Simple Harmonic Motion


 Doris Cooper
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1 Simple Hamonic Motion Intoduction Simple hamonic motion occus when the net foce acting on an object is popotional to the object s displacement fom an equilibium position. When the object is at an equilibium position, the net foce acting on it is zeo. The displacement of the object will be sinusoidal as a function of time and will not damp out. This is an idealized model of oscillatoy motion, but it can effectively descibe numeous physical scenaios while still being simple enough that physical intuition can be developed by studying the model. On the small scale, vibating atoms in a solid can be modeled as simple hamonic oscillatos whee the inteatomic bonds between atoms geneate estoing foces on the atoms that ae linea in thei displacement fom thei equilibium positions. On a lage scale, the swinging pendulum in a gandfathe clock can also be modeled as as simple hamonic oscillato. In this case, the gavitational foce geneates a estoing foce that is linea in the angula displacement of the pendulum fo small angula displacements. In this lab, we will be using a mass on a sping system to study simple hamonic motion. You will be using two independent techniques to deduce physical popeties of spings fom you knowledge of Newton s 2 nd law. Sping Equipment: Mass hange Slotted mass LabPo Inteface Set of Spings (#3 and #4) Motion senso Plumb Bob Slotted masses 50 g mass hange Motion senso Backgound You know fom expeience that the moe you stetch a sping the hade it will pull back on you  ubbe bands, bungee cods, and slingshots exhibit this behavio. In one dimension, a foce of this natue can be descibed mathematically by F x = "k#x (1) whee k is the sping constant that descibes the stiffness of the sping and Δx is the displacement of the sping fom its elaxed o equilibium length. Robet Hooke is cedited fo fist obseving this linea elationship between the 1
2 foce exeted by a sping and the displacement of the sping fom equilibium. Thus, Equation (1) is sometimes efeed to as Hooke s Law. Any foce that has this geneal fom is called a estoing foce because the foce that aises due to stetching/compessing the sping seeks to estoe the sping to its elaxed length. The minus sign in the above equation ensues this behavio. If the sping is stetched in the +x diection, Δx will be positive, and the foce exeted by the sping will be in the x diection. Stetching the sping causes the sping to geneate a foce that ties to etun it to its elaxed length. It is this tendency to etun the sping to its equilibium length that esults in simple hamonic motion. Hooke s law is valid as long as the sping is not ovely stetched o distoted. If the stetch exceeds the elastic limit of the sping, it becomes pemanently distoted, and it cannot etun to its elaxed length. Conside a sping attached to a fixed post at one end and to a block that is fee to slide on a low fiction suface as shown in Figue 1. Let s look at the foces acting on the block when it is at diffeent positions. When the sping is elaxed (block at x = 0) as in Figue 1(b), thee is no hoizontal foce acting on the block, so the net foce on the block in this diection is zeo. This means that the velocity (speed and diection) of the block is not changing when it is at x = 0. If the block is displaced to the ight by stetching the sping [see Figue 1(c)], the sping exets a leftwad foce on the block. When the block is eleased, the foce exeted by the sping on the block will cause the block to move to the left. Since it is moving when it eaches x = 0, it will oveshoot the equilibium position, and begin to compess the sping. Once the sping is compessed, it exets a ightwad foce on the block [see Figue 1(a)]. This will slow down the block, bing it to est, and then change its diection sending it back towad equilibium at x = 0. In the absence of fiction, this oscillatoy motion continues indefinitely. In eal oscillatos, fiction will dissipate enegy fom the system, and eventually bing the block to est. Sping exets a foce on the block to the ight (a) (b) Sping exets no foce on the block. (c) Sping exets a foce on the block to the left. x = 0 Figue 1: The foce the sping exets on a block when the sping is (a) compessed, (b) elaxed, and (c) stetched. Note that the foce the sping exets on the block is such that it ties to etun the block to the equilibium position at x =0. 2
3 In lab today, we will be studying a mass on a sping system, but the entie setup will be otated 90 so that the mass hangs vetically as shown in Figue 2 below. If we apply Newton s 2 nd law to the mass attached to the sping in Figue 2, we see that the net foce acting on the mass is due to a sping estoing foce and a gavitational foce = F sp + F g. (2) (a) (b) (c) +x x = 0 x 0 x = 0 x 1 x= 0 The mass hange stetches the sping by an amount x o when the system is in equilibium. The mass hange and additional masses stetch the sping by (x 0 + x 1 ) when the system is in equilibium. Figue 2: Attaching mass to the sping causes the sping to stetch fom its elaxed length. When the mass hange is attached to the sping, its mass causes the sping to stetch by an amount x 0. When additional slotted masses ae added to the mass hange, the sping is stetched by an additional amount x 1. Once the mass comes to est, the net foce on it is zeo, and the system is in equilibium. In this state, the upwad estoing foce of the sping balances the downwad gavitational foce of the mass attached to the sping. 3
4 Using the coodinate systems in Figue 2, we can explicitly wite out the net foce on the total mass attached to the sping in Figue 2(c): = (!k x"" ) ˆx + ( m h + m)gˆx =!k ( x 0 + x 1 + x) ˆx + ( m h + m)gˆx = #$!kx! k ( x 0 + x 1 ) + ( m h + m)g% & ˆx whee x "" is the stetch of the sping fom its elaxed state at any instant in time (not just when it is at its equilibium position), m h is the mass of the hange, and m is the additional mass added to the hange. Note that x "" is a signed quantity that is positive when the bottom of the sping is below x "" = 0 and negative when it is above x "" = 0. Taking a close look at Equation (3), you may be suspicious about whethe this vetical mass on a sping will exhibit simple hamonic motion since the net foce acting on the mass has an additional constant tem that is not popotional to the displacement of the sping. Howeve, oienting the mass on a sping vetically simply sets a new equilibium position about which the mass will oscillate. We can see this most clealy by consideing the equilibium condition shown in Figue 2(c). When the system is in equilibium (i.e. the net foce is zeo), Newton s 2 nd law tells us that = F { + sp Fg { = 0 (4)!k( x 0 +x 1 )ˆx ( m h +m)gˆx so k(x 0 + x 1 ) = (m h + m)g. Inseting this esult into Equation (3), we see that the net foce acting on the mass is indeed popotional to the mass s displacement fom an equilibium position, =!kx ( ) ˆx, (5) so this system, too, exhibits simple hamonic motion. Howeve, the mass now oscillates about a new equilibium position ( x = 0). The displacement of this oscillating mass fom its equilibium position as a function of time can be descibed mathematically by x(t) = Asin[(2"t /T)+#)] (6) whee A is the amplitude, T is the peiod, and ϕ is the phase shift of the oscillation. The amplitude is a positive constant that descibes the maximum displacement fom equilibium, and the phase shift detemines the displacement of the mass at t=0. The peiod is the length of time it takes the mass to complete one full cycle of its motion, and it depends on the amount of oscillating mass, M, and the sping constant, k, of the sping, (3) T = 2! M k. (7) This expession fo the peiod is obtained by solving a diffeential equation that aises by applying Newton s 2nd law to the system of inteest. In the fist pat of this expeiment, you will use a static technique whee you vay the amount of mass hung fom a sping and use knowledge about the equilibium condition fo the system to detemine the sping constant. In the second pat of this expeiment, you will employ a dynamic technique to study the oscillatoy motion of the mass on a sping and use measuements of its peiod to independently infe the sping constant. Expeiment I: A Static Technique to Measue Sping Constants When mass is attached to a sping as shown in Figue 2(c), the upwad estoing foce of the sping balances the downwad gavitational foce acting on the mass once the system comes to est and eaches equilibium, 4
5 Fom Figue 2, we can ewite this expession as = F sp + F g = 0. (8) k ( x 0 + x 1 ) = ( m h + m)g. (9) Fo a given amount of slotted masses, m, placed on the mass hange, we can use the Venie motion senso to measue the displacement of the bottom of the mass hange, so we will vay m and measue x 1. By applying Newton s 2 nd law to the mass hange in Figue 2(b), we see that kx 0 = m h g, so Equation (9) can be ewitten as kx 1 = mg. (10) You goal fo this expeiment is to ceate a plot of x 1 vs. m fo each of the two spings and then detemine the the sping constant of each sping fom a linea fit to this data. To pepae to make these measuements: Hang sping #3 fom the hoizontal suppot ba that is about 1.25 m above the table top. The sping should be suspended fom the goove in this ba to keep the sping fom sliding along the ba duing measuements. Use the plumb line to place the motion senso on the table diectly below the sping. Then emove the plumb line fom the sping. Attach the mass hange to the fee end of the sping. Plug the motion senso into the Lab Po inteface, and make sue that the LabPo inteface has powe and is connected to a lab compute. Launch LoggePo on the lab compute, and open the file SHM_FL08_template.cmbl fom the LoggePo Templates folde on the desktop. Befoe you do anything else, pefom a Save As and save a copy of this file to the Documents folde fo you pesonal use. Replace template in the file name with the initials of you and you lab patne to uniquely identify you vesion of the LoggePo file. Familiaize youself with the two pages of the LoggePo file. Page 1 (Static Data) contains a data table, a digital mete to measue the displacement of the mass hange, and a gaph to plot you data. Page 2 (Dynamic Data) has a data table and two stacked gaphs which will be used fo the second pat of the expeiment. To acquie the data: Zeo the motion senso on the bottom of the mass hange when no additional mass is added to the hange, and then set up the motion senso to display a live eadout (Expeiment Live Readout). In this state, the motion senso will continually measue the displacement of the mass hange. Use the motion senso to detemine the displacement of the mass hange when additional masses fom 50 g to 250 g ae added to the mass hange, taking data in 50 g incements. Ente you displacements in the data table next to the mass data. Afte you have ecoded displacements fo all the masses, stop the live eadout fo the motion senso and save the data un (Expeiment Stoe Latest Run). In the data table, double click on the name of the stoed data set and ename it with a name that includes the numbe of the sping you ae studying. Repeat this pocess fo sping #4 but add 100 to 300 gams of slotted mass to the mass hange in 50 gam incements. To analyze the data: Plot the displacement vesus mass data fo both spings on the same gaph on Page 1 of you LoggePo file. Apply a linea fit to both data sets, and detemine the sping constant of both spings fom the fits to you data. Be sue to include dawings of you expeimental setup, desciptions of you methods, as well as you data tables and gaphs fom LoggePo in you lab notebook. 5
6 Answe the following questions about you data: 1) Based on you obsevations while acquiing data, do spings with lage sping constants stetch moe o less than those with smalle sping constants when the same mass is hung fom them? The sping constant is elated to the stiffness of the sping. Based on you obsevations and calculations, is the sping constant popotional o invesely popotional to the stiffness? Explain you easoning. 2) Theoetically, what should the vetical intecept be in you plots of displacement vesus mass? How well do you data agee with the theoetical value? What factos in you expeimental technique could explain any discepancies between theoy and expeiment? Explain you easoning. Expeiment II: A Dynamic Technique to Measue Sping Constants The second technique fo detemining sping constants is a dynamic one. That is, one in which the motion of the oscillating mass is obseved ove time, and the stiffness of the sping is detemined fom measuements associated with the timedependent motion of the mass on a sping. Equations (6) and (7) elate the stiffness of a sping to the timevaying displacement of the oscillating mass. In this pat of the expeiment, you will measue the displacement of the mass as a function of time, and use this infomation to detemine the peiod of the oscillation and ultimately the sping constant. To pepae to make these measuements: Switch to Page 2 (Dynamic Data) of you LoggePo wokbook. You should see a data table and two gaphs of position vesus time stacked vetically. Hang sping #3 vetically fom the hoizontal suppot ba. The motion senso should again be placed on the table diectly below the sping. Use the plumb line to confim you alignment. Attach the mass hange to the fee end of the sping, and place 100 g of additional mass on the mass hange. You will be measuing the displacement of the bottom of the mass hange elative to this equilibium position, so zeo the motion senso once the system is at est. In the Expeiment Data Collection menu, adjust the settings to acquie a 10 sec ecod digitizing 30 samples/sec. To acquie the data: Pull the hanging mass down a couple centimetes (do not ovestetch the sping!) and elease it to set the mass into motion. Click on the Collect button to acquie a 10 sec sample of the mass s motion. You can begin acquiing data at any point in the mass s motion. Save the data un (Expeiment Stoe Latest Run). In the data table, double click on the name of the stoed data set and ename it with a name that includes the numbe of the sping you ae studying. Repeat this study using sping #4 but this time place 200 g of additional mass on the mass hange instead of 100 g. To analyze the data: On the top gaph in Page 2, display the Position of sping #3 as a function of Time. Plot the same quantities fo sping #4 on the lowe gaph. Note: If you click on the vetical axis label of one of the gaphs and select Moe, you will see all of you data sets. You will have multiple Position data uns fo the diffeent spings you have studied. Check the Position box within the data set fo the appopiate sping to display Position on the vetical axis. Apply a sinusoidal custom cuve fit of the fom x(t) = A sin(bt + C) to detemine the peiod of the oscillating mass. If this function is not listed in the Geneal Equations list, define a new function of this fom. The thee fit paametes (A, B, and C) have physical meanings fo the motion of ou oscillating mass [see Equation (6)]. While the amplitude of the oscillation is conventionally chosen to be positive, it can also be negative fo a diffeent choice of the phase shift. The quantity (2π/T), howeve, should be positive. The cuve fitting outine does not account fo physical constaints, so if you get values fo you coefficients that 6
7 do not make physical sense, ty poviding an initial estimate fo coefficient B that has the coect sign and ode of magnitude to help the cuve fitting outine convege with physically meaningful coefficients. Fom the thee fit paametes (A, B, and C), detemine the peiod, and use this infomation to calculate the sping constant of the sping. Think caefully about what masses ae needed to calculate the sping constant. Calculate the pecent diffeence between the sping constant values you obtained with the static and dynamic measuements. Take the static value to be the gold standad to which you ae compaing the dynamic value. Repeat this analysis fo sping #4. Answe the following questions about you data: 1) The dynamic method fo estimating the stiffness of spings can be susceptible to eos associated with the fact that, depending on the specific type of sping used, a significant faction of the sping can also oscillate up and down along with the hanging mass, effectively inceasing the amount of oscillating mass. Would you expect a sping with a small o lage sping constant to be moe susceptible to this sot of eo (assuming the spings have compaable masses)? Do you pecent diffeence calculations fo measuing the sping constant two diffeent ways suppot you answe? Explain. 2) Fo dynamic measuements of the sping constant, will you undeestimate o oveestimate the sping constant if you assume that the sping does not contibute to you oscillating mass? Explain. Concluding Questions When esponding to the questions/execises below, you esponses need to be complete and coheent. Full cedit will only be awaded fo coect answes that ae accompanied by an explanation and/o justification. Include enough of the question/execise in you esponse that it is clea to you teaching assistant to which poblem you ae esponding. 1. If we have two spings suspended vetically, the fist of which is fou times as stiff as the second, detemine which sping will be stetched moe and by how much if the same mass is hung on both spings. 2. If the peiod of one oscillating sping is six times shote than that of anothe when the same mass is hanging fom each sping, which sping is the least stiff and by how much compaed to the othe? 3. Descibe thee eal wold examples of objects that undego simple hamonic motion (the motion of you examples may be damped out ove time due to fiction). You may not use any examples mentioned in the lab manual o by you TA. Explain what it is about the object that leads you to believe it is a hamonic oscillato. 7
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