CHAP 11 Oscillations Edited PDF

CHAPTER#11 OSCILLATIONS Q. Explain the concept of periodic motion to oscillatory motion. Discuss the terms frequency, amplitude, vibration, time period and angular frequency. Periodic Motion A motion which repeats itself in equal intervals of time is called pe...

CHAPTER#11 OSCILLATIONS Q. Explain the concept of periodic motion to oscillatory motion. Discuss the terms frequency, amplitude, vibration, time period and angular frequency. Periodic Motion A motion which repeats itself in equal intervals of time is called periodic motion. Vibratory Motion (Oscillatory Motion) A kind of periodic motion which is performed between two fixed points also called to and fro motion about mean position. Vibratory motion uses the same path to come back to the same point form where it starts the motion. Instantaneous displacement (x) In vibrational motion, the distance from the mean position at any instant is known as instantaneous displacement. It is zero at the instant when the body is at mean position and it is maximum at the extreme position. Amplitude (𝒙𝒐 ) In Vibrational motion, the maximum distance from the mean position to either extreme position is known as amplitude. The SI unit of Amplitude is meter. Vibration One complete round trip of a body during its vibrational motion is called vibration. For example, when the body starts its motion from its first extreme position (-x) to the second extreme position (x) and then from the second extreme position (x) to the first extreme position (-x) crossing the mean position (o) is called one vibration. Time Period Time period is defined as the time taken to complete one vibration or one cycle. It is represented by "T" and its SI unit is second’s’. Frequency Frequency is defined as the number of vibrations completed by the vibrating body in one second. It is expressed in terms of the reciprocal of time period. The unit of frequency is hertz (Hz) and it is equal to per second. Angular Frequency Angular frequency is defined as the number of revolutions per unit time. It is represented by '𝝎' and it can be expressed as 𝝎 = 𝟐𝝅𝒇 𝟐𝝅 𝝎= 𝑻 Q. Define simple harmonic motion (SHM). Discuss the key characteristics of a system undergoing SHM.  Simple Harmonic Motion (S.H.M): A kind of vibratory motion in which acceleration of a body is directly proportional to the displacement of a body and it is always directed toward the mean position is called simple harmonic motion. 𝒂 ∝ −𝒙 Where, x represents displacement and negative sign shows that the direction ofacceleration is towards means position. Characteristics of S.H.M: A body executing simple harmonic motion shows the following characteristics. 1. Its motion is vibratory. 2. Some restoring force acts on the vibratory system. 3. Acceleration of the body executing simple harmonic motion is directly proportional to its displacement “ x ” and always directed towards the mean position. 4. Energy system oscillates between kinetic energy and potential energy but the total energy of a frictionless system remains constant. Examples of S.H.M 1. The motion of a mass attached to an elastic spring over a frictionless surface is an example of S.H.M. 2. The motion of projection of a particle in a circular path is an example of S.H.M. 3. The motion of a simple pendulum is also a simple harmonic motion. 4. The motion of a swing is also S.H.M. 5. The vibratory motion of a string of a musical instrument. Hooke’s Law: Force required to stretch a spring is directly proportional to the displacement “x”. If elastic limit is not violated i.e. 𝐹∝𝑥 𝐹 = 𝑘𝑥 Where k = force constant or spring constant. Q. Derive equation of motion for a mass spring system in SHM, illustrating each step of the derivation. MOTION OF THE MASS-SPRING SYSTEM IS S.H.M Motion under Elastic Restoring Force: Consider a body of mass “m” attached with a spring of spring’s constant “k” lying on a smooth surface of table as show in figure. Initially, the body is at rest position ‘O’ called mean position or equilibrium position. Now we apply some force F on body and we displace the body from ‘O’ to ‘A’.  The spring will exert the force on body due to elastic restoring force. 𝐹𝑎𝑝𝑝𝑙𝑖𝑒𝑑 = 𝑚𝑎 Hence we release the body it will move towards ‘B’ and will cross the mean position ‘O’ due to inertia and reaches point B compresses the spring it returns and start oscillation between A & B. 𝐹𝑟𝑒𝑠𝑡𝑜𝑟𝑖𝑛𝑔 = −𝑘𝑥 Therefore; 𝐹𝑎𝑝𝑝𝑙𝑖𝑒𝑑 = 𝐹𝑟𝑒𝑠𝑡𝑜𝑟𝑖𝑛𝑔 𝑚𝑎 = −𝑘𝑥 𝑘 𝑎=− 𝑥 𝑚 Whew, k and m are constants so, 𝑎 = −𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 𝑥 𝑎 α−𝑥 Hence, it is proved that the motion of mass spring system is SHM. Time period: The time required for one vibration of a simple harmonic oscillator is 2𝜋 𝑇= − − − (𝑖) 𝜔 But we know that 𝑎 = −𝜔2 𝑥 − − − (𝑎) And, 𝑘 𝑎 = − 𝑥 − − − (𝑏) 𝑚 Comparing above two equations (a) and (b) 𝑘 − 𝑥 = −𝜔2 𝑥 𝑚 𝑘 𝜔2 = 𝑚 𝑘 𝜔=√ 𝑚 Substitute above in equation no (i) 2𝜋 𝑇= √𝑘 𝑚 𝑚 𝑇 = 2𝜋√ 𝑘 Frequency:  1 𝑓= 𝑇 1 𝑓= 𝑚 2𝜋√ 𝑘 1 𝑘 𝑓= √ 2𝜋 𝑚 Q. A particle in a state of uniform circular motion. Show that its projection along one of its diameter executes simple harmonic motion. Simple Harmonic Motion Connected with Circular Motion: Consider a body which is moving in a circle with a constant linear speed “𝑣” and constant angular speed “ω”. ̅̅̅̅ Let the linear speed of a particle at point “P” is “𝑣” where its projection along the Diameter 𝐴𝐵 is “Q” Then, Centripetal acceleration of the body at point P is 𝑣2 𝑎𝑐 = − 𝑟 Minus sign indicates that 𝒂 ⃗ c is opposite to the position vector 𝒓 ⃗ of the body. But 𝑣 = 𝑟𝜔 Where ω is the angular speed of the body, therefore (𝑟𝜔)2 𝑎𝑐 = − 𝑟 𝑟 2 𝜔2 𝑎𝑐 = − 𝑟 So, 𝑎𝑐 = −𝜔2 𝑟 The acceleration of the projection at point “Q” is 𝑎 = 𝑎𝑐 𝑐𝑜𝑠𝜃 Putting the value of 𝑎𝑐 we get 𝒂 = −𝜔2 𝒓 𝑐𝑜𝑠𝜃 But from ∆𝑂𝑃𝑄 𝒙 𝐶𝑜𝑠𝜃 = 𝒓 Where 𝑥 is the displacement vector of the projection. Using this relation we get, a = −ω2x-------------------- eq (1) If angular speed ω is constant, Then a − x This equation indicates that: i. The magnitude of acceleration is directly proportional to the displacement ii. The direction of acceleration always remains towards the mean position. Therefore, we conclude that the motion of the projection is the SHM. MAXIMUM ACCELERATION: The acceleration will be maximum 𝑎𝑚𝑎𝑥 at extreme position, where 𝑥 = 𝑥𝑜 2 Then the equation 𝑎 = −𝜔 𝑥 will become, 𝑎𝑚𝑎𝑥 = −𝜔2 𝑥𝑜 MINIMUM ACCELERATION: The acceleration will be minimum 𝑎𝑚𝑖𝑛 at mean position, where 𝑥=0 2 Then the equation 𝑎 = −𝜔 𝑥 will become, 𝑎𝑚𝑖𝑛 = 0 VELOCITY OF PROJECTION OF A UNIFORM CIRCULAR OBJECT Consider a body which is orbiting in a circle, and its projection is oscillating on the diameter as shown. If velocity of the body at point P is V, then velocity of its projection at point Q is: 𝑣 = 𝑣 𝑠𝑖𝑛𝜃 But 𝑣 = 𝑟 𝜔 Where ω is the angular velocity of the body, therefore we have 𝑣 = 𝑟 𝜔 𝑠𝑖𝑛𝜃 − − − −(1) Since sin2θ =1− cos2θ base x and cosθ = = hyp r 2 2 x sin 𝜃 = 1 − ( ) r x2 𝑟 2 −𝑥 2 1 Or Sinθ = 1 − 2 = √ 2 = r2 − x2 r 𝑟 𝑟 Putting the values of sinθ in the above equation (1) we get 𝒗 = ω r2 − x2 here r = xo  𝒗 = ω xo 2 − x 2 ---------------- (2) Maximum Velocity: The velocity of projection “Q” is maximum at mean position Where x = 0, Putting x = 0 in (1)𝒗 = ω xo 2 − 02 = ω xo 2 𝑣𝑚𝑎𝑥 = 𝑥𝑜 𝜔 Minimum Velocity: The velocity of projection “Q” is minimum at extreme position Where x = xo Putting x = xo in (1) 𝒗 = ω xo 2 − xo 2 = ω 0 𝑣𝑚𝑖𝑛 = 0 Q. Define simple pendulum and prove that the motion of simple pendulum executes simple harmonic motion. Also calculate its time period. SIMPLE PENDULUM A point mass object (bob) is attached by inextensible string whose other end is attached by fixed frictionless support and move about mean position is called a simple pendulum. MOTION OF SIMPLE PENDULUM IS SHM DERIVATION:- Consider a simple pendulum having mass m and length of pendulum “L” which is displaced at angle “θ” from its mean position as shown. The weight of pendulum is divided into two components. The component W cosθ balances the tension: Therefore T=W cosθ Or T = mg cosθ Whereas the component W sinθ behaves as the restoring force, which is responsible for the oscillations of the pendulum: Therefore 𝐹𝑟𝑒𝑠 = W sinθ Or 𝐹𝑟𝑒𝑠 = 𝑚𝑔𝑠𝑖𝑛𝜃 Since angle θ in simple pendulum is kept very small x Therefore sinθ = tanθ = L Where L = length of the pendulum x = displacement of the bob from the mean position 𝑥 Therefore 𝐹𝑟𝑒𝑠 =𝑚𝑔 𝐿 𝑚𝑔 Or 𝐹𝑟𝑒𝑠 = x 𝐿 Since the force ⃗𝑭res and the displacement “𝒙 ⃗ ” remain opposite to each other during the oscillations, therefore we place a minus sign in the equation i.e. 𝑚𝑔 𝐹𝑟𝑒𝑠 = −( ) x … (i) 𝐿 This force produces acceleration in the bob, which is given by: 𝑭𝑟𝑒𝑠 = 𝒎𝒂 ……… (ii) Comparing the two equations (i) and (ii) we get 𝑚𝑔 𝑚𝑎 = −( ) x 𝐿 𝑔 𝑎 = −( )𝑥 𝐿 Since g and L are constant: Therefore a  −x This equation indicates that the motion of the simple pendulum is S.H.M. Angular frequency: We know that acceleration of a body executing SHM is given by 𝑎 = −𝜔2 𝑥 − − − (𝑎) 𝑔 𝑎 = − ( ) 𝑥 − − − (𝑏) 𝐿 Comparing this equation (a) with the equation (b) we get 𝑔 ( ) 𝑥 = 𝜔2 𝑥 𝐿 So, g 𝑔 𝜔2 = or ω=√ 𝐿 𝐿 Frequency: Since the angular frequency and the frequency are related as: ω 1 g ω = 2π f therefore f= or f= 2π 2 l Time period: We know that the time period and the frequency are related as: 1 l T= Therefore T = 2π f g Q. Discuss the concept of energy in SHM. Explain how kinetic energy and potential energy vary throughout the motion of a particle in SHM and how the total mechanical energy is conserved? ENERGY CONSERVATION IN SHM Consider a mass spring system, when the mass is pulled towards right and released so it moves towards the equilibrium position as show in figure. It can be seen that the speed is greatest as the object passes through the equilibrium position and the object slows down as it reaches to the end points. This phenomenon indicates the interconversion of kinetic and potential energies of the system at different points. The total mechanical energy of the system at any instant shall remain constant Energy = k=Kinetic energy (K.E) + Elastic potential energy (E.P.E) ----(A) Inter conversion of kinetic and potential energies during SHM: KINETIC ENERGY (K) OF THE OSCILLATOR: Since the kinetic energy at any instant of the system is given by, 1 𝐾. 𝐸 = 𝑚 𝑣 2 2 𝑥2 Where 𝑣 = 𝑥𝑜 𝜔√1 − 𝑥𝑜2 2 1 𝑥2 𝐾. 𝐸 = 𝑚 (𝑥𝑜 𝜔√1 − 2 ) 2 𝑥𝑜 𝑘 Where 𝜔 = √ 𝑚 2 1 𝑘 𝑥2 𝐾. 𝐸 = 𝑚 (𝑥𝑜 √ √1 − 2 ) 2 𝑚 𝑥𝑜 𝟏 𝑲. 𝑬 = 𝒌 (𝒙𝟐𝒐 − 𝒙𝟐 ) − −(𝒊) 𝟐 Above expression represents the instantaneous kinetic energy of the object executing SHM. MAXIMUM KINETIC ENERGY: Since the speed is maximum at equilibrium position, then the K.E at mean position or equilibrium position is also maximum i.e., 𝑥 = 0 Equation (i) becomes, 𝟏 𝑲. 𝑬 = 𝒌 (𝒙𝟐𝒐 − 𝟎𝟐 ) − −(𝒊) 𝟐 𝟏 𝑲. 𝑬 = 𝒌𝒙𝟐𝒐 𝟐 MINIMUM KINETIC ENERGY: Since the speed is maximum at equilibrium position, then the K.E at mean position or equilibrium position is also maximum i.e., 𝑥 = 0 The speed of the object is zero at extreme position, then the K.E at extreme position is also minimum where 𝑥 = 𝑥𝑜 𝟏 Equation (i) becomes, 𝑲. 𝑬 = 𝒌 (𝒙𝟐𝒐 − 𝒙𝟐𝒐 ) 𝟐 𝑲. 𝑬 = 𝟎 POTENTIAL ENERGY (E.P.E) OF THE OSCILLATOR: The net force on the oscillator at equilibrium position x = 0, 𝐹𝑜 = 0 and at extreme point, i.e., at A its 𝐹𝐴 = 𝑘𝑥 The average applied force exerted on the system in displacing it from 0 to A is. 0 + 𝑘𝑥 𝐹𝑎𝑣𝑔 = 2 1 𝐹𝑎𝑣𝑔 = 𝑘𝑥 2 The work done in moving the object from O to a, against the elastic restoring force. 𝑤𝑜𝑟𝑘𝑑𝑜𝑛𝑒 = 𝐹𝑎𝑣𝑔. 𝑥 1 𝑤 = ( 𝑘𝑥). 𝑥 2 1 𝑤 = 𝑘𝑥 2 2 This work is stored in the spring mass system as its elastic potential energy E.P.E as shown in fig, above equation is rewritten as: 𝟏 𝑬. 𝑷. 𝑬 = 𝒌𝒙𝟐 − − − (𝒊𝒊) 𝟐 Above equation expresses the instantaneous elastic potential energy of the object executing SHM. MAXIMUM POTENTIAL ENERGY: Since the speed is maximum at equilibrium position, i.e., 𝑥 = 0 Equation (ii) becomes, 𝟏 𝑬. 𝑷. 𝑬 = 𝒌𝟎𝟐 𝟐 𝑬. 𝑷. 𝑬 = 𝟎 MINIMUM POTENTIAL ENERGY: The speed of the object is instantaneously at rest on extreme position where 𝑥 = 𝑥𝑜 Equation (ii) becomes, 𝟏 𝑬. 𝑷. 𝑬 = 𝒌𝒙𝒐 𝟐 𝟐 TOTAL ENERGY OF THE OSCILLATOR: Substitute equation (i) and (ii) in equation (A) 𝟏 𝟏 𝑬 = 𝒌 (𝒙𝟐𝒐 − 𝒙𝟐 ) + 𝒌𝒙𝟐 𝟐 𝟐 𝟏 𝟐 𝟏 𝟏 𝑬 = 𝒌𝒙𝒐 − 𝒌𝒙𝟐 + 𝒌𝒙𝟐 𝟐 𝟐 𝟐 𝟏 𝟐 𝑬 = 𝒌𝒙𝒐 𝟐 The graphs show that the elastic Potential energy is zero where the displacement is zero and maximum at extreme positions. Contrary to P.E, kinetic energy is maximum at zero displacement. i.e., at equilibrium position and minimum at extreme positions. FREE OSCILLATIONS: Consider a body or a system capable of oscillating, which is displaced from its mean position to its extreme position and then left free. Due to the restoring force, it starts oscillation with certain frequency which is called its natural frequency and the corresponding period is called its natural time period. If a body is oscillating with its own natural frequency and it is free from all the external resistive forces then such oscillations of the body are called free oscillations. For example, oscillations of a simple pendulum, vibrations of prongs of a tuning fork, vibrations of string of musical instrument etc. In free oscillations, the total energy of the body remains constant ie. It is conserved. As we are assuming the absence of resistance force therefore the amplitude of the oscillation remains constant. Graphically, the free oscillations of a body with constant amplitude are shown. Q. Explain the concept of damping and its effect on oscillatory motion. Discuss the types of damping such as over damping, critical damping and under damping. DAMPED OSCILLATIONS: In practical, when a body is oscillating with its natural frequency, the amplitude of the oscillation gradually decreases with time and finally it comes to rest. This is due to the presence of resistive forces such as; air resistance, friction etc. The oscillation with decreasing amplitude in the presence of various resistive forces is called damped oscillations and the resistive forces are called damping forces. Graphically the damped oscillation of the oscillating body is shown in Fig. Now the damped oscillation can be studied under the following three different cases. 1. When Damping force > oscillating force, the body does not oscillate. i.e., without performing any oscillation, the body quickly comes at rest position. Such motion is called over damping. 2. When Damping force = oscillating force, then the motion of the body is called critical damping. In this case, the body returns to the equilibrium position with uniform speed along a curved path without performing oscillation. 3. When Damping force < oscillating force, then the body is set into oscillation and is called under damping.  Forced oscillations and resonance: In damped oscillation, the oscillator cannot maintain its natural frequency for long duration due to the resistive forces and the amplitude of the oscillation decreases gradually with time. But we can maintain constant amplitude by applying a periodic external force which is called a driving force. Thus, when the oscillating body is subjected to a periodic driving force then such oscillation is called forced oscillation and its frequency is called driving frequency. The vibration of a vehicle caused by the running of engine is an example of forced vibration. In forced oscillation, the amplitude of the oscillation depends upon the relation between the driving frequency and the natural frequency of the body." Q. Discuss the concept of resonance in SHM. Explain how resonance occurs and its effects on the amplitude and energy transfer in a driven oscillating system. If the frequency of the driving force is equal or integral multiple of natural frequency of the oscillating body such that 𝑓𝑒𝑥𝑡𝑒𝑟𝑛𝑎𝑙 = 𝑓1 𝑜𝑟 = 2𝑓1 𝑜𝑟 = 3𝑓1 … … 𝑜𝑟 𝑛𝑓1 , the amplitude of vibration is very much increased. This phenomenon is known as resonance and the oscillations of large amplitude are called resonant oscillations. The resonance phenomenon can further be explained by some examples: 1. The soldiers are advised to break their steps while crossing a bridge. If the soldiers march in steps then it is possible that the frequency of their footsteps become equal to the natural frequency of the bridge and the bridge may be set into vibrations with large amplitude due to the resonance. 2. During earthquake, when the frequency of earthquake is equal to the natural frequency of a building then the building will be set into vibrations with large amplitude due to the resonance and the building may collapse. 3. In communication system, all the transmitting signals can be received by receivers due to the resonance phenomenon when the frequency of the receiver is made equal to the frequency of incoming signal. 4. Microwave ovens generate super high frequency electromagnetic waves (3GHz-30GHz and wavelength of about 12 cm) and scatter them throughout the oven. The frequency of microwave excites water molecules into resonance and causes them to collide with one another. Friction generated by the collisions changes the kinetic energy of the water into heat that warms the food. Food containing water molecules can only be heated by the microwave oven. 5. The amplitude of a swing can be increased by applying a suitable periodic force on it. Q. Discuss the factor that affects the sharpness of resonance in an oscillatory system. Explain how damping and quality factor influence the width and peak of the resonance curve. SHARPNESS OF RESONANCE: We have studied in the resonance phenomenon that the amplitude of the Oscillation is maximum when the frequency of the driving force is nearly equal to the natural frequency of the oscillating body. The amplitude can be decreased by changing the frequency of driving force. If the amplitude of oscillation increases rapidly at a frequency f slightly different that from the resonant frequency 𝑓𝑜 , then the resonance is said to be sharp. Amplitude of the resonance oscillation and its sharpness depend upon damping that is, smaller the damping, greater will be the amplitude and sharper will be the resonance. Similarly, for greater damping, the amplitude of the resonant oscillation will be small and such resonance is called flat resonance. However, the amplitude may become very large if the damping is small and the applied frequency is close to the natural frequency. The sharpness of resonance depends mainly on two factors: amplitude and damping. The Q-factor quantifies the sharpness of resonance It is defined as “the number of oscillations an oscillator makes before its amplitude decays to zero” The higher the Q-factor means that the resonance peak is sharper. The system with more damping will have a low Q-factor. Q-factor is a dimensionless quantity and it is 𝐸𝑠𝑡𝑜𝑟𝑒𝑑 𝑄= 𝐸𝑙𝑜𝑠𝑡 The resonance effect is very important in the design of bridges and other civil engineering projects. On July, 1940 the newly constructed Tacoma Narrow Bridge (Washington) was opened for traffic. Only four months after this, a mild wind set up the bridge in resonant vibrations. In a few hours the amplitude became so large that the bridge could not stand the stress and a part broke off and went into the water below. After this incident the engineers considered the resonance phenomenon in the design and construction of long span bridges.