Example Physics Essay

Example Physics EssayThe consummation of a Mass Spring SystemThe example of a intensity attached to the end of a leap out is a powerful cocksucker in physics due to the fact that it is analogous to many personal phenomena. To be able to use this example to elegantly describe otherwise much complex strategys it is crucial to first fully understand how this undecomposable system works itself.The durability acting on a fortune attached to a jump-start at a given model is given by (Serway, 2003 437).F= -kx (Eqn 1)This equation may be derived from due norths Second Law of interrogative, which states that the trace on a thr adept is proportional to the rate of change in whim.A hatful and spring system bed be described as a artless Harmonic Oscillator and at that place be some fundamental equations that g everywheren the motion of such a system (Serway, 2003 436).= (k/m) (Eqn 2)Equation twain shows how the angular oftenness (=2*frequency) of an object vacillate du e to it being fixed to a spring that is inversely proportional to the loudness of the object. k is a constant, cognise as the spring constant that is defined by the properties of the spring. k cease be easily de bournined experimentally for a given spring by changing the band attached to the spring and measuring the frequency. imputable to the periodic nature of such a system as that which discharge be described using equations unity and two, they are known to be in innocent harmonic motion.The motion of a particle oer clock is described effectively by a cosine drift (Serway, 2003 436 Hayek , 2003 562).x(t)=Acos(t+) (Eqn 3)Equation three shows how this motion tail end be mapped over time where A is the amplitude of the cycle per second, and is a term to correct the phase. This female genitals be plotted to show how the smokestack and spring system bequeath move over time.The figure preceding(prenominal) shows how a mass on a spring go away be rich person in a fri ctionless universe while obeying Newtons First Law of motion. A useful trait of the Simple Harmonic Oscillator is that the equations for the upper and the acceleration are easily derived from that of the invest equation (Serway, 2003 436).V(t)=-A.sin(t) (Eqn 4)a(t)= -A cos(t) (Eqn 5)For brevity the phase term () has been omitted from these two equations as it can be assumed that phase is the same. These two equations give valuable insight into the nature of as mass on a spring and how its amphetamine and acceleration is linked. The interesting thing to tint is that velocity is governed by a sine wave produce, yet acceleration is pendent on the cosine waveform. What this means is that when the particle on the spring has stripped-down velocity it will have maximum acceleration, it also means that when the mass is travelling at its maximum velocity it has minimum (possibly nil) acceleration.Realistically til now, or so situations where a simple harmonic oscillator may be appli ed will involve a resistive force of some description, such as friction in the case of a mass on a spring. The effect this has on the motion of the mass and the spring system can be impinge onn in the figure below.It is clear to see on the above figure that the presence of friction causes the amplitude of the cps to declension over time. This effect is known as damping. In a damped system that has no external force driving the oscillation itself, the rate at which the oscillation decreases is directly proportional to the resistive force being applied to it. The damping force is at its greatest when the particle is moving at its fastest velocity and at a minimum when the acceleration is at a maximum. on that bit are three types of damping in an oscillatory system, underdamping, over-damping and critically damped. Underdamped is where the amplitude of the decay envelope does not decay rapidly. Critically damped systems are the fastest to call back to equilibrium and will have a decay envelope that allows one oscillation over the entire damping period and will decay towards zero rapidly during this period. Over-damping occurs when there are no oscillations (as seen in critical damping) however there is an infinite time to return to equilibrium (Hayek , 2003 567). The equation which describes this damped oscillation is given byF= -kx-l dx/dt (Eqn 6)Here the original equation for the force is extended by a first order derived function term relating to the change in the velocity due to the damping constant l. Equation hexad is able to be expanded into a more useful form by applying Newtons Second Law, which gives(d x)/(dt )+2D_0 dx/dt+_0 x=0 (Eqn 7)This equation now contains a first and second order differential equation relating to the velocity and acceleration respectively of the particle. Equation seven looks to be much more complex than that of equation six, however it is now in a considerably more useful form as it allows to see equation six in terms of the angular frequency of the system. D is the damping ratio and is given by D= l/(2mk) , taking into bet the damping coefficient, the spring constant and the mass of the particle.To fully understand the motion of the mass and spring system there must be consideration of the energy within the system. This may be done with the help of some simple drawings. The blue lines indicate the spring and the solid red block with a blue border indicates the mass.From the figures it is possible to imagine stretching the spring, this means that there is a force acting on the mass and if it is held at this stretched point (x) the mass will have a emf energy U. should the mass be released from this point it will have a maximum velocity Vmax and a maximum Kinetic energy KEmax. The total energy in the system at any one point in time is the sum of the potential and kinetic energies.E(t)=KE+U (Eqn 8)E(t)=1/2 mv(t)+1/2 kx(t) (Eqn 9)By substituting the formulae for velocity and position (equations thr ee and four) into the energy equation it is possible to simplify this further.E(t)= 1/2 kA sin(t)+cos(t) (Eqn 10)E(t)=1/2 kA (Eqn 11)The elegance of this simple algebra is that for an oscillating mass on a spring the energy in the system at any given point in time is solely independent of time. If there are dissipative or driving affects occurring during the oscillations then the come of energy in the system will change, however for a close system this fact holds true.This equation occurs in many areas of physics, for much more complex systems than a single mass on a spring. These equations can be applied to pendulums, resonant electrical circuits (RLC circuits) (Mispelter, 2006 35) such as those utilize to detect Radio and TV signals, or even in quantum mechanics and the time independent Schrodinger equation, where it is found that a quantum harmonic oscillator, such as a particle in a potential well (Schrdinger, 1926 1054), is one of the few quantum mechanical problems that it is possible to find analytical answers for. If the Hamiltonian for such a system is examined it is shown that its structure is very similar to that of equation nine (Schrdinger, 1926 1057 Levitt, 2012 144).H = p /2m+1/2 m x (Eqn 12)p is the momentum operator that forms the kinetic half of the Hamiltonian and x is the position operator which calculates the potential part of the Hamiltonian (Schrdinger, 1926 1052). It is obvious that the simple classical physics still applies to this quantum system.These are some of the situations where this type of motion is observed and the table shows how the equations are manipulated to barrack the system under examination ( Hayek , 2003 562 Mispelter, 2006 38 Dirac, 1958 108 Boylsestad, 2010 871).In conclusion the mass and spring systems motion is elegantly described by some simple mathematics that can be manipulated to suit systems that have external forces acting on the motion of the system. The beauty of this is that the mathematics can then be applied to much more complex systems.ReferencesBoylestad, Robert. (2010) Intrductory locomote Analysis, 12th edition. Pearson.Dirac, P. A. M. (1958). The Principles of Quantum mechanics, 4th edition. Oxford University PressHayek, S. I. 2003. Mechanical Vibration and Damping. Encyclopaedia of apply Physics.Levitt, M. (2012). Spin Dynamics, Basics of Nuclear Magnetic Resonance 2nd edition. magic trick Wiley and Sons Ltd.Mispelter, J. (2006) NMR probeheads for biophysical and biomedical experiments theoretical principles practical guidelines. Imperial College Press.Schrdinger, E. (1926). An Undulatory Theory of the Mechanics of Atoms and Molecules. Phys. Rev. 28 (6) pp 1049-1070.Serway, R. A., Jewett, J. W., Serway, R. A. (n.d.). Physics for scientists and engineers, with modern physics. Belmont, CA Thomson-Brooks/Cole.