PY07 Oscillations using Springs Experient Physics 07 Oscillations using Springs Experient Prelab Read the following bacground/setup and ensure you are failiar with the concepts and theory required for the experient. Bacground/Setup Mass oscillating on a single spring Springs are very failiar objects that have a range of everyday applications. When a force is applied to a spring it is either copressed or stretched, depending on the direction in which the force is applied. The original length of a spring is nown as its unstrained length and it can be shown that for sall displaceents x the force applied to the spring, F, is directly proportional to that displaceent (c.f. Figure ). This can be represented as: F applied = x, () where is the spring constant and has units N. Any spring that behaves according to expression () is an ideal spring. The spring constant is related to the stiffness of the spring i.e. a large value of eans the spring Figure : Deonstration of copression and stretching of a spring. is stiff (and hard to copress/stretch) whereas a sall value for eans the spring is easy to copress/stretch. If a spring of length L with spring constant is cut in half, the two half springs with length L/ will each have a spring constant of. The spring exerts a restoring force that is equal in agnitude but opposite in direction to the applied force: F x = x. () The ter restoring force is used because it restores the spring bac to its equilibriu (unstrained) position as soon as the applied force is reoved. In an ideal environent (i.e. one without friction or air resistance) the spring would oscillate around the equilibriu position with oscillations of aplitude A. The displaceent of a ass on the end of the spring would follow siple haronic otion (c.f. Figure ) and the displaceent at any tie t would be given by: ( t) = Acos( ω t + θ ), x o (3) whereω o is the natural angular frequency of the ass on the end of the spring and θ is a rando phase. The period of the oscillations (in seconds) can be defined by: T π =. (4) ω o 3.
PY07 Oscillations using Springs Experient It is soeties ore convenient to discuss the otion in ters of the frequency of the oscillations (i.e. the nuber of cycles of otion per second) f o = T and it is given in units of Hz. Figure : An object oving in siple haronic otion oscillates such that its position as a function of tie follows a sinusoidal pattern with aplitude A. In reality, the otion will be daped due to air resistance and the aplitude of the oscillations will decrease exponentially with tie and the frequency of oscillation will also vary slightly fro the natural frequency. In this case the otion can be described by: x βt ( t) = Ae cos( ωt + φ), where β is the daping factor, ω is the daped angular frequency and φ is soe arbitrary phase. (5) Prelab Read the rest of this lab description and set-up tables for your data collection. You should have identified the goals of the experient and the easureents that you ust ae before coing into the lab. Part : Deterine the Spring Constant,, using Hooe s Law Using a single spring, hang a ass of nown value on the end of the spring and easure the spring s extension, x. Ensure you start with a ass that gives a easurable extension for the spring. Repeat the easureent for five different values of ass, each tie deterining the extension of the spring x fro the unstretched position. Using the standard forula for the weight pulling on the spring (and ignoring the ass of the spring itself), deterine F = g for each ass. Fro Hooe s Law we have that the spring restoring force, F: Fx = x F = g = x. x (6) 3.
PY07 Oscillations using Springs Experient Plot a graph of the restoring force F x against the extension of the spring x. The graph should yield a straight line with a slope equal to the spring constant, for the spring you are using. Mass (g) Weight (N) Extension () = F = x = = F = x = = F = x = 5 5 5 Fro the graph deterine. Ensure your answer includes estiates for the axiu error on your value for. Value for the spring constant ( ) N = ± Part : Deterine the Spring Constant,, fro the frequency of oscillation The spring syste consists of a agnetic (B-) field sensor, a agnet and a spring bar attached to the spring being tested. As the spring oscillates, so does the agnet, thereby changing the agnitude of the B- field easured by the sensor. The sensor is interfaced to the PC and using the e-prolab software a plot of B-field agnitude as a function of tie can be obtained. The Fourier transfor of the tie varying signal will give the frequency of oscillation of the spring. We will assue that this is the spring s natural frequency. Instructions on using the e-prolab software Double clic on the e-prolab icon on the destop Select the HiScope option (which you will have used in first year). Failiarise yourself with HiScope. In particular recall how to change the tie scale etc. in order to suit the signal you are recording. Reeber to only use a single (y ) variable. If you are getting two traces, switch off y. Within HiScope, clic on the fourth icon (Choose Connected Sensors) Fro the list of available sensors select B-field (MG) High Clic Add Next clic on y:variables on the sae screen Select B-field (MG) High, V in, 0 Clic Choose One 3.3
PY07 Oscillations using Springs Experient Clic OK Clic on the t on the top of the screen to set the conditions for data accuulation. Select the saple tie (e.g. 0. s). You ll find that it ay be necessary to change this value to suit the easureents you are aing. Choose nuber of saples (e.g. 00). Again you ay find that you have to change this value. Choose trigger repeat to start. Choose source B field Clic OK. Clic on the y(t) icon in order to open the easureent screen. Select y(t): clic Add and then OK Press the sall green arrow on the top of the screen to start data accuulation In order to get the Fourier transfor of your signal clic on the FT sybol and choose the Display Values option. Procedure Hang a single ass onto the spring (50 g or 00 g is suitable) and easure the frequency of oscillation. Neglecting the ass of the spring and using ω = o, deterine for the spring. Note: it is iportant that the spring only undergoes up-down otion. Please ensure this is the case. Value for the spring constant ( ) N = ± Coent on how the value you obtain using this ethod copares to that obtained in Part. Part 3: Investigation of the daping of a syste consisting of one ass and one spring Using a single ass,, suspended on a single spring easure the logarithic decreent of the syste i.e. observe how the oscillation aplitude decreases with βt tie, given by eqn. (5) x() t = Ae cos( ωt + φ). Assue that ω ωo and, using the value for deterined in Part, deterine ω = o. Thence, deterine the daping factor β for the otion. Coent on the value obtained. 3.4
PY07 Oscillations using Springs Experient Part 4: Two springs and one ass: Springs in series and parallel Deterine the spring constants for the two possible spring cobinations (by following the sae ethod as given in Part ) and copare with that for a single spring. Chec the values you obtain against those expected fro theory. Part 5: Two springs and two asses: Investigate the otion of two coupled oscillators When two asses and two springs are coupled as shown, the otion appears to be quite coplex. In fact, this otion is a siple ixture of two fundaental odes of angular frequency ω and ω, where: and ω = o. ω 3 + 5 3 5 = ω0 and ω = ω0, (7) The siplest setup uses two identical springs and two equal asses. Here it is relatively easy to find the forula giving an expression for the two frequencies. You 3.5
PY07 Oscillations using Springs Experient should copare your experiental values to those predicted fro theory (research this topic in the library in order to obtain the theoretical prediction for this scenario). You can use the fast Fourier transfor (FFT) facility on the software pacage to extract the two fundaental odes. The FFT allows you to separate frequencies when there are two or ore present in a coplex signal. In this experient, you use this facility on data that you have already collected rather than on live data (as was the case for the sound experient last wee). Do the following: (i) Using two identical springs and two equal asses deterine the tow fundaental frequencies experientally. Copare these values to those expected fro theory (equation 7). (ii) Use two springs and two different asses e.g. 50 g and 00 g. First deterine the fundaental odes for the heavier ass on top and then the reverse. Do you get the sae results? (iii) Use three springs and three identical asses. How does this behave? (iv) Finally, briefly exaine the case of a single spring that is undergoing fro otion as well as up-down. to-and- The expressions for the frequencies in (ii) and (iii) are ore coplex but you should be able to wor the out. Ensure you ae a reasonable estiate on the axiu error in each of your easureents. All data that you record should show the easured value and its associated axiu error (and, of course, units). Relevant Literature. J. Cutnell and K.W. Johnson, Physics, 6 th Ed., Wiley, London, (004). (ISBN: 978-0-47-583-8). T.W.B. Kibble and F.H. Bershire, Classical Mechanics, 5 th Ed., Iperial College Press, London, (004). ISBN 978--86094-44-6 3. http://www.scar.utoronto.ca/~pat/fun/newtd/pdf/coupled.pdf 3.6