Simple Harmonic Motion
Short Description
Physics Lab...
Description
Physics Experiment 10 Okorie Esonu
SIMPLE HARMONIC MOTION
Abstract In this lab we studied examples of simple harmonic motion. Focusing on the spring system
Introduction Simple harmonic motion is the motion of a simple harmonic oscillator, a motion that is neither driven nor damped. he motion is periodic, as it repeats itself at standard intervals 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$uency which is the number of cycles per p er unit time, and its phase, which determines the starting point on the sine wave. %e notice in our everyday lives that a number of ob&ects oscillate about their e$uilibrium positions. ' swing goes back and forth, a chandelier oscillates about a mean position, a weed swings back and forth, and a plucked string oscillates about its e$uilibrium position, and so on. he atoms and molecules in solids vibrate about their mean positions as well. Oscillating electric and magnetic fields result in electromagnetic radiation. 'll these relate to simple harmonic motion oscillation.
In the lab that we did, we made a simple harmonic oscillator by attaching a mass to the end of a spring and then set it into motion. %hat happened( he mass execu tes repetitive motion, moving back and forth between two points. o describe describe this) *ooking at the system before we set it into motion, we see the mass at rest at a position known as its e$uilibrium e$u ilibrium position. If we tap on the mass while it is in its e$uilibrium position, the oscillations begin. In words, the mass first moves away from e$uilibrium in one direction "we+ll call that the positive direction#, reaches a maximum displacement from e$uilibrium where it changes its direction of motion "instantaneously coming to rest#, speeds up as it moves back towards the e$uilibrium position "going in the opposite direction compared to when we tapped it#, slows down as it passes the e$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 o f 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 ; 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$uation was obtained
Discussion
; kg ; kg ; kg ; kg ; kg
; kg .;>; kg .-;; kg .->; kg .3;; kg
Force due to additional weights "F# ;: .D- : .- : -.D3 : -.H3 :
Table 2) Force 8onstant of Spring = 3 "continued # Gun = x "xn B x;# kF/x ; :/' 3 .;33 m 33.? ? .;D m 3;. D .;- m 3;. > .;> m 3;.
Questions -) 8onsidering the mass undergoing simple harmonic motion in the figure. he velocity of the particle can be calculated by differentiating the displacement. So that when the displacement is at a maximum the velocity is at a minimum and when the displacement is !ero "minimum# the velocity has its greatest value "maximum#.
Jifferentiating the velocity with respect to time we obtain the acceleration. he maximum acceleration occurs at the extreme displacement "maximum#.
3) Kes I expect them to have different phases. %hen the mass is at its highest point, the velocity is !ero as it changes direction and begins to fall back down. %hen it reaches its lowest position, it again slows and changes d irection in the oscillatory cycle. herefore, the velocity curve should be out of phase with the position curve. %hen the position vs. time curve is at a maximum or minimum, the velocity curve will be crossing !ero, when the velocity is at its maximum, the position will be crossing !ero, or we can say that they are out of phase. %hen the displacement is at its maximum, the restoring force and therefore the acceleration will be maximum in the opposite direction. herefore, it is out of phase with the velocity and the position curve. ?) the functional relationship between the fre$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
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