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7/30/2019 Final DOT (Expt. #1) .2
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SIMPLE HARMONIC MOTION
Simple harmonic motion can serve as a mathematical model of a variety of motions, such as the oscillation of a spring.
Additionally, other phenomena can be approximated by simple harmonic motion, including the motion of a simple pendulum and
molecular vibration.
Simple harmonic motion provides the basis of the characterization of more complicated motions through the techniques of
Fourier analysis.
Simple harmonic motion had shown both in real space and phase space. The orbit is periodic. (Here the velocity and
position axes have been reversed from the standard convention in order to align the two diagrams)
A simple harmonic oscillator is attached to the spring, and the other end of the spring is connected to a rigid support such
as a wall. If the system is left at rest at the equilibrium position then there is no net force acting on the mass. However, if the mass isdisplaced from the equilibrium position, a restoring elastic force which obeys Hooke's law is exerted by the spring.
Mathematically, the restoring force F is given by:
where F is the restoring elastic force exerted by the spring (in SI units: N), k
is the spring constant (Nm1
), and x is the displacement from the
equilibrium position (in m).
For any simple harmonic oscillator:
When the system is displaced from its equilibrium position, arestoring force which obeys Hooke's law tends to restore the system to
equilibrium.
Once the mass is displaced from its equilibrium position, itexperiences a net restoring force. As a result, it accelerates and starts
going back to the equilibrium position. When the mass moves closer to the
equilibrium position, the restoring force decreases. At the equilibrium
position, the net restoring force vanishes. However, at x= 0, the
momentum of the mass does not vanish due to the impulse of the restoring force that has acted on it. Therefore, the mass
continues past the equilibrium position, compressing the spring. A net restoring force then tends to slow it down, until its velocity
vanishes, whereby it will attempt to reach equilibrium position again. As long as the system has no energy loss, the mass will
continue to oscillate. Thus, simple harmonic motion is a type ofperiodic motion.
Dynamics of simple harmonic motion
For one-dimensional simple harmonic motion, the equation of motion, which is a second-order linear ordinary differential
equation with constant coefficients, could be obtained by means ofNewton's second law and Hooke's law.
where m is the inertial mass of the oscillating body, xis its displacement from the equilibrium (or mean) position, and kis the spring
constant.
Therefore,
Solving the differential equation above, a solution which is a sinusoidal function is obtained.
Where
In the solution, c1 and c2 are two constants determined by the initial conditions, and the origin is set to be the equilibrium
position.[A]
Each of these constants carries a physical meaning of the motion: A is the amplitude (maximum displacement from the
equilibrium position), = 2fis the angular frequency, and is the phase.[B]
http://en.wikipedia.org/wiki/Mathematical_modelhttp://en.wikipedia.org/wiki/Pendulumhttp://en.wikipedia.org/wiki/Molecular_vibrationhttp://en.wikipedia.org/wiki/Fourier_analysishttp://en.wikipedia.org/wiki/Phase_spacehttp://en.wikipedia.org/wiki/Orbit_(dynamics)http://en.wikipedia.org/wiki/Harmonic_oscillatorhttp://en.wikipedia.org/wiki/Mechanical_equilibriumhttp://en.wikipedia.org/wiki/Forcehttp://en.wikipedia.org/wiki/Elasticity_(physics)http://en.wikipedia.org/wiki/Hooke%27s_lawhttp://en.wikipedia.org/wiki/International_System_of_Unitshttp://en.wikipedia.org/wiki/Newton_(unit)http://en.wikipedia.org/wiki/Hooke%27s_lawhttp://en.wikipedia.org/wiki/Newton_(unit)http://en.wikipedia.org/wiki/Displacement_(vector)http://en.wikipedia.org/wiki/Accelerationhttp://en.wikipedia.org/wiki/Momentumhttp://en.wikipedia.org/wiki/Impulse_(physics)http://en.wikipedia.org/wiki/Velocityhttp://en.wikipedia.org/wiki/Energyhttp://en.wikipedia.org/wiki/Frequencyhttp://en.wikipedia.org/wiki/Ordinary_differential_equationhttp://en.wikipedia.org/wiki/Ordinary_differential_equationhttp://en.wikipedia.org/wiki/Newton%27s_second_lawhttp://en.wikipedia.org/wiki/Hooke%27s_lawhttp://en.wikipedia.org/wiki/Mass#Inertial_masshttp://en.wikipedia.org/wiki/Displacement_(vector)http://en.wikipedia.org/wiki/Mechanical_equilibriumhttp://en.wikipedia.org/wiki/Hooke%27s_law#The_spring_equationhttp://en.wikipedia.org/wiki/Hooke%27s_law#The_spring_equationhttp://en.wikipedia.org/wiki/Sine_wavehttp://en.wikipedia.org/wiki/Simple_harmonic_motion#cnote_Ahttp://en.wikipedia.org/wiki/Simple_harmonic_motion#cnote_Ahttp://en.wikipedia.org/wiki/Simple_harmonic_motion#cnote_Ahttp://en.wikipedia.org/wiki/Angular_frequencyhttp://en.wikipedia.org/wiki/Simple_harmonic_motion#cnote_Bhttp://en.wikipedia.org/wiki/Simple_harmonic_motion#cnote_Bhttp://en.wikipedia.org/wiki/Simple_harmonic_motion#cnote_Bhttp://en.wikipedia.org/wiki/File:Simple_Harmonic_Motion_Orbit.gifhttp://en.wikipedia.org/wiki/File:Simple_Harmonic_Motion_Orbit.gifhttp://en.wikipedia.org/wiki/File:Simple_Harmonic_Motion_Orbit.gifhttp://en.wikipedia.org/wiki/File:Simple_Harmonic_Motion_Orbit.gifhttp://en.wikipedia.org/wiki/File:Simple_Harmonic_Motion_Orbit.gifhttp://en.wikipedia.org/wiki/File:Simple_Harmonic_Motion_Orbit.gifhttp://en.wikipedia.org/wiki/File:Simple_Harmonic_Motion_Orbit.gifhttp://en.wikipedia.org/wiki/File:Simple_Harmonic_Motion_Orbit.gifhttp://en.wikipedia.org/wiki/Simple_harmonic_motion#cnote_Bhttp://en.wikipedia.org/wiki/Angular_frequencyhttp://en.wikipedia.org/wiki/Simple_harmonic_motion#cnote_Ahttp://en.wikipedia.org/wiki/Sine_wavehttp://en.wikipedia.org/wiki/Hooke%27s_law#The_spring_equationhttp://en.wikipedia.org/wiki/Hooke%27s_law#The_spring_equationhttp://en.wikipedia.org/wiki/Mechanical_equilibriumhttp://en.wikipedia.org/wiki/Displacement_(vector)http://en.wikipedia.org/wiki/Mass#Inertial_masshttp://en.wikipedia.org/wiki/Hooke%27s_lawhttp://en.wikipedia.org/wiki/Newton%27s_second_lawhttp://en.wikipedia.org/wiki/Ordinary_differential_equationhttp://en.wikipedia.org/wiki/Ordinary_differential_equationhttp://en.wikipedia.org/wiki/Frequencyhttp://en.wikipedia.org/wiki/Energyhttp://en.wikipedia.org/wiki/Velocityhttp://en.wikipedia.org/wiki/Impulse_(physics)http://en.wikipedia.org/wiki/Momentumhttp://en.wikipedia.org/wiki/Accelerationhttp://en.wikipedia.org/wiki/Displacement_(vector)http://en.wikipedia.org/wiki/Newton_(unit)http://en.wikipedia.org/wiki/Hooke%27s_lawhttp://en.wikipedia.org/wiki/Newton_(unit)http://en.wikipedia.org/wiki/International_System_of_Unitshttp://en.wikipedia.org/wiki/Hooke%27s_lawhttp://en.wikipedia.org/wiki/Elasticity_(physics)http://en.wikipedia.org/wiki/Forcehttp://en.wikipedia.org/wiki/Mechanical_equilibriumhttp://en.wikipedia.org/wiki/Harmonic_oscillatorhttp://en.wikipedia.org/wiki/Orbit_(dynamics)http://en.wikipedia.org/wiki/Phase_spacehttp://en.wikipedia.org/wiki/Fourier_analysishttp://en.wikipedia.org/wiki/Molecular_vibrationhttp://en.wikipedia.org/wiki/Pendulumhttp://en.wikipedia.org/wiki/Mathematical_model7/30/2019 Final DOT (Expt. #1) .2
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Using the techniques ofdifferential calculus, the velocity and acceleration as a function of time can be found:
Acceleration can also be expressed as a function of displacement:
Then since = 2f,
And since T= 1/fwhere T is the time period,
These equations demonstrate that the simple harmonic motion is isochronous (the period and frequency are independent of the
amplitude and the initial phase of the motion).
Energy of simple harmonic motion
The kinetic energy Kof the system at time tis
And the potential energy is
The total mechanical energy of the system therefore has the constant value
The following physical systems are some examples ofsimple harmonic oscillator.
Mass on a spring
(DRAWING?)
A mass m attached to a spring of spring constant kexhibits simple harmonic motion in space. The equation
Shows that the period of oscillation is independent of both the amplitude and gravitational acceleration.
Uniform circular motion
Simple harmonic motion can in some cases be considered to be the one-dimensional projection ofuniform circular motion.
If an object moves with angular velocity around a circle of radius rcentered at the origin of the x-yplane, then its motion along
each coordinate is simple harmonic motion with amplitude rand angular frequency .
Mass on a simple pendulum
In the small-angle approximation, the motion of a simple pendulum is approximated by simple harmonic motion. The
period of a mass attached to a string of length with gravitational acceleration g is given by
This shows that the period of oscillation is independent of the amplitude and mass of the pendulum but not the acceleration due to
gravity (g), therefore a pendulum of the same length on the Moon would swing more slowly due to the Moon's lower gravitational
acceleration.
This approximation is accurate only in small angles because of the expression for angular acceleration being proportional to
the sine of position:
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Where I is the moment of inertia. When is small, sin and therefore the expression becomes
This makes angular acceleration directly proportional to , satisfying the definition of simple harmonic motion.
Pendulum
A pendulum is a weight suspended from a pivot so that it can swing freely.[1]
When a pendulum is displaced from its resting
equilibrium position, it is subject to a restoring force due to gravity that will accelerate it back toward the equilibrium position.
When released, the restoring force combined with the pendulum's mass causes it to oscillate about the equilibrium position,
swinging back and forth. The time for one complete cycle, a left swing and a right swing, is called the period. A pendulum swings
with a specific period which depends (mainly) on its length.
From its discovery around 1602 by Galileo Galilei the regular motion of pendulums was used for timekeeping, and was the
world's most accurate timekeeping technology until the 1930s.[2]
Pendulums are used to regulate pendulum clocks, and are used in
scientific instruments such as accelerometers and seismometers. Historically they were used as gravimeters to measure the
acceleration of gravity in geophysical surveys, and even as a standard of length. The word 'pendulum' is new Latin, from the Latin
pendulus, meaning 'hanging'.[3]
The simple gravity pendulum is an idealized
mathematical model of a pendulum. This is a weight (or
bob) on the end of a massless cord suspended from a pivot,
without friction. When given an initial push, it will swing
back and forth at constant amplitude. Real pendulums are
subject to friction and air drag, so the amplitude of their
swings declines.
A simple pendulum is one which can be considered to be apoint mass suspended from a string or rod of negligible
mass. It is a resonant system with a single resonant
frequency. For small amplitudes, the period of such a
pendulum can be approximated by:
If the rod is not of negligible mass, then it must be treated as a physical pendulum
Pendulum Motion
The motion of a simple pendulum is like simple harmonic motion in that the equation for the angular displacement is
which is the same form as the motion of a mass on a spring:
The anglular frequency of the motion is then given by
compared to for a mass on a spring.
"Simple gravity
pendulum" model
assumes no friction or
air resistance.
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The frequency of the pendulum in Hz is given by
and the period of motion is then
Period of Simple Pendulum
A point mass hanging on a massless string is an idealized example of a simple pendulum. When displaced from its equilibrium point,
the restoring force which brings it back to the center is given by:
For small angles , we can use the approximation
in which case Newton's 2nd law takes the form
Even in this approximate case, the solution of the equation uses calculus and differential equations. The
differential equation is
and for small angles the solution is:
http://hyperphysics.phy-astr.gsu.edu/hbase/pend.html#c1http://hyperphysics.phy-astr.gsu.edu/hbase/torq.html#equihttp://hyperphysics.phy-astr.gsu.edu/hbase/newt.html#fmahttp://hyperphysics.phy-astr.gsu.edu/hbase/pend.html#c5http://hyperphysics.phy-astr.gsu.edu/hbase/pend.html#c5http://hyperphysics.phy-astr.gsu.edu/hbase/newt.html#fmahttp://hyperphysics.phy-astr.gsu.edu/hbase/torq.html#equihttp://hyperphysics.phy-astr.gsu.edu/hbase/pend.html#c1