a spring of k= 25 n/m was stretched from 5cm to 10cm what is the energy stored in the spring
Motion of a Mass on a Spring
In a previous part of this lesson, the motion of a mass fastened to a spring was described as an instance of a vibrating arrangement. The mass on a spring motility was discussed in more detail as we sought to empathize the mathematical properties of objects that are in periodic motion. Now we volition investigate the movement of a mass on a spring in even greater item every bit nosotros focus on how a variety of quantities change over the class of time. Such quantities will include forces, position, velocity and energy - both kinetic and potential energy. We will begin our discussion with an investigation of the forces exerted by a jump on a hanging mass. Consider the system shown at the right with a spring attached to a support. The jump hangs in a relaxed, unstretched position. If y'all were to hold the lesser of the spring and pull downward, the spring would stretch. If you lot were to pull with just a petty force, the leap would stretch simply a petty flake. And if you were to pull with a much greater force, the leap would stretch a much greater extent. Exactly what is the quantitative relationship betwixt the amount of pulling forcefulness and the amount of stretch? To determine this quantitative relationship between the amount of force and the amount of stretch, objects of known mass could be attached to the spring. For each object which is added, the corporeality of stretch could exist measured. The strength which is applied in each instance would be the weight of the object. A regression analysis of the strength-stretch data could be performed in club to determine the quantitative relationship between the forcefulness and the amount of stretch. The information table below shows some representative data for such an experiment. Mass (kg) Forcefulness on Spring (N) Amount of Stretch (m) 0.000 0.000 0.0000 0.050 0.490 0.0021 0.100 0.980 0.0040 0.150 1.470 0.0063 0.200 1.960 0.0081 0.250 ii.450 0.0099 0.300 ii.940 0.0123 0.400 3.920 0.0160 0.500 4.900 0.0199 Past plotting the strength-stretch information and performing a linear regression assay, the quantitative relationship or equation can be determined. The plot is shown below. A linear regression analysis yields the following statistics: slope = 0.00406 1000/N The equation for this line is Stretch = 0.00406•Forcefulness + iii.43x10-five The fact that the regression constant is very close to 1.000 indicates that at that place is a potent fit between the equation and the data points. This strong fit lends credibility to the results of the experiment. This relationship between the strength applied to a spring and the corporeality of stretch was first discovered in 1678 by English scientist Robert Hooke. As Hooke put it: Ut tensio, sic vis. Translated from Latin, this means "Every bit the extension, so the forcefulness." In other words, the corporeality that the jump extends is proportional to the amount of force with which information technology pulls. If we had completed this study about 350 years ago (and if we knew some Latin), we would be famous! Today this quantitative human relationship betwixt force and stretch is referred to as Hooke's law and is ofttimes reported in textbooks as Fbound = -k•x where Fspring is the force exerted upon the leap, x is the corporeality that the spring stretches relative to its relaxed position, and k is the proportionality abiding, oftentimes referred to as the spring constant. The spring constant is a positive constant whose value is dependent upon the spring which is being studied. A stiff jump would have a high spring abiding. This is to say that it would take a relatively large amount of force to crusade a niggling deportation. The units on the jump constant are Newton/meter (Due north/m). The negative sign in the above equation is an indication that the direction that the bound stretches is opposite the management of the force which the spring exerts. For instance, when the spring was stretched beneath its relaxed position, x is downward. The spring responds to this stretching past exerting an up force. The x and the F are in reverse directions. A last comment regarding this equation is that it works for a spring which is stretched vertically and for a leap is stretched horizontally (such as the one to be discussed below). Earlier in this lesson nosotros learned that an object that is vibrating is acted upon by a restoring force. The restoring force causes the vibrating object to boring downwardly as information technology moves away from the 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 strength for a mass on a bound? We will begin our discussion of this question past considering the organization in the diagram below. The diagram shows an air track and a glider. The glider is attached by a jump to a vertical back up. At that place is a negligible amount of friction between the glider and the air track. Every bit such, there are three ascendant forces acting upon the glider. These iii forces are shown in the free-body diagram at the right. The force of gravity (Fgrav) is a rather predictable force - both in terms of its magnitude and its management. The force of gravity Let's suppose that the glider is pulled to the correct of the equilibrium position and released from rest. The diagram below shows the direction of the bound forcefulness at five different positions over the class of the glider's path. Every bit the glider moves from position A (the release indicate) to position B so to position C, the bound 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 forcefulness decreases, consistent with Hooke's Constabulary. Despite this decrease in the spring strength, there is still an acceleration caused by the restoring force for the entire span from position A to position C. At position C, the glider has reached its maximum speed. In one case the glider passes to the left of position C, the spring strength acts rightward. During this phase of the glider'south wheel, the spring is existence compressed. The farther past position C that the glider moves, the greater the amount of compression and the greater the leap forcefulness. This bound force acts every bit a restoring strength, slowing the glider downwards as it moves from position C to position D to position Eastward. Past the time the glider has reached position E, it has slowed down to a momentary rest position before irresolute its direction and heading dorsum towards the equilibrium position. During the glider's motion from position E to position C, the amount that the spring is compressed decreases and the spring force decreases. There is still an dispatch for the unabridged distance from position E to position C. At position C, the glider has reached its maximum speed. At present the glider begins to move to the right of point C. As it does, the jump forcefulness acts leftward upon the rightward moving glider. This restoring strength causes the glider to slow downwardly during the entire path from position C to position D to position East. Previously in this lesson, the variations in the position of a mass on a spring with respect to time were discussed. At that time, information technology was shown that the position of a mass on a spring varies with the sine of the time. The discussion pertained to a mass that was vibrating upwards and down while suspended from the spring. The discussion would be just every bit applicative to our glider moving along the air track. If a motion detector were placed at the right end of the air track to collect information for a position vs. time plot, the plot would look like the plot below. Position A is the right-about position on the air rails when the glider is closest to the detector. The labeled positions in the diagram above are the same positions used in the discussion of restoring strength above. You might call back from that give-and-take that positions A and E were positions at which the mass had a zero velocity. Position C was the equilibrium position and was the position of maximum speed. If the same motion detector that collected position-time data were used to collect velocity-time information, then the plotted data would look similar the graph below. Observe that the velocity-time plot for the mass on a leap is too a sinusoidal shaped plot. The only difference betwixt the position-time and the velocity-time plots is that one is shifted ane-quaternary of a vibrational cycle away from the other. Also detect in the plots that the absolute value of the velocity is greatest at position C (respective to the equilibrium position). The velocity of any moving object, whether vibrating or not, is the speed with a direction. The magnitude of the velocity is the speed. The direction is frequently expressed equally a positive or a negative sign. In some instances, the velocity has a negative management (the glider is moving leftward) and its velocity is plotted below the time axis. In other cases, the velocity has a positive management (the glider is moving rightward) and its velocity is plotted in a higher place the fourth dimension centrality. You will also notice that the velocity is zero whenever the position is at an extreme. This occurs at positions A and Due east when the glider is beginning to change direction. And so just as in the instance of pendulum motion, the speed is greatest when the displacement of the mass relative to its equilibrium position is the least. And the speed is to the lowest degree when the displacement of the mass relative to its equilibrium position is the greatest. On the previous page, an energy analysis for the vibration of a pendulum was discussed. Here we will conduct a like analysis for the motion of a mass on a spring. In our give-and-take, we will refer to the motion of the frictionless glider on the air runway that was introduced above. The glider will be pulled to the right of its equilibrium position and be released from residuum (position A). As mentioned, the glider then accelerates towards position C (the equilibrium position). Once the glider passes the equilibrium position, it begins to irksome down equally the spring strength pulls it backwards against its motion. By the fourth dimension it has reached position East, the glider has slowed downwards to a momentary suspension before irresolute directions and accelerating dorsum towards position C. In one case once again, after the glider passes position C, it begins to slow down as it approaches position A. Once at position A, the cycle begins all over once more ... and again ... and again. The kinetic energy possessed past an object is the free energy it possesses due to its move. It is a quantity that depends upon both mass and speed. The equation that relates kinetic free energy (KE) to mass (grand) and speed (5) is KE = ½•1000•vii The faster an object moves, the more than kinetic free energy that it will possess. We tin combine this concept with the discussion higher up about how speed changes during the course of move. This blending of the concepts would pb usa to conclude that the kinetic energy of the mass on the bound increases as information technology approaches the equilibrium position; and information technology decreases as it moves away from the equilibrium position. This information is summarized in the table below: Phase of Bicycle Change in Speed Change in Kinetic Energy A to B to C Increasing Increasing C to D to Eastward Decreasing Decreasing Due east to D to C Increasing Increasing C to B to A Decreasing Decreasing Kinetic energy is but i grade of mechanical energy. The other form is potential energy. Potential energy is the stored energy of position possessed by an object. The potential free energy could be gravitational potential energy, in which case the position refers to the summit above the basis. Or the potential energy could be elastic potential energy, in which instance the position refers to the position of the mass on the bound relative to the equilibrium position. For our vibrating air track glider, in that location is no change in height. And then the gravitational potential energy does not change. This form of potential energy is not of much involvement in our analysis of the energy changes. In that location is all the same a alter in the position of the mass relative to its equilibrium position. Every time the spring is compressed or stretched relative to its relaxed position, there is an increase in the elastic potential free energy. The corporeality of elastic potential energy depends on the amount of stretch or compression of the spring. The equation that relates the amount of rubberband potential energy (PEspring) to the amount of compression or stretch (x) is PEspring = ½ • one thousand•102 where thousand is the spring constant (in N/thousand) and x is the distance that the bound is stretched or compressed relative to the relaxed, unstretched position. When the air rail glider is at its equilibrium position (position C), it is moving it's fastest (every bit discussed above). At this position, the value of x is 0 meter. And so the amount of rubberband potential free energy (PEspring) is 0 Joules. This is the position where the potential energy is the to the lowest degree. When the glider is at position A, the jump is stretched the greatest distance and the rubberband potential energy is a maximum. A similar statement can exist made for position East. At position E, the spring is compressed the most and the elastic potential energy at this location is as well a maximum. Since the spring stretches as much equally compresses, the rubberband potential energy at position A (the stretched position) is the same as at position E (the compressed position). At these two positions - A and E - the velocity is 0 m/s and the kinetic energy is 0 J. Then just like the instance of a vibrating pendulum, a vibrating mass on a spring has the greatest potential free energy when it has the smallest kinetic energy. And it also has the smallest potential energy (position C) when it has the greatest kinetic energy. These principles are shown in the animation below. When conducting an free energy analysis, a common representation is an energy bar chart. An energy bar chart uses a bar graph to represent the relative amount and form of energy possessed by an object as it is moving. It is a useful conceptual tool for showing what form of energy is present and how it changes over the course of time. The diagram below is an energy bar chart for the air track glider and leap system. The bar chart reveals that as the mass on the spring moves from A to B to C, the kinetic energy increases and the elastic potential energy decreases. Yet the total amount of these ii forms of mechanical energy remains constant. Mechanical free energy is being transformed from potential form to kinetic grade; yet the full amount is existence conserved. A similar conservation of energy phenomenon occurs as the mass moves from C to D to E. As the bound becomes compressed and the mass slows downwards, its kinetic energy is transformed into elastic potential energy. As this transformation occurs, the total amount of mechanical energy is conserved. This very principle of free energy conservation was explained in a previous chapter - the Energy chapter - of The Physics Classroom Tutorial. Equally is likely obvious, not all springs are created equal. And not all spring-mass systems are created equal. One measurable quantity that can be used to distinguish one spring-mass system from some other is the menstruum. As discussed earlier in this lesson, the period is the time for a vibrating object to make 1 complete wheel of vibration. The variables that effect the period of a jump-mass system are the mass and the spring abiding. The equation that relates these variables resembles the equation for the menstruation of a pendulum. The equation is T = ii•Π•(m/k).v where T is the period, m is the mass of the object attached to the spring, and g is the spring constant of the spring. The equation tin can be interpreted to mean that more massive objects will vibrate with a longer menses. Their greater inertia means that it takes more than time to consummate a cycle. And springs with a greater spring constant (stiffer springs) take a smaller period; masses attached to these springs take less time to complete a cycle. Their greater spring constant means they exert stronger restoring forces upon the attached mass. This greater force reduces the length of time to complete i cycle of vibration. As nosotros have seen in this lesson, vibrating objects are wiggling in place. They oscillate dorsum and along about a fixed position. A simple pendulum and a mass on a jump are classic examples of such vibrating motion. Though not axiomatic by elementary observation, the use of motion detectors reveals that the vibrations of these objects have a sinusoidal nature. In that location is a subtle moving ridge-like behavior associated with the mode in which the position and the velocity vary with respect to time. In the next lesson, we will investigate waves. As we will soon discover out, if a mass on a bound is a jerk in time, then a moving ridge is a drove of wigglers spread through space. As we begin our study of waves in Lesson 2, concepts of frequency, wavelength and amplitude will remain important. 1. A strength of 16 N is required to stretch a leap a distance of 40 cm from its remainder position. What force (in Newtons) is required to stretch the same leap … a. … twice the altitude? 2. Perpetually disturbed by the habit of the lawn squirrels to raid his bird feeders, Mr. H decides to use a little physics for amend living. His current plot involves equipping his bird feeder with a spring organisation that stretches and oscillates when the mass of a squirrel lands on the feeder. He wishes to take the highest aamplitude of vibration that is possible. Should he employ a spring with a large spring abiding or a pocket-size bound constant? 3. Referring to the previous question. If Mr. H wishes to accept his bird feeder (and attached squirrel) vibrate with the highest possible frequency, should he use a spring with a large spring abiding or a small spring constant? 4. Use energy conservation to fill in the blanks in the following diagram. five. Which of the following mass-bound systems will have the highest frequency of vibration? Case A: A bound with a g=300 N/g and a mass of 200 g suspended from it. vi. Which of the following mass-spring systems volition have the highest frequency of vibration? Case A: A spring with a k=300 N/m and a mass of 200 g suspended from it.
y-intercept = 3.43 x10-5 (pert near shut to 0.000)
regression constant = 0.999
Force Analysis of a Mass on a Leap
Sinusoidal Nature of the Motion of a Mass on a Bound
Free energy Assay of a Mass on a Spring
Period of a Mass on a Leap
Looking Forward to Lesson ii
Nosotros Would Like to Suggest ...
Why but read about it and when you could exist interacting with information technology? Interact - that's exactly what you lot exercise when you use ane of The Physics Classroom's Interactives. Nosotros would similar to advise that you combine the reading of this page with the apply of our Mass on a Jump Interactive. You can find it in the Physics Interactives section of our website. The Mass on a Spring Interactive provides the learner with a unproblematic surround for exploring the effect of mass, bound abiding and duration of motion upon the period and amplitude of a vertically-vibrating mass. Check Your Agreement
b. … iii times the distance?
c. … one-one-half the altitude?
Case B: A spring with a thou=400 North/one thousand and a mass of 200 g suspended from it.
Instance B: A spring with a k=300 North/1000 and a mass of 100 g suspended from it.
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Source: https://www.physicsclassroom.com/class/waves/Lesson-0/Motion-of-a-Mass-on-a-Spring
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