Physics
Physics
Experiment 10 Experiment 10 Okorie Esonu Okorie EsonuSIMPLE HARMONIC MOTION SIMPLE HARMONIC MOTION
Abstract
Abstract
In this lab we studied examples of
In this lab we studied examples of simple harmonic motion. Focusing on the springsimple harmonic motion. Focusing on the spring system
system
Introduction
Introduction
Simple harmonic motion is the motion of a
Simple harmonic motion is the motion of a simple harmonic oscillator, a motion that issimple harmonic oscillator, a motion that is neither driven nor damped. he motion
neither driven nor damped. he motion is periodic, as it repeats itself at standard intervalsis periodic, as it repeats itself at standard intervals in a specific manner with constant amplitude. It is characteri!ed by its amplitude "which in a specific manner with constant amplitude. It is characteri!ed by its amplitude "which is always positive#, its period which is the time for a single oscillation, its fre$uency is always positive#, its period which is the time for a single oscillation, its fre$uency which is the number of cycles p
which is the number of cycles per unit time, and its phase, which er unit time, and its phase, which determines the startingdetermines the starting point on the sine wave.
point on the sine wave. %e
%e notice in our everyday lives that notice in our everyday lives that a number of ob&ects oscillate about their e$uilibriuma number of ob&ects oscillate about their e$uilibrium positions. '
positions. ' swing goes back and forth, a chandelier oscillates about a mean position, aswing goes back and forth, a chandelier oscillates about a mean position, a weed swings back and
weed swings back and forth, and a plucked string oscillates about its e$uilibriumforth, and a plucked string oscillates about its e$uilibrium
position, and so on. he atoms and molecules in solids vibrate about their mean positions position, and so on. he atoms and molecules in solids vibrate about their mean positions
as well. Oscillating electric and
as well. Oscillating electric and magnetic fields result in electromagnetic magnetic fields result in electromagnetic radiation. 'llradiation. 'll these relate to simple harmonic motion oscillation.
these relate to simple harmonic motion oscillation.
In the lab that we did,
In the lab that we did, we made a simple harmonic oscillator by attaching we made a simple harmonic oscillator by attaching a mass to thea mass to the end of a spring and
end of a spring and then set it into motion. %hat happened( he mass executhen set it into motion. %hat happened( he mass execu tes repetitivetes repetitive motion, moving back and forth between two points.
motion, moving back and forth between two points. o descri
o describe this) *ooking at the system before be this) *ooking at the system before we set it into motion, we see the we set it into motion, we see the mass atmass at rest at a position known as its e$u
rest at a position known as its e$uilibrium position. If we tap on the mass while it is in itsilibrium position. If we tap on the mass while it is in its e$uilibrium position, the oscillations begin. In words, the mass first moves away from e$uilibrium position, the oscillations begin. In words, the mass first moves away from e$uilibrium in one direction "we+ll call that the positive direction#, reaches a maximum e$uilibrium in one direction "we+ll call that the positive direction#, reaches a maximum displacement from e$uilibrium where it changes its direction of motion "instantaneously displacement from e$uilibrium where it changes its direction of motion "instantaneously coming to rest#, speeds up as it moves back towards the e$uilibrium position "going in the coming to rest#, speeds up as it moves back towards the e$uilibrium position "going in the opposite direction compared to when we tapped it#, slows down as it passes the
opposite direction compared to when we tapped it#, slows down as it passes the e$uilibrium position until it reaches its maximum negative displacement "the same e$uilibrium position until it reaches its maximum negative displacement "the same
distance from the origin as the maximum positive displacement# and then heads back to the origin. %hat Ive described is one cycle of its oscillation. he oscillation cycles repeat.
*et me define a few terms related to oscillatory motion. he distance x "t# of the ob&ect from its e$uilibrium position is the displacement. he maximum displacement is called the amplitude. One oscillation cycle as Ive explained earlier corresponds to a complete to and fro motion of the ob&ect from some initial position returning to the same position moving in the same direction. he time it takes to complete one oscillation is called the time period "t#. he number of oscillations in a unit time "- second# is known as the fre$uency "f#. he fre$uency is measured in oscillations per second or simply hert! "h!#. If the ob&ect "or the field# is oscillating at regular intervals of time then it is called
periodic motion, which is also known as harmonic motion. If the periodic motion is
sinusoidal, it is called simple harmonic motion "shm#. In the following, we present d etails of the simple harmonic motion.
he time period - / f
%here f is the fre$uency
he displacement of a particle x "t# " x as a function of time# executing simple harmonic motion may be expressed as a sine or a cosine function of time t,
x"t# ' sin "0t 1 2#
where x "t# is the displacement, which is a function of time, ' is the amplitude, 0 is the angular fre$uency, t is the time, and 2 is the phase constant. he angular fre$uency 0 34f
has the units of rad/s. it turns out that most of the natural systems execute harmonic motion when they are perturbed from their e$uilibrium position.
he velocity
5"t# dx / dt ' 0 cos "0t 1 2# 'nd acceleration
'"t# 6'07 sin "0 t 1 2# 6 07 x "t#
his is a basic e$uation characteristic of the simple harmonic motion. he instantaneous acceleration a "t# is e$ual to the instantaneous displacement x "t# times the s$uare of the angular fre$uency "07# and is oppositely directed.
8onsider a mass m on a frictionless hori!ontal surface connected to a spring. If we stretched the spring by a small distance "x# from its unstretched po sition and release, it will execute oscillatory motion. If we further assume that the spring is mass less and ideal "internal frictional forces of the spring are negligible#, the spring exerts a force on the mass
F"x# 6kx "9ookes law#
%here x is the displacement of the mass from its e$uilibrium position, and k is the force constant of the spring. he value of k depends on the stiffness of the spring. he negative sign indicates that the force exerted by the spring is opposite in direction to the
displacement of the mass. %hen the mass is pulled, the wo rk done in stretching the spring is stored as potential energy of the spring. 's the mass is released, the spring force pulls the mass towards the unstretched position. %hen the mass reaches the unstretched position, all the potential energy of the spring is converted into the kinetic energy of the
mass. he kinetic energy of the mass is converted into the potential energy of the spring as the spring is compressed again.
Procedures
%e opened the motion detector software. For this part of the experiment, we did not set the mass m oscillating and we measured the elongation of the spring as a function of the applied "f#. if the mass of the hanger plus the weights is m"kg#, the force will be mg ":#. first we put some weights say m";# so that the spring will be straight and has no kinks. %e measured the positioned "x# of the weight hanger using the motion detector. Since the mass is stationary, we expected to get approximately hori!ontal line on the displacement versus time plot. %e added weights to the hanger and for each additional weight we measured the new position of the weight hanger using the motion detector.
In <art I of the experiment, two springs was used. One is a thick "larger in diameter# short spring "spring = 3#, and a thin "smaller in diameter# long spring "spring =3#. he shorter spring is stiffer than the longer spring. he following procedures were done for both springs. ' short spring was attached to a hanger, and hung vertically. he distance from the table to the bottom part of the spring, &ust above the hook is measured. his is the initial poison. >; grams of weight is attached to the hook, and the spring elongates. he distance form the table to the spring is measured, which is a shorter distance. his is done with more weight added, for ? more trials. his was also done for the long spring "spring = 3#. In both cases, the force due to the addition of weights was calculated. Fmg. he spring constant was calculated in each case, using 9ookes *aw.
In part 3 we suspended a mass m ";# to the spring such that there are no kinks in the
spring. %e held the mass m ";#, raised it upward vertically by a couple of cm, and release it. It had an oscillatory motion. @y keeping the motion detector as shown, we collected data for certain duration of time. ' motion detector was placed directly under spring = 3A the motion detector program was set up. >; grams of weight was added, and the spring began to oscillate. he program captured the spring oscillating on three graphs, distance
vs. time, velocity vs. time, and acceleration vs. time. For each graph, the e$uation was obtained
Discussion
<art
-In part - of the experiment, we attached weights to a vertically suspended spring and measured the elongation of the spring as a function of the force applied. From these data we calculated the force constant of the spring.
%e suspended the spring vertically and attached a mass m at its end. %e held the mass by hand and released it gradually at its e$uilibrium position. %e let the elongation of the spring from its mean position be x. since the mass m is in e$uilibrium, the net force acting on the mass was e$ual to !ero. he gravitational force acting on the mass "mg# was e$ual in magnitude and opposite to the restoring force "f# of the spring.
If the applied force is within the elastic limit of the spring, the restoring force of the spring f is proportional to the elongation "x# of the spring and is oppositely directed with respect to x. this is known as hookes law.
F 6 kx "hookes law#
%here f is the restoring force of the spring and x is the displacement of the spring from its e$uilibrium "unstretched# position. he proportionality constant k is called the spring constant. If f is in :ewton and x in meters, the spring constant k will be in :ewton per meter ":/m#
he e$uilibrium condition may be written as 1 Fspring B mg ;
<art 3
In part 3, we set the mass to oscillating mass. he oscillation of a mass attached to a
Employing such a setup we measured and studied a number of physical parameters related to the simple harmonic motion.
8onsidering a mass m attached to a spring suspended vertically. Since the weight of the mass was within the elastic limit of the spring, the spring was stretched and the mass reached an e$uilibrium condition. he forces acting on the mass were the gravitational force acting vertically downwards and the restoring force due to the spring acting in the opposite direction. %e let the e$uilibrium position be ', then we lifted the mass to the position b with our hand and gently released it, the mass executed oscillations. 't b, the
kinetic energy was at maximum. 't a, the kinetic energy of the mass was maximum whereas potential energy of the spring was e$ual to !ero. 't c, the kinetic energy of the mass was !ero, the potential energy of the spring was at maximum and the gravitational potential energy was negative with respect to the mean position. 't intermediate
positions, part of the energy was in the form of kinetic energy and part of the energy was in the form of potential energy of the spring and the gravitational potential energy.
he mass m was acted on by two forces) - the force due to gravity "mg# and the restoring force due to the spring. In this experiment, we lifted the mass vertically by a couple of cm and left it so that it executes oscillatory motion. 's the mass oscillates, the gravitational potential energy of the mass as well as the p otential energy of the spring changed as a function of time. 9owever, it can be shown that if we measured the
displacement from the e$uilibrium position of the mass after it was suspended from the spring, the motion was still S9C.
Conclusion
he data shows that the spring elongated as more weight was added, because the force acting on the spring increased. he spring constant is supposed to be the same, but the data shows the constant changing for spring = - by one "from D. to ?.# for the first and last trials. he spring constants for spring = 3 were closer. he last three values were 3;., 3;., and 3;. which are all about the same. he results for spring = - were
probably caused by experimental errors. here can be several errors that can skew the results of this experiment. hese sources of error can come from the spring being more elastic is some areas, or from human error. 9uman error would result from misreading the distance from the table to the spring before and after the weight was added. 8alculation errors can also affect the result of this experiment
Results
Table 1) Force 8onstant of Spring = -Gun = m; 'dditional %eights "m# Force due to additional weights Initial <osition of the Final <osition of the mass
"F# mass "x;# "xn# - ; kg ; kg ; : .H m .H m 3 ; kg .;>; kg .D- : .H m .H> m ? ; kg .-;; kg .- : .H m .3> m D ; kg .->; kg -.D3 : .H m .> m > ; kg .3;; kg -.H3 : .H m .DD m
Table 1) Force 8onstant of Spring = - "continued #
Gun = x "xn B x;# k F /x - ; :/' 3 .-;? m D. ? .3D? m D.;D D .?- m ?.H > .>3; m ?.
Table 2) Force 8onstant of spring = 3 Gun = m; 'dditional %eights "m# Force due to additional weights "F# Initial <osition of the mass "x;# Final <osition of the mass "xn# - ; kg ; kg ; : .?; m .?; m 3 ; kg .;>; kg .D- : .?; m .; m ? ; kg .-;; kg .- : .?; m .? m D ; kg .->; kg -.D3 : .?; m .> m > ; kg .3;; kg -.H3 : .?; m .?> m
Table 2) Force 8onstant of Spring = 3 "continued #
Gun = x "xn B x;# k F /x - ; :/' 3 .;33 m 33.? ? .;D m 3;. D .;- m 3;. > .;> m 3;.
Questions
-) 8onsidering the mass undergoing simple harmonic motion in the figure. he velocity of the particle can be calculated by differentiating the displacement. So that when the displacement is at a maximum the velocity is at a minimum and when the displacement is !ero "minimum# the velocity has its greatest value "maximum#.
Jifferentiating the velocity with respect to time we obtain the acceleration. he maximum acceleration occurs at the extreme displacement "maximum#.
3) Kes I expect them to have different phases. %hen the mass is at its highest point, the velocity is !ero as it changes direction and begins to fall back down. %hen it reaches its lowest position, it again slows and changes direction in the oscillatory cycle. herefore, the velocity curve should be out of phase with the position curve. %hen the position vs. time curve is at a maximum or minimum, the velocity curve will be crossing !ero, when the velocity is at its maximum, the position will be crossing !ero, or we can say that they are out of phase.
%hen the displacement is at its maximum, the restoring force and therefore the
acceleration will be maximum in the opposite direction. herefore, it is out of phase with the velocity and the position curve.
?) the functional relationship between the fre$uency f of the spring mass system and the mass m is that the fre$uency the mass on the spring is determined by the mass m and the stiffness of the spring expressed in terms of a spring constant k. 'lso a mass on a spring has a single resonant fre$uency determined by its spring constant k and the mass m. D) Ideally, if it is assumed that the spring has no mass, it will also be assumed that the restoring force of the spring is only used to move the attached mass, but in fact, part of the restoring force is used to move the spring back to its e$uilibrium position. 's a result, the mass of the spring cannot be neglected
>) frictional forces act to retard the motion. If the frictional force exceeded the restoring force, the LoscillatorL would never oscillateA when displaced by a small distance the frictional force would exceed the restoring force, and energy would stay stored in the stretched spring. or!s cited www.physicsforums.com www.utk.edu www.gmu.edu ww.splung.com www.indiana.edu
