JOURNAL OF PHYSICAL AND CHEMICAL SCIENCES

Transcription

1 JOURNAL OF PHYSIAL AND HEMIAL SIENES Journal hoepage: Review Open Access A Review of Siple Haronic Motion for Mass Spring Syste and Its Analogy to the Oscillations in L ircuit Getahun Getachew * Departent of Physics, Wolaita Sodo University, Wolaita Sodo, Ethiopia * orresponding Author: Getahun Getachew; Eail: getahungetachewt@yahoo.co Received: July 5, 08, Accepted: August 3, 08, Published: August 3, 08. ABSTRAT Siple haronic otion coes up in any places in physics and provides a generic first approxiation to odels of oscillatory otion. The ass oscillating on a spring and the oscillation of L circuit appear to have little in coon because the two systes are apparently different systes but both can be described in ters of second order differential equations with constant coefficients of the sae for. With the absence of friction in the ass-spring syste, the oscillations would continue indefinitely and we obtain equations for the ways in which the displaceent, velocity and acceleration of a siple haronic oscillator vary with tie and the ways in which the kinetic and potential energies of the oscillator vary. Siilarly, the oscillations of an L circuit with no resistance would continue forever if undisturbed and we obtain equations for tie varying charge, current, energy stored in inductor and energy stored in capacitor. Therefore, the ai of this article is to discuss the properties of oscillations of ass-spring syste and provide interesting analogies with oscillations in L circuit. Keywords: Mass-spring syste, L circuit, siple haronic otion, Analogy INTRODUTION Oscillations occur in any branches of physics, but in this article I discuss the oscillations of ass-spring syste without friction and L circuit oscillations without resistance. A ass oscillating on a spring and the oscillation of L circuit appear to have little in Figure : A block attached to a spring oving on a frictionless surface. (a) When the block is displaced to the right of equilibriu (x> 0), the force exerted by the spring acts to the left. (b) When the block is at its equilibriu position (x = 0), the force exerted by the coon; but the atheatics odels the is alost spring is zero. (c) When the block is displaced to the left of indistinguishable and both can be described in ters of a second order differential equations with constant coefficients. In this case the second order differential equation with constant coefficients for equation of otion is equilibriu (x < 0), the force exerted by the spring acts to the right [3]. When the spring is neither stretched nor copressed, the block is at the position called the equilibriu position of the syste, which we d X ω X () identify as x = 0. If the block is displaced to a position x, the spring exerts a force on the block that is proportional to the position where, X represents the sall displaceent (x) fro equilibriu and given by Hooke s law position in the ass - spring syste or the charge (q) in the L F circuit and ω represents constant coefficients [, ]. This equation s = kx (3) This force is called a restoring force because it is always directed of otion has a general solution which is given by: toward the equilibriu position and therefore opposite the X(t) = Acos(ωt) + B sin(ωt ) = cos(ωt + φ)- () displaceent fro equilibriu. is called the aplitude of the oscillation, ω is the angular The syste ust obey Newton s second law of otion which frequency and ϕ is the phase []. states that the force is equal to ass ties acceleration a. Furtherore in this article I also briefly describe soe of the Applying Newton s second law F x = a x to the otion of the conditions in which such equations arise and then investigate the block, with F reasons why such apparently different systes exhibit very siilar s = kx providing the net force in the x direction, we obtain [, 3, 4, 5, 6] behavior. kx = a I. Mass-Spring Syste x (4) onsider a block of ass attached to the end of a spring, with the a x = k x (5) block free to ove on a horizontal, frictionless surface as shown in the figure below [3]. That is, the acceleration is proportional to the position of the block, and its direction is opposite the direction of the displaceent fro equilibriu. Systes that behave in this way are said to exhibit siple haronic otion. Modeling the block as a particle subject to the force F s = kx, and assuing that the oscillation occurs along the x-axis, equation (5) can be rewritten as d x = k x (6) Recall that, by definition, a x = dv x a x = d (dx) = d x and v x = dx this iplies that

2 If we denote the ratio k with the sybol ω, ω = k ω = k (7) Accordingly, d x = ω x (8) The general solution to the above differential equation is x(t) = Acos(ωt + ϕ) (9) Where A, ω, and ϕ are constants. In order to give physical significance to these constants, it is convenient to for a graphical representation of the otion by plotting x as a function of t, as in figure below. is at its equilibriu. It is also observed that both graphs position vs. tie and velocity vs. tie are periodic waves of the sae frequency just shifted by 90 or π/. Furtherore, note that the phase of the acceleration differs fro the phase of the position by π radians, or 80 [7]. Position Velocity acceleration Figure : position versus tie graph for an object undergoing siple haronic otion [4]. A is called the aplitude of the otion, and it is siply the axiu value of the position of the particle in either the positive or negative x direction. The period T of the otion is the tie interval required for the particle to go through one full cycle of its otion. T = π (0) ω The inverse of the period is called the frequency f of the otion. f = = ω () T π The units of f are cycles per second, or hertz (Hz). Rearranging the above equation gives ω = πf = π () T In ters of the characteristics and k, the period and frequency of the otion for the particle spring syste can be expressed as follows. T = π = ω π (3) k f = = k (4) T π Thus the period and frequency depend only on the ass of the particle and the force constant of the spring. The velocity of the particle undergoing siple haronic otion becoes v x = dx = A d cos(ωt + ϕ) v x = ωasin(ωt + ϕ) (5) a x = d x = dv d( ωasin(ωt + ϕ)) = a x = ω Acos(ωt + ϕ) (6) Since the sine and cosine functions oscillate between±, the extree values of the velocity v are ±ωa. Siilarly, the extree values of the acceleration a are±ω A. Therefore, the axiu values of the agnitudes of the velocity and acceleration are Figure 3: Graphs of position, velocity and acceleration as a function of tie [7]. Let us exaine the echanical energy of the block spring syste illustrated in the figure below. The only horizontal force on the block spring syste is the conservative force exerted by an ideal spring. The vertical forces do no work, so the total echanical energy of the syste is conserved [3, 4]. We assue a ass less spring, so the kinetic energy of the syste corresponds only to that of the block. Therefore, the kinetic energy of the block is K = v, but v = ωasin(ωt + ϕ) v = ω A sin (ωt + ϕ) K = ω A sin (ωt + ϕ) (9) The elastic potential energy stored in the spring for any elongation x is given by U = kx, but x(t) = Acos(ωt + ϕ) x = A cos (ωt + ϕ) U = ka cos (ωt + ϕ) (0) Since ω = k, we can express the total echanical energy of the siple haronic oscillator as E = K + U = ka [sin (ωt + ϕ) + cos (ωt + ϕ)] () But, fro trigonoetric identity, sin (ωt + ϕ) + cos (ωt + ϕ) = Therefore, the equation reduces to E = ka () That is, the total echanical energy of a siple haronic oscillator is a constant of the otion and is proportional to the square of the aplitude. Energy is continuously being transfored between potential energy stored in the spring and kinetic energy of the block. When the body reaches the point x = A its axiu displaceent fro equilibriu, it oentarily stops as it turns back toward the equilibriu position. That is, when x = ±A, because v = 0 at these points the energy is entirely potential, and E = ka. Because E is v ax = ωa = A k (7) constant, it is equal to E = ka at any other point. At the a ax = ω A = A k (8) equilibriu position (x= 0), the energy is entirely kinetic because U = 0, the total energy, all in the for of kinetic energy is again E = Graphs of position, velocity and acceleration as a function of tie are displayed in real tie in the sae window, illustrated in Figure ka.we can use the principle of conservation of energy to obtain 3. As shown in the top and iddle plots the axiu and iniu the velocity for an arbitrary position by expressing the total energy values of the position occur when the velocity is zero. Likewise the at soe arbitrary position x as E = K + U v + kx = axiu and iniu values of velocity occur when the position

3 ka. Solving for v gives v = ± k (A x ) = ±ω A x (3) potential energy kinetic energy Figure 4: Kinetic energy and potential energy versus tie for a siple haronic oscillator with ϕ = 0 [8]. II. Oscillations in L ircuits L circuit is the siplest exaple of an oscillating electrical circuit consists of an inductor L and capacitor connected together in series with a switch []. Unlike a resistor, which always resists the flow of current, an inductor tends to oppose changes to the flow of electric current. Figure 5: A siple L circuit [3]. A circuit containing an inductor and a capacitor shows an entirely new ode of behavior, characterized by oscillating current and charge [, 3, 4, 5, 6]. The voltage drop v L across an inductor is given by the forula v L = L di (4) And the voltage drop v across a capacitor is given by the forula v = q (5) To study this oscillation in detail, we apply Kirchhoff s loop rule to the circuit in Fig.5.which states that the su of the voltages around any loop of the circuit is zero v L + v = (6) L di q = (7) Since i = dq di it follows that = d q Substituting this expression into equation (7) and dividing by L one can obtain d q + q L If we denote the ω = L Accordingly, = (8) with the sybol ω, L ω = L (9) d q = ω q (30) The Matheatical solution to the above differential equation is q(t) = q ax cos(ωt + ϕ) (3) where q ax, ω, and ϕ are constants. The period T of the oscillation in L circuit is T = π L (3) The inverse of the period is called the frequency f of the oscillation. f = = T π L (33) That is, the period and frequency depend only on the charge across the capacitor and the capacitance of the capacitor. For the given haronically oscillating charge, the voltage and the current in the L circuit also oscillate according to eq (5). v = q ax cos (ωt + ϕ) v = v ax cos (ωt + ϕ) (34) i = dq(t) d = q ax cos(ωt + ϕ) i = ωq ax sin(ωt + ϕ) (35) Where ωq ax is axiu current and is given by i ax = ωq ax = q ax L (36) Roughly speaking, if we assue that the capacitor begins charged, then the capacitor begins by discharging through the inductor, slowly at first but picking up speed as the inductor lets ore current through. Once the capacitor is fully discharged, the inductor continues pushing current through the circuit, which drains even ore charge fro the capacitor, leaving it with a negative total charge. The capacitor then reverses the flow of current to regain the lost charge, but the sae thing happens again, with the inductor continuing to push current through in the reverse direction until the capacitor is back to its initial charged state. The cycle thus continues indefinitely. Thus the charge and current in an L- circuit oscillate sinusoidally with tie, with an angular frequency deterined by the values of L and. charge current Figure 6: Graphs of charge versus tie and current versus tie for a résistance less, non radiating L circuit [9]. Since the L- circuit is a conservative syste, we can analyze the circuit using an energy approach and we expect that the total energy stored in the syste to be constant. When the capacitor is fully charged, the energy U in the circuit is stored in the electric field of the capacitor and is equal to q ax. At this tie, the current in the circuit is zero, and therefore no energy is stored in the inductor. After the switch is closed, the rate at which charges leave or enter the capacitor plates (which is also the rate at which the charge on the capacitor changes) is equal to the current in the circuit. As the capacitor begins to discharge after the switch is closed, the energy stored in its electric field decreases. The discharge of the capacitor represents a current in the circuit, and hence soe energy is now stored in the agnetic field of the inductor [3, 4]. The electric potential energy stored in the capacitor is given by 3

4 U = q ax cos (ωt + ϕ) (37) The agnetic energy stored in the inductor, U L = Li sin (ωt + ϕ) (38) obining equations (37) and (38) the total electrical energy of the L oscillator can be expressed as U = U + U L = q ax [sin (ωt + ϕ) + cos (ωt + ϕ)] (39) But, fro trigonoetric identity, sin (ωt + ϕ) + cos (ωt + ϕ) = Therefore, the equation reduces to U = q ax (40) That is, the total electrical energy of a L oscillator is a constant of the otion and is proportional to the square of the aplitude charge. Energy is continuously being transfored between potential energy stored in the capacitor and inductor. We can use the principle of conservation of energy to obtain the expression for electric current as U = U + U L = q + Li = q ax Solving for i gives i = ± L (q ax q ) = ±ω q ax q (4) electrical energy agnetic energy Figure.7: Graphs of agnetic energy stored in inductor and electric potential energy stored in capacitor versus tie for oscillations in L ircuit with ϕ = 0 [9]. III. oparing Siple haronic otion for a ass-spring syste with the oscillations in L ircuit oparing equations (8-3) with equations (30-4), the siple haronic otion for a ass-spring syste and oscillations in L ircuit have the sae for. Thus all the discussions about the siple haronic otion for a ass-spring syste can be carried over to oscillations in L ircuit. Moreover one can see a direct correspondence between the two sets of physical quantities involved: Displaceent (x) corresponds to the charge(q); Velocity (v x ) corresponds to the current(i ); Inductance(L) corresponds to the ass (); Spring constant (k) corresponds to the Inverse of capacitance (/). Force(F x ) corresponds to the voltage(v) ; Kinetic energy of oving block corresponds to the agnetic energy stored in inductor and Potential energy stored in a stretched spring corresponds to the electric potential energy stored in the capacitor. oparing graphs in figure (3) with figure (6), for oscillating ass spring syste the axiu and iniu values of velocity occur when the displaceent is zero and the axiu and iniu values of displaceent occur when the velocity is zero; and in L circuit oscillation, the axiu and iniu values of current occur when the charge on the capacitor is zero and the axiu and iniu values of charge occur when the current in the circuit is zero. Thus the displaceent of a stretched spring is analogous to the charge on the capacitor and the velocity of the oving block of ass is analogous to the current in the inductor. oparing graphs in figure (4) with figure (7), for oscillating ass spring syste, the total energy is equal to the axiu potential energy stored in the spring when x = ±A, and at x= 0 the total energy is equal to the kinetic energy and is equal to E = ka ; and in L circuit oscillation, the total energy is equal to axiu electric potential energy stored in the electric field of the capacitor when the current in the circuit is zero and at q= 0, the total energy is all in the for the agnetic energy stored in the inductor and is equal to U = q ax. This iplies that the potential energy stored in a stretched spring is analogous to the electric potential energy stored in the capacitor and the kinetic energy of the oving block is analogous to the agnetic energy stored in the inductor. Therefore, the properties oscillations of ass-spring syste are very iportant and provide interesting analogies with oscillations in L circuit. The following table suarizes the physical quantities involved in siple haronic otion for ass-spring syste and L- circuit oscillations and the analogies between the. Mass-spring syste L circuit Quantities Equations Quantities Equations Position(x) x(t) = Acos(ωt + ϕ) harge q(t) = q ax cos(ωt + ϕ) Velocity(v x ) v x (t) = ωasin(ωt + ϕ) urrent i(t) = ωq ax sin(ωt + ϕ) Acceleration (a x ) a x (t) = ω Acos(ωt + ϕ) Rate of change of di(t) current = ω q ax cos(ωt + ϕ) Force (F x ) F x (t) = kacos(ωt + ϕ) Voltage Mass() (k=spring k constant) = k ω k = ω Inductance apacitance v(t) = q ax cos(ωt + ϕ) L = ω = Lω 4

5 Angular frequency Period ω = k T = π ω = π k Angular frequency Period ω = L T = π L Frequency Kinetic energy of oving block Potential energy stored in spring f = T = π k K = v axsin (ωt + ϕ) U = ka cos (ωt + ϕ) Frequency Magnetic energy stored in inductor Electric energy stored in capacitor f = π L U L = Li axsin (ωt + ϕ) U = q ax cos (ωt + ϕ) Total energy E = ka Total energy U = q ax ONLUSION Generally speaking the ass-spring syste and the L circuit are the two very different physical systes but both can be described by siilar second order differential equations with constant coefficients because atheatical odel is the sae in both cases. oparing equation (8) with equation (30), the two equations have the sae for and the general solutions for displaceent and charge take the sae for as indicated in equations (9) and (3).This shows that the displaceent oscillation of the ass-spring syste driven by an externally supplied sinusoidal force is analogous to the charge oscillation in the L circuit driven by an externally supplied sinusoidal voltage. Thus the siilar discussions can be carried over the siple haronic otion for a ass-spring and oscillations in L ircuit. Moreover there is a direct correspondence between charge (q) and displaceent (x); current (i) and velocity (v x ) ; inductance (L) and ass (); inverse of capacitance (/) and spring constant (k) ;force(f x ) and voltage(v) ; kinetic energy of oving block and agnetic energy stored in inductor; potential energy stored in a stretched spring and electric potential energy stored in the capacitor. Therefore, the properties of oscillations of ass-spring syste are very iportant and provide interesting analogies with oscillations in L circuit. Acknowledgent I a very grateful to y colleagues and friends Belachew Desalegn( MSc) and Kassaye Bewketu(MSc) for their incredibly valuable suggestions and coents in copiling this paper. REFERENES. Flap, Module P 5.5 the atheatics of oscillations, Open University (998). George. Kings, Vibrations and waves, A John Wiley and Sons, Ltd., Publication (009). 3. Rayond A. Serway, Physics: For Scientists & Engineers, 6th ed., Thoson Bruke, (004) 4. Hugh D. Young and Roger A. Freedann, University Physics with Modern Physics th ed., (008) 5. Douglas. Giancoli, Physics for scientists and engineers, Printice Hall, 4th, (005) 6. Robert Resnick and David Halliday, Fundaentals of Physics Extended, HRW 8 th ed., (008) 7. Physics lab for scientists and Engineers, Advanced Instructional Systes, Inc. and North arolina State University 3 rd edition (03) 8. ecture0.pdf 9. h3a.pdf. itation: Getahun Getachew ( 08). A Review of Siple Haronic Motion for Mass Spring Syste and Its Analogy to the Oscillations in L ircuit. j. of Physical and heical Sciences.V6I3-05. DOI: 0.58/zenodo opyright: 08 Getahun Getachew, This is an open-access article distributed under the ters of the reative oons Attribution License, which perits unrestricted use, distribution, and reproduction in any ediu, provided the original author and source are credited. 5