quantity has a specific direction. So the displacement in this example is vector AB. Again, the other vector, i.e., vector BA should not be confused with vector AB. Though the two vectors have the same magnitude but they have different directions, i.e., vector AB has direction from A to B whereas vector BA has direction from B to A. Therefore, when the particle moved from point A to B, we are to calculate displacement vector AB. 2.3 Velocity While studying kinematics, the most common terms we come across are ‘speed’ and ‘velocity’. Often these two quantities are considered to be the same in general. But doing so while studying kinematics would be a ‘blunder’ as these two are ‘very different quantities’. Let’s have a look at their properties. Speed Velocity 1. It is the distance covered per 1. It is the displacement per unit unit time. time. 2. It can never be zero or 2. It can be zero or negative. negative. 3. It is a ‘vector’ quantity, i.e., it 3. It is a scalar quantity, i.e., it has both the direction and has only the magnitude but no magnitude associated with it. direction associated with it. To get a better understanding of the two quantities above, i.e., speed and velocity, let’s once again take a look at Illustration 2. Suppose that the particle takes 5 second (sec) to cover the total distance from P to Q, then the speed = distance/time = 15.7/5 = 3.14 m/sec. And velocity = displacement/time = 10/5 = 2 m/sec. Clearly, speed and velocity have different values for the same case. 4 Now, several quantities are related with the motion of a particle describing ‘how fast’ or ‘how slow’ it is moving. Two of them, i.e., speed and velocity have already been discussed. Technically, the terms speed and velocity are known as ‘average speed’ and ‘average velocity’. However, ‘instantaneous velocity’ and ‘instantaneous speed’ are also used to define motion. The basic difference between these two is that average velocity tells us about ‘how fast an object or a body is moving in a given ‘interval of time’. However, as the name itself suggest instantaneous velocity and instantaneous speed tell us about motion at a particular instant of time, e.g., at time t = 5 sec, or t = 6 sec, etc. Instantaneous velocity is also defined as ‘the rate of change of displacement with time’ and its value can be found by differentiating the displacement with respect to time, i.e., if x is the displacement of a body at t = 5 sec then the instantaneous velocity is = dx/dt. Here and below, whenever we are describing through numerical examples or equations, the magnitudes of the vector quantities have been used. Illustration 3: The magnitude of the displacement ‘x’ of a particle (given in m) at any time ‘t’ is given by the equation x = 5t2 – 2t. Find the velocity at t = 3 sec. Solution: Note that we are required to find velocity at a particular instant of time, i.e., at t = 3 sec. so, we differentiate the given equation so that dx d d 5 t 2 2 t dt dt dt 5 (2t) 2 (1) 10t 2 Or the magnitude ofvelocity v 10t 2 .