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3/1/2017 Motion of a Mass on a Spring Motion of a Mass on a Spring Vibrational Motion Properties of Periodic Motion Pendulum Motion Motion of a Mass on a Spring In a previous part of this lesson, the motion of a mass attached to a spring was described as an example of a vibrating system. The mass on a spring motion was discussed in more detail as we sought to understand the mathematical properties of objects that are in pe
  3/1/2017 Motion of a Mass on a Spring 1/7   Motion of a Mass on a Spring  Vibrational MotionProperties of Periodic MotionPendulum MotionMotion of a Mass on a Spring In a previous part of this lesson, the motion of a mass attached to a spring was described as anexample of a vibrating system. The mass on a spring motion was discussed in more detail as we soughtto understand the mathematical properties of objects that are in periodic motion. Now we willinvestigate the motion of a mass on a spring in even greater detail as we focus on how a variety of quantities change over the course of time. Such quantities will include forces, position, velocityand energy - both kinetic and potential energy.   Hooke's Law We will begin our discussion with an investigation of the forces exerted by a springon a hanging mass. Consider the system shown at the right with a spring attachedto a support. The spring hangs in a relaxed, unstretched position. If you were tohold the bottom of the spring and pull downward, the spring would stretch. If youwere to pull with just a little force, the spring would stretch just a little bit. And if you were to pull with a much greater force, the spring would stretch a muchgreater extent. Exactly what is the quantitative relationship between the amount of pulling force and the amount of stretch?To determine this quantitative relationship between the amount of force and theamount of stretch, objects of known mass could be attached to the spring. For each object which isadded, the amount of stretch could be measured. The force which is applied in each instance would bethe weight of the object. A regression analysis of the force-stretch data could be performed in order todetermine the quantitative relationship between the force and the amount of stretch. The data tablebelow shows some representative data for such an experiment. Mass (kg) Force on Spring (N) Amount of Stretch (m) 0.000 0.000 0.00000.050 0.490 0.00210.100 0.980 0.00400.150 1.470 0.00630.200 1.960 0.00810.250 2.450 0.00990.300 2.940 0.01230.400 3.920 0.01600.500 4.900 0.0199 By plotting the force-stretch data and performing a linear regression analysis, the quantitativerelationship or equation can be determined. The plot is shown below.  3/1/2017 Motion of a Mass on a Spring 2/7    A linear regression analysis yields the following statistics:slope = 0.00406 m/Ny-intercept = 3.43 x10 (  󰁰󰁥󰁲󰁴     near close to 0.000)regression constant = 0.999The equation for this line is Stretch = 0.00406ãForce + 3.43x10 The fact that the regression constant is very close to 1.000 indicates that there isa 󰁳󰁴󰁲󰁯󰁮󰁧󰁦󰁩󰁴     between the equation and the data points. This 󰁳󰁴󰁲󰁯󰁮󰁧󰁦󰁩󰁴     lends credibility tothe results of the experiment.This relationship between the force applied to a spring and the amount of stretch wasfirst discovered in 1678 by English scientist Robert Hooke. As Hooke put it: U󰁴󰁴󰁥󰁮󰁳󰁩󰁯,󰁳󰁩󰁣󰁶󰁩󰁳    . Translatedfrom Latin, this means As the extension, so the force. In other words, the amount that the springextends is proportional to the amount of force with which it pulls. If we had completed this study about350 years ago (and if we knew some Latin), we would be famous! Today this quantitative relationshipbetween force and stretch is referred to as Hooke's law and is often reported in textbooks as F = -kãx where Fspring is the force exerted upon the spring,  x is the amount that the spring stretches relative toits relaxed position, and k is the proportionality constant, often referred to as the spring constant. Thespring constant is a positive constant whose value is dependent upon the spring which is being studied. A stiff spring would have a high spring constant. This is to say that it would take a relatively largeamount of force to cause a little displacement. The units on the spring constant are Newton/meter(N/m). The negative sign in the above equation is an indication that the direction that the springstretches is opposite the direction of the force which the spring exerts. For instance, when the springwas stretched below its relaxed position, x is 󰁤󰁯󰁷󰁮󰁷󰁡󰁲󰁤     . The spring responds to this stretching byexerting an 󰁵󰁰󰁷󰁡󰁲󰁤      force. The x and the F are in opposite directions. A final comment regarding thisequation is that it works for a spring which is stretched vertically and for a spring is stretchedhorizontally (such as the one to be discussed below).  Force Analysis of a Mass on a Spring Earlier in this lesson we learned that an object that is vibrating is acted upon by a restoring force. The restoring force causes the vibrating object to slow down as it moves away fromthe equilibrium position and to speed up as it approaches the equilibrium position. It is this restoring force which is responsible for the vibration. So what is the restoring force for a mass on a spring? We will begin our discussion of this question by considering the system in the diagram below. -5 -5spring  3/1/2017 Motion of a Mass on a Spring 3/7   The diagram shows an air track and a glider. The glider is attached by a spring to a vertical support.There is a negligible amount of friction between the glider and the air track. As such, there are threedominant forces acting upon the glider. These three forces are shown in the free-body diagram at theright. The force of gravity ( Fgrav ) is a rather predictable force - both in terms of its magnitude and itsdirection. The force of gravity always acts downward; its magnitude can be foundas the product of mass and the acceleration of gravity ( mã9.8 N/kg ). Thesupport force ( Fsupport ) balances the force of gravity. It is supplied by the airfrom the air track, causing the glider to 󰁬󰁥󰁶󰁩󰁴󰁡󰁴󰁥      about the track's surface. The finalforce is the spring force ( Fspring ). As discussed above, the spring force varies inmagnitude and in direction. Its magnitude can be found using Hooke's law. Itsdirection is always opposite the direction of stretch and towardsthe equilibrium position. As the air track glider does 󰁴󰁨󰁥󰁢󰁡󰁣󰁫󰁡󰁮󰁤󰁦󰁯󰁲󰁴󰁨     , the springforce ( Fspring ) acts as the restoring force. It acts leftward on the glider when it is positioned to theright of the equilibrium position; and it acts rightward on the glider when it is positioned to the left of the equilibrium position.Let's suppose that the glider is pulled to the right of the equilibrium position and released from rest. Thediagram below shows the direction of the spring force at five different positions over the course of theglider's path. As the glider moves from position A (the release point) to position B and then to positionC, the spring force acts leftward upon the leftward moving glider. As the glider approaches position C,the amount of stretch of the spring decreases and the spring force decreases, consistent with Hooke'sLaw. Despite this decrease in the spring force, there is still an acceleration caused by the restoringforce for the entire span from position A to position C. At position C, the glider has reached its maximumspeed. Once the glider passes to the left of position C, the spring force acts rightward. During this phaseof the glider's cycle, the spring is being compressed. The further past position C that the glider moves,the greater the amount of compression and the greater the spring force. This spring force acts asa restoring force, slowing the glider down as it moves from position C to position D to position E. By thetime the glider has reached position E, it has slowed down to a momentary rest position beforechanging its direction and heading back towards the equilibrium position. During the glider's motionfrom position E to position C, the amount that the spring is compressed decreases and the spring forcedecreases. There is still an acceleration for the entire distance from position E to position C. At positionC, the glider has reached its maximum speed. Now the glider begins to move to the right of point C. Asit does, the spring force acts leftward upon the rightward moving glider. This restoring force causes theglider to slow down during the entire path from position C to position D to position E.  3/1/2017 Motion of a Mass on a Spr ing 4/7     Sinusoidal Nature of the Motion of a Mass on a Spring Previously in this lesson, the variations in the position of a mass on a spring with respect to time were discussed. At that time, it was shown that the position of a mass on a spring varies with the sine of thetime. The discussion pertained to a mass that was vibrating up and down while suspended from thespring. The discussion would be just as applicable to our glider moving along the air track. If a motiondetector were placed at the right end of the air track to collect data for a position vs. time plot, the plotwould look like the plot below. Position A is the right-most position on the air track when the glider isclosest to the detector.The labeled positions in the diagram above are the same positions used in the discussion of restoringforce above. You might recall from that discussion that positions A and E were positions at which themass had a zero velocity. Position C was the equilibrium position and was the position of maximumspeed. If the same motion detector that collected position-time data were used to collect velocity-timedata, then the plotted data would look like the graph below.Observe that the velocity-time plot for the mass on a spring is also a sinusoidal shaped plot. The only difference between the position-time and the velocity-time plots is that one is shifted one-fourth of a
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