Download Does law of conservation of energy really hold good in

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