How To Find Spring Constant With Mass And Distance

Motion of a Mass on a Spring

In a previous part of this lesson, the motility of a mass fastened to a spring was described as an example of a vibrating organisation. The mass on a leap motility was discussed in more than particular equally we sought to understand the mathematical properties of objects that are in periodic move. Now we will investigate the movement of a mass on a spring in even greater detail as nosotros focus on how a variety of quantities change over the grade of fourth dimension. Such quantities volition include forces, position, velocity and energy - both kinetic and potential energy.

Hooke'southward Law

We volition begin our give-and-take with an investigation of the forces exerted by a spring on a hanging mass. Consider the arrangement shown at the correct with a spring attached to a support. The spring hangs in a relaxed, unstretched position. If you lot were to hold the bottom of the spring and pull downwards, the bound would stretch. If y'all were to pull with but a little forcefulness, the spring would stretch just a piffling bit. And if you were to pull with a much greater force, the bound would stretch a much greater extent. Exactly what is the quantitative human relationship between the amount of pulling force and the amount of stretch?

To decide this quantitative relationship between the amount of strength and the amount of stretch, objects of known mass could be attached to the leap. For each object which is added, the amount of stretch could be measured. The force 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 order to determine the quantitative human relationship between the force and the amount of stretch. The data tabular array below shows some representative information for such an experiment.

Mass (kg)	Forcefulness on Spring (N)	Corporeality 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	2.450	0.0099
0.300	two.940	0.0123
0.400	3.920	0.0160
0.500	four.900	0.0199

Past plotting the force-stretch information and performing a linear regression analysis, the quantitative relationship or equation tin exist determined. The plot is shown below.

A linear regression analysis yields the following statistics:

slope = 0.00406 m/N
y-intercept = iii.43 x10^-5 (pert almost shut to 0.000)
regression constant = 0.999

The equation for this line is

Stretch = 0.00406•Force + 3.43x10^-5

The fact that the regression abiding is very close to 1.000 indicates that there is a stiff fit between the equation and the data points. This strong fit lends credibility to the results of the experiment.

This human relationship between the force applied to a spring and the amount of stretch was first discovered in 1678 by English scientist Robert Hooke. As Hooke put information technology: Ut tensio, sic vis. Translated from Latin, this means "Equally the extension, and then the strength." In other words, the corporeality that the spring extends is proportional to the corporeality of force with which information technology pulls. If we had completed this study about 350 years ago (and if we knew some Latin), nosotros would be famous! Today this quantitative human relationship between force and stretch is referred to as Hooke'due south law and is often reported in textbooks as

F_spring = -1000•x

where Fspring is the forcefulness exerted upon the leap, x is the corporeality that the bound stretches relative to its relaxed position, and k is the proportionality constant, ofttimes referred to equally the spring abiding. The spring constant is a positive constant whose value is dependent upon the spring which is beingness studied. A potent jump would accept a loftier spring constant. This is to say that information technology would take a relatively large corporeality of force to cause a picayune displacement. The units on the bound constant are Newton/meter (Due north/m). The negative sign in the in a higher place equation is an indication that the direction that the bound stretches is reverse the direction of the force which the spring exerts. For example, when the leap was stretched beneath its relaxed position, 10 is downwardly. The spring responds to this stretching by exerting an upward strength. The ten and the F are in opposite directions. A final comment regarding this equation is that information technology works for a spring which is stretched vertically and for a spring is stretched horizontally (such every bit the ane 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 past a restoring force. The restoring strength causes the vibrating object to slow down every bit it moves away from the equilibrium position and to speed upwards as it approaches the equilibrium position. It is this restoring forcefulness which is responsible for the vibration. And so what is the restoring forcefulness for a mass on a spring?

We volition begin our discussion of this question by considering the system in the diagram beneath.

The diagram shows an air track and a glider. The glider is fastened by a spring to a vertical back up. In that location is a negligible amount of friction between the glider and the air track. As such, there are three dominant forces interim upon the glider. These iii forces are shown in the complimentary-body diagram at the right. The force of gravity (Fgrav) is a rather anticipated strength - both in terms of its magnitude and its direction. The force of gravity always acts downward; its magnitude can exist establish every bit the product of mass and the dispatch of gravity (m•9.8 North/kg). The back up force (Fsupport) balances the force of gravity. Information technology is supplied past the air from the air rails, causing the glider to levitate virtually the track's surface. The final force is the spring strength (Fspring). As discussed above, the leap force varies in magnitude and in direction. Its magnitude can be found using Hooke's constabulary. Its management is always opposite the direction of stretch and towards the equilibrium position. As the air track glider does the dorsum and forth, the leap force (Fspring) acts equally the restoring force. It acts leftward on the glider when information technology is positioned to the right of the equilibrium position; and it acts rightward on the glider when it is positioned to the left of the equilibrium position.

Allow's suppose that the glider is pulled to the right of the equilibrium position and released from rest. The diagram beneath shows the direction of the leap strength at v different positions over the course of the glider'due south path. Equally the glider moves from position A (the release point) to position B and then to position C, the spring force acts leftward upon the leftward moving glider. Every bit the glider approaches position C, the amount of stretch of the bound decreases and the bound force decreases, consistent with Hooke's Police force. Despite this decrease in the spring forcefulness, at that place is however an dispatch caused by the restoring force for the unabridged span from position A to position C. At position C, the glider has reached its maximum speed. Once the glider passes to the left of position C, the jump force acts rightward. During this stage of the glider's cycle, the spring is being compressed. The farther past position C that the glider moves, the greater the amount of pinch and the greater the spring force. This spring force acts as a restoring force, slowing the glider downward every bit it moves from position C to position D to position E. Past the time the glider has reached position Due east, it has slowed down to a momentary remainder position before changing its direction and heading back towards the equilibrium position. During the glider'due south motion from position East to position C, the corporeality that the leap is compressed decreases and the spring strength decreases. There is still an acceleration for the entire distance from position E to position C. At position C, the glider has reached its maximum speed. Now the glider begins to movement to the right of signal C. As it does, the spring force acts leftward upon the rightward moving glider. This restoring force causes the glider to tiresome down during the unabridged path from position C to position D to position Eastward.

Sinusoidal Nature of the Motility 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 the time. The discussion pertained to a mass that was vibrating up and downwardly while suspended from the spring. The word would be just as applicable to our glider moving along the air track. If a motility detector were placed at the correct stop of the air track to collect data for a position vs. time plot, the plot would wait like the plot below. Position A is the right-near position on the air track when the glider is closest to the detector.

The labeled positions in the diagram above are the aforementioned positions used in the discussion of restoring force above. You might recall from that word 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 aforementioned motion detector that nerveless position-time data were used to collect velocity-time data, then the plotted data would look similar the graph beneath.

Discover 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 1 is shifted 1-fourth of a vibrational cycle away from the other. As well observe in the plots that the accented value of the velocity is greatest at position C (corresponding 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 every bit a positive or a negative sign. In some instances, the velocity has a negative direction (the glider is moving leftward) and its velocity is plotted below the time centrality. In other cases, the velocity has a positive direction (the glider is moving rightward) and its velocity is plotted above the time centrality. Yous will also find that the velocity is nil whenever the position is at an farthermost. This occurs at positions A and Due east when the glider is kickoff to modify direction. And so but as in the instance of pendulum move, 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.

Free energy Analysis of a Mass on a Leap

On the previous folio, an energy assay for the vibration of a pendulum was discussed. Here we will acquit a like assay for the motion of a mass on a spring. In our discussion, we will refer to the motion of the frictionless glider on the air track that was introduced above. The glider will be pulled to the correct of its equilibrium position and be released from rest (position A). As mentioned, the glider then accelerates towards position C (the equilibrium position). Once the glider passes the equilibrium position, information technology begins to deadening down as the spring force pulls it backwards against its movement. Past the time it has reached position E, the glider has slowed down to a momentary pause before changing directions and accelerating dorsum towards position C. Once again, after the glider passes position C, it begins to slow down as it approaches position A. In one case at position A, the cycle begins all over once more ... and again ... and once more.

The kinetic free energy possessed by an object is the energy it possesses due to its motion. It is a quantity that depends upon both mass and speed. The equation that relates kinetic free energy (KE) to mass (m) and speed (v) is

KE = ½•m•v²

The faster an object moves, the more than kinetic free energy that it will possess. Nosotros can combine this concept with the discussion above virtually how speed changes during the course of move. This blending of the concepts would lead u.s. to conclude that the kinetic energy of the mass on the spring increases every bit it approaches the equilibrium position; and information technology decreases every bit it moves away from the equilibrium position.

This data is summarized in the table below:

Stage of Wheel	Change in Speed	Change in Kinetic Free energy
A to B to C	Increasing	Increasing
C to D to E	Decreasing	Decreasing
E to D to C	Increasing	Increasing
C to B to A	Decreasing	Decreasing

Kinetic energy is only i class of mechanical energy. The other form is potential energy. Potential free energy is the stored free energy of position possessed by an object. The potential energy could be gravitational potential free energy, in which instance the position refers to the height above the basis. Or the potential energy could exist rubberband potential energy, in which example the position refers to the position of the mass on the spring relative to the equilibrium position. For our vibrating air rail glider, at that place is no change in meridian. So the gravitational potential energy does not modify. This form of potential energy is not of much interest in our analysis of the energy changes. At that place is however a change in the position of the mass relative to its equilibrium position. Every time the bound is compressed or stretched relative to its relaxed position, at that place is an increase in the elastic potential energy. The amount of rubberband potential energy depends on the amount of stretch or compression of the bound. The equation that relates the amount of elastic potential energy (PEspring) to the amount of compression or stretch (10) is

PE_spring = ½ • k•x²

where 1000 is the spring abiding (in N/m) and x is the distance that the spring is stretched or compressed relative to the relaxed, unstretched position.

When the air track glider is at its equilibrium position (position C), it is moving it's fastest (as discussed higher up). At this position, the value of x is 0 meter. So the corporeality of elastic potential energy (PEspring) is 0 Joules. This is the position where the potential energy is the least. When the glider is at position A, the spring is stretched the greatest distance and the elastic potential energy is a maximum. A similar statement can be fabricated for position E. At position Eastward, the leap is compressed the almost and the elastic potential energy at this location is also a maximum. Since the leap stretches as much equally compresses, the elastic potential energy at position A (the stretched position) is the same every bit at position Due east (the compressed position). At these 2 positions - A and East - the velocity is 0 m/southward and the kinetic free energy is 0 J. So just like the case of a vibrating pendulum, a vibrating mass on a spring has the greatest potential energy when it has the smallest kinetic energy. And information technology as well has the smallest potential energy (position C) when information technology has the greatest kinetic energy. These principles are shown in the animation below.

When conducting an energy analysis, a common representation is an energy bar chart. An energy bar chart uses a bar graph to represent the relative amount and class of energy possessed by an object as it is moving. It is a useful conceptual tool for showing what form of free energy is present and how it changes over the course of fourth dimension. The diagram below is an free energy bar chart for the air rails 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. Still the full amount of these ii forms of mechanical energy remains constant. Mechanical free energy is being transformed from potential form to kinetic form; nonetheless the total corporeality is being conserved. A like conservation of energy phenomenon occurs as the mass moves from C to D to E. As the bound becomes compressed and the mass slows down, its kinetic energy is transformed into elastic potential energy. As this transformation occurs, the total amount of mechanical free energy is conserved. This very principle of free energy conservation was explained in a previous chapter - the Energy affiliate - of The Physics Classroom Tutorial.

Menstruum of a Mass on a Leap

As is probable obvious, not all springs are created equal. And not all spring-mass systems are created equal. One measurable quantity that tin be used to distinguish i spring-mass system from another is the menstruum. Equally discussed earlier in this lesson, the flow is the time for a vibrating object to brand one complete cycle of vibration. The variables that issue the menstruation of a spring-mass system are the mass and the spring constant. The equation that relates these variables resembles the equation for the period of a pendulum. The equation is

T = two•Π•(thou/thou)^.5

where T is the menstruum, one thousand is the mass of the object attached to the spring, and k is the leap constant of the spring. The equation tin can exist interpreted to mean that more massive objects will vibrate with a longer period. Their greater inertia means that it takes more time to complete a cycle. And springs with a greater spring abiding (stiffer springs) have a smaller period; masses attached to these springs take less time to complete a bike. Their greater spring constant means they exert stronger restoring forces upon the attached mass. This greater force reduces the length of fourth dimension to consummate ane cycle of vibration.

Looking Forward to Lesson 2

As we accept seen in this lesson, vibrating objects are fluctuant in place. They oscillate back and forth virtually a stock-still position. A unproblematic pendulum and a mass on a spring are classic examples of such vibrating motion. Though non evident by simple observation, the utilise of motion detectors reveals that the vibrations of these objects accept a sinusoidal nature. At that place is a subtle moving ridge-like beliefs associated with the fashion in which the position and the velocity vary with respect to fourth dimension. In the next lesson, nosotros volition investigate waves. As we will soon find out, if a mass on a bound is a wiggle in fourth dimension, then a wave is a collection of wigglers spread through space. As nosotros begin our study of waves in Lesson ii, concepts of frequency, wavelength and amplitude will remain important.

We Would Like to Suggest ...

Why merely read well-nigh information technology and when you could exist interacting with information technology? Interact - that'southward exactly what yous exercise when y'all use ane of The Physics Classroom's Interactives. We would similar to advise that you combine the reading of this page with the use of our Mass on a Spring Interactive. You can find information technology in the Physics Interactives section of our website. The Mass on a Leap Interactive provides the learner with a uncomplicated environs for exploring the consequence of mass, leap constant and elapsing of motion upon the period and amplitude of a vertically-vibrating mass.

Check Your Understanding

1. A force of 16 N is required to stretch a spring a distance of 40 cm from its residuum position. What forcefulness (in Newtons) is required to stretch the same spring …

a. … twice the distance?
b. … three times the distance?
c. … 1-half the altitude?

2. Perpetually disturbed by the habit of the backyard squirrels to raid his bird feeders, Mr. H decides to utilise a little physics for better living. His current plot involves equipping his bird feeder with a bound system that stretches and oscillates when the mass of a squirrel lands on the feeder. He wishes to have the highest aamplitude of vibration that is possible. Should he use a spring with a large spring abiding or a small spring 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 leap constant or a small spring constant?

4. Use free energy conservation to fill in the blanks in the following diagram.

v. Which of the following mass-spring systems will accept the highest frequency of vibration?

Case A: A bound with a g=300 N/1000 and a mass of 200 1000 suspended from it.
Instance B: A jump with a g=400 N/thou and a mass of 200 1000 suspended from it.

6. Which of the following mass-spring systems volition take the highest frequency of vibration?

Case A: A spring with a chiliad=300 N/m and a mass of 200 1000 suspended from it.
Example B: A spring with a k=300 N/k and a mass of 100 g suspended from it.