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Aakash Educational Services Limited - Regd. Office : Aakash Tower, 8, Pusa Road, New Delhi-110005 Ph. 011-47623456 Chapter Contents Introduction Periodic and Oscillatory Motions Period and Frequency Harmonic and Non Harmonic Motion Displacement Simple Harmonic Motion and Uniform Circular Motion Displacement, Velocity and Acceleration in Simple Harmonic Motion Force Law for Simple Harmonic Motion Energy in Simple Harmonic Motion Some Systems Executing Simple Harmonic Motion Damped Oscillations Forced Oscillations and Resonance Introduction When we look around us, we often find objects moving back and forth repeatedly. During an earthquake, buildings may be set oscillating so strongly that they are shaken apart. When an airplane is in flight, wings may oscillate due to turbulence of the air, resulting in metal fatigue and even failure. Resonance appears to be one reason behind collapse of buildings. Aircraft designers ensure that none of the natural angular frequencies at which a wing can oscillate matches the angular frequency of the engine in flight (you will find reasons for these two in the section on forced oscillations and resonance-chapter in daily life). In this chapter we shall study simple harmonic motion, periodic motion of pendulum and spring mass system, damped and forced oscillations and many more interesting topics. PERIODIC AND OSCILLATORY MOTIONS Periodic Motion Periodic motion is defined as the motion which repeats itself after equal intervals of time. The interval of time is called the time period of periodic motion. The motion of second, minute and hour arms of a clock is periodic motion. Oscillatory Motion Oscillatory or vibratory motion is defined as a periodic and bounded motion of a body about a fixed point. O is the equilibrium position for the ball placed in hemispherical bowl. When it is displaced to A, it moves as O  A  O  B  O. B A O Chapter 14 Oscillations
2 Oscillations NEET Aakash Educational Services Limited - Regd. Office : Aakash Tower, 8, Pusa Road, New Delhi-110005 Ph. 011-47623456 In case of a simple pendulum, O is the equilibrium or mean position. As the bob is displaced a little to A, it oscillates about O. Time taken by the bob from O  A  O  B  O is the time period of the oscillation. B A O Note : The body is confined within well defined limits (called extreme position) on either side of mean position. Difference between Periodic and Oscillatory Motion Every oscillatory motion is periodic, but every periodic motion need not be oscillatory. e.g., circular motion (or the motion of planets around the sun) is a periodic motion, but it is not oscillatory, because the basic concept of to and fro motion about the mean position for oscillatory motion is not present here. Note : There is no significant difference between oscillations and vibrations. When the frequency is small, we call it oscillation (like the oscillation of a branch of a tree), when the frequency is high (like the vibration of a string of a musical instrument), we call it vibration. PERIOD AND FREQUENCY Period is the smallest interval of time after which the motion is repeated. It is denoted by the symbol T. Its SI unit is second. Frequency is defined as the number of oscillations per unit time. It is the reciprocal of time period T. It is represented by the symbol . The relation between  and T is 1 T   Its SI unit is hertz (abbreviated as Hz), 1 hertz = 1 Hz = 1 oscillation per second = 1 s–1 Note : Frequency,  is not necessarily an integer. HARMONIC AND NON HARMONIC MOTION Harmonic oscillation is that oscillation which can be expressed in terms of single harmonic function (i.e., sine function or cosine function). A harmonic oscillation of constant amplitude and of single frequency is called simple harmonic motion. Simple harmonic oscillation is the simplest form of oscillatory motion. This motion arises when the force on the oscillating body is directly proportional to its displacement from the mean position (or the equilibrium position). Further at any point, in its oscillation, this force is directed towards the mean position.
NEET Oscillations 3 Aakash Educational Services Limited - Regd. Office : Aakash Tower, 8, Pusa Road, New Delhi-110005 Ph. 011-47623456 Mathematically, a SHM can be expressed as 2 sin sin    t xA tA T ...(1) or 2 cos cos    t xA tA T ...(2) Here, x = displacement of body from mean position at any instant t. A = maximum displacement or amplitude of oscillation.  = angular frequency (= 2) =   2     T  = frequency, T = time period Graph of = sin xA t  Figure-b T 2 3 2 T T t x +A –A T 0 Figure-a T 4 T 2 3 4 T 3 2 T T +A –A T t x 0 Graph of = cos xA t  Non-harmonic oscillation is that oscillation which cannot be expressed in terms of single harmonic function. It is a combination of two or more than two harmonic oscillations. Mathematically, it may be expressed as x = Asint + Bsin2t ...(3)      or 2 4 sin sin     t xA tB T T Graphically, it can be represented by a curve of the type shown in given figure. T T 2 O x t Example 1 : Categorize the motion as periodic or oscillatory motion (i) Motion of planets around the sun (ii) A weighted test tube floating in a liquid pressed down and released (iii) Motion of hands of a clock Solution : Periodic motion – (i), (ii) and (iii) Oscillatory motion – (ii)
4 Oscillations NEET Aakash Educational Services Limited - Regd. Office : Aakash Tower, 8, Pusa Road, New Delhi-110005 Ph. 011-47623456 DISPLACEMENT In general, it refers to the change with time of any physical property under consideration. Consider an oscillating simple pendulum. The angle with the vertical as a function of time is the displacement variable.   Consider a block attached to a spring, whose other end is fixed to a rigid wall. Here, it is convenient to measure displacement of the body from its equilibrium position. Note : The term displacement is not always to be referred in the context of position only. There can be other kinds of displacement variables, e.g., the voltage across a capacitor, changing with time in an A.C. circuit, pressure variations in time in the propagation of sound wave, the changing electric and magnetic fields in a light wave. The displacement variable may take both positive and negative values. The displacement can be represented as a mathematical function of time. In case of periodic motion, this function is periodic with time. One of the simplest periodic functions is given by f(t) = Acost If the argument of this function, t is increased by an integral multiple of 2 radians, the value of the function will be same. In one revolution, the angle covered by the reference particle is 2 radian and time period is T. If  is uniform angular velocity of the reference particle, Then, 2 2 or T T      Thus, the function f(t) is periodic with period T f(t) = f(t + T) This result is also correct for f(t) = Asint A linear combination of sine and cosine functions like f(t) = Asint + Bcost ...(1) is also a periodic function with the same period T. Take A = Dcos ...(2) and B = Dsin ...(3) Putting these values in equation (1), f(t) = Dcos sint + Dsin cost = D(sint cos + cost sin) f(t) = Dsin(t + ) ...(4) using sinAcosB + cosAsinB = sin(A + B) Here, D and  are constant found in the following way