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Does law of conservation of energy really hold good in case of attractive forces that obey inverse square law? Charanjeet Singh Bansrao [email protected] Abstract It is shown that while studying the energy conversion in an approximated simple harmonic motion without damping of two masses of a beam balance an anomaly is observed. The total energy calculated out at equilibrium position where the kinetic energy of both masses is maximum is greater than the energy spent in setting the system in simple harmonic motion which is contradictory to the first law of thermodynamics. This anomaly is found only in case where conservative force is attractive in nature and follows the inverse square law such as gravity, attractive magnetic and electric forces. 1. Introduction Principle of conservation of energy is applied while studying every natural phenomenon. In the end of any process energy is found to remain constant. This law states that energy of an isolated system remains constant but it can change its forms. Simple harmonic motion is often considered for explaining this law where energy changes from potential energy to kinetic energy and vice versa but at any moment of the cycle the total energy remains constant. In case of LC (inductance and capacitor) circuits the energy stored in electric field is changed into energy stored in magnetic field and vice versa. Following is very brief explanation of the nature of different forces. (a) Gravity: Gravity is the force that is attractive in nature. It follows the inverse square law. (b) Restoring force: It is the force that makes a flexible body to attain its initial form. For instance a stretched or compressed spring is always ready to attain its initial state. Further this force can be studied as follows: (i) Repulsive restoring force: A compressed spring is an example of repulsive restoring force. It increases as the spring is compressed further and further and decreases as the spring is allowed to expand. Thus this force can be compared with repulsive electrical and magnetic forces. (ii) Attractive restoring force: A stretched spring exerts an attractive restoring force. This force decreases as the spring approaches its initial state or increases as it is stretched further and further. Thus this force cannot be compared with attractive gravitational, electrical and magnetic forces. (c) Magnetic force: Magnetic force of unlike poles is attractive and like poles is repulsive. It follows the inverse square law. (d) Electric force: Electric force of unlike charges is attractive whereas like charges repel each other. This force follows the inverse square law. 2. Energy conversion in different oscillators: (A) Oscillators where gravity is the acting force Now we will study energy conversion that takes place in different oscillators and show that an anomaly is found where attractive force that obeys inverse square law is acting like a driving force for oscillations. (a) Two masses at the ends of a massless beam balance: In our first configuration two equal masses are fitted at ends of massless beam balance. The pivot is in middle of the beam as shown in Fig 1. Thus beam is in neutral equilibrium. The centre of mass of individual sphere is 5 metres from the ground. If each mass is 1kg then total potential energy of both masses is 10g where g is acceleration due to gravity. Push the left mass upwards up to height of 10 meters as shown in Fig2. The potential energy of lifted mass is therefore 10g whereas the potential energy of mass touching the ground is 0. Now leave the beam. The higher mass moves down whereas lower mass moves up. Both masses attain maximum kinetic energy at mean position of 5 meters. The right mass will cross this point and reach the height of ten meters. These oscillations will keep on going forever under ideal conditions. Let us consider the mean position where kinetic energy of both masses is maximum as shown in Fig 3. Thus the total energy of the system is as follows if we ignore the signs of angular velocities of both the masses---E = ½1w2 + ½1w2 + 5g +5g = w2 + 10g ( 1) Where ‘w’ is the angular velocity of masses. Suppose these masses are suddenly stopped at the mean position by any means. Which means their kinetic energy is lost in some way and they come to rest suddenly at this position. Therefore the energy of the system according to equation 1 at this position after losing the kinetic energy is 10g which is purely potential energy and thus these masses establish the neutral equilibrium at this position as shown in Fig 1. Now the question arises— “Can we use this 10g energy for restarting the cycle or not?” The answer to the above question is obviously ‘yes’ according to the law of conservation of energy. Let us join both masses and use their combined potential energy for generating electricity or for running a fly wheel. After this operation both masses are on ground. The generated electricity or rotational energy of fly wheel can be used to lift one of the masses up to height of 10 meters and other mass remains on the ground. Connect both masses with beam balance as shown in Fig 2. Restart the cycle. In this way we can extract w2 energy according to equation 1 in each cycle for infinite time period. (B) Oscillators where restoring force is acting force 1. A mass attached to two springs: A mass is attached to two massless springs on a horizontal plane. Three cases of neutral equilibrium can be considered. (a) When both springs are in original state: In this case both springs are neither compressed nor stretched. Thus they do not have any initial potential energy. When mass is moved one spring is stretched and another is compressed. Thus if we set the system in oscillations, no inconsistency is found at the equilibrium position or at any other position. Hence the oscillator works according to law of conservation of energy. At equilibrium position the kinetic energy of mass will be the equal to the potential energy of both springs at extreme position. (b) When both springs are in stretched condition: In this case springs are exerting attractive restoring force as shown in Fig 4. Thus they have some initial potential energy. When mass is moved as shown in Fig 5 one spring is stretched even further and its displacement is 2x whereas another restores its original state. Suppose at extreme position right spring does not have any potential energy. Then total energy of the system is the potential energy of stretched spring. E = ½k(2x)2 = 2kx2 (2) When mass is released it gains kinetic energy and at equilibrium position its kinetic energy is maximum and both the springs establish initial stretch. Total energy at equilibrium is E = ½mv2 + ½kx2 +1/2kx2 The combined potential energy of both springs--E = ½kx2 +1/2kx2 = kx2 This combined potential energy is less than the potential energy of stretched spring in equation 2. Therefore we cannot stretch the spring up to extreme position by the use of combined potential energy of both springs only. The kinetic energy of mass has to be used. Thus there is no surplus energy left. Hence this system also obeys law of conservation of energy. ( c )When both springs are in compressed condition: In this case springs exert a repulsive restoring force on mass. The mathematical argument for this system is same as in the previous case. Thus this system also obeys the law of conservation of energy. From this mass-spring system we can conclude that attractive restoring force that increases with the displacement does not violate law of energy conservation. Neither the repulsive restoring force that imitates magnetic and electric repulsive forces violates the law. This result can be generalised to any other mass spring system. ( C ) Oscillators where electric and magnetic forces are acting forces: Consider two capacitors of equal capacitance C connected to an inductor as shown in fig 6. Capacitor C1 is charged at some voltage V whereas capacitor C2 is not charged. Thus the energy of the system is: E = ½ CV2 When switch is dropped the charge flows from C1 to C2 through inductor L. When both capacitors are in equilibrium their individual voltage is V/2 and magnetic flux in inductor is maximum. The total energy of the system in equilibrium is: E = ½ C(V/2) 2 + ½ C(V/2)2 + 1/2 Li 2 It is obvious from the above equation that combined electrical energy of both capacitors is not enough in order to charge a single capacitor up to voltage V. The energy stored in the magnetic field of the inductor has to be used. Thus this system also obeys law of conservation of energy. This system can be compared with the mass spring system where both springs are in compressed condition. In this system capacitors are like springs and inductor is like mass that produces inertia in the electric current. 3. Conclusion: From this whole discussion we can draw the following conclusions: 1. In case of attractive forces that obey inverse square law, the law of conservation of energy does not hold good in oscillators where the force on the oscillating body is acting at its ends such as the beam balance attached to two masses at its ends. We can use magnets instead of masses where the furculum is fixed to a magnet with the opposite pole faced towards the hanging magnets as shown in Fig 7. In case of simple pendulums no anomaly is found. 2. The law of conservation of energy holds good in case of repulsive forces. 3. The law of conservation of energy holds good in case of restoring force. 4. In case where two forces are acting such as gravity and restoring force on hanging mass-spring system. In this case law of conservation of energy holds good because one of the force is restoring force. 4. Future outlook: (a) Perpetual motion: When there is violation of any of the laws of thermodynamics, the first thing that comes into mind is the possibility of perpetual motion. We know that this paper is not enough to make strong generalizations but according to reasoning given by considering the beam balance system there is possibility of perpetual motion of first kind. (b) Reconsideration of conservative forces: The anomaly found in case of beam balance system asks for the reconsideration of conservative forces that are attractive and obey inverse square law. For instance:- Gravity, attractive magnetic and electric forces. References: (i) K. Samson : “Classical Mechanics” ( 2006) ISBN 81-7625697-8; pp 57-64, 100-110,122-135 (ii) Li Shi-Song , Lan Jiang Han Bing, Tan Hong, Li ZhengKun: “Nonlinearity and periodic solution of a standardbeam balance oscillation system” Chin. Phys. B Vol. 21, No. 6 (2012) 064601