SOFT MASSIVE SPRING Objectives: ) T deterine the spring cnstant and the ass crrectin factr fr the given sft assive spring by static (equilibriu extensin) ethd. 2) T deterine the spring cnstant and the ass crrectin factr fr the given sft assive spring by dynaic (spring ass scillatins) ethd. 3) T deterine the frequency f scillatins f the spring with ne end fixed and the ther end free i.e. zer ass attached. 4) T study the lngitudinal statinary waves and t deterine the fundaental frequency f scillatins f the spring with bth the ends fixed. Apparatus: ) A sft assive spring 2) A lng and heavy retrt stand with a clap at the tp end 3) A set f calibrated ass with hks (including fractinal ass) 4) A functin generatr with its cnnecting crd 5) A dual utput pwer aplifier with the cnnecting crds 6) A echanical vibratr unted n the retrt stand 7) A digital ultieter (t be used as frequency cunter) 8) A digital stpwatch 9) A easuring tape (3.0 ) 0) Tw easuring scales (.0 and 0.6 ) ) A tissue paper Intrductin: A spring is a flexible elastic device, which stres ptential energy n accunt f straining f the bnds between the ats f the elastic aterial f the spring. A variety f springs are available which are designed and fabricated t suit the varius echanical systes. Mst cn types f springs are cpressin springs, extensin springs and trsin spring. There are se special types f springs like leaf spring, V-spring, spiral spring etc. The cil r helical type f springs can have cylindrical r cnical shape. Rbert Hke, a 7 th century physicist studied the behavir f springs under different lads. He established an equatin, which is nw knwn as Hke s law f elasticity. This law states that the aunt by which a aterial bdy is defred (the strain) is linearly prprtinal t the frce causing the defratin (the stress). Thus, when applied t a spring, Hke s law iplies that the restring frce is linearly prprtinal t the equilibriu extensin. F = K x, where F is the restring frce exerted by the spring, x is the equilibriu extensin and K is called the spring cnstant. (The negative sign indicates that the frce F is ppsite in directin t the extensin x. Hence als the ter restring frce.) Fr this equatin t be valid, x needs t be belw the elastic liit f the spring. If x is re than the elastic liit, the spring will exhibit plastic behavir, where in the atic bnds in the aterial f the spring get brken r rearranged and the spring des nt return t its riginal state. It ay be nted that the ptential energy U stred in a spring is given by U = ½ K x 2. Depending n the value f the spring cnstant, a spring can be called as a sft r hard spring. A spring can be called ass-less r assive, depending n the ass, which needs t be
attached t get a cnsiderable extensin in the spring. Seties springs are als categrized by the rati f spring cnstant t the ass f the spring (K/ s ). A sft assive spring has a lw spring cnstant and its ass can nt be neglected. In the deterinatin f the spring cnstant f a spring, we generally neglect the effect f the ass f the spring n the equilibriu extensin r the tie perid f scillatin f the spring fr a given ass attached. In the case f sft assive springs the ass f the spring cannt be neglected. These types f springs have extensin under their wn weight and therefre need a crrectin fr the extensin. Siilarly, they scillate withut any attached ass, which iplies that the standard frula fr the tie perid f scillatins f a spring needs dificatin. Peple have theretically wrked ut the dificatin and crrected the frula fr the equilibriu extensin and als the tie perid f scillatins. Interestingly, ne finds that the ass crrectin factrs in these tw cases are nt the sae. In this prble, we will experientally study and verify the dified frulae. An extended sft assive spring claped at bth the ends can be assued t be a unifrly distributed ass syste. It has its wn natural frequencies f scillatins (crrespnding t different nral des) like a hllw pipe clsed at bth the ends. Using the ethd f resnance we will excite and study different nral des f vibratins f the spring. Here the lngitudinal statinary waves will be set up n the extended sft assive spring. Descriptin: In Part A, we will use the static ethd, where the equilibriu extensin f a given spring will be easured fr different attached ass and the spring cnstant and the ass crrectin factr will be deterined. In Part B, we will use the dynaic ethd, where different ass will be attached t the lwer end f the spring with its upper end fixed and crrespnding tie perid f scillatins fr such a spring-ass syste will be easured. Als the frequency f scillatins f the spring with the upper end fixed and the lwer end free i.e. the zer attached ass will be deterined graphically. In Part C, we will use a echanical vibratr t frce scillatins n the spring and excite different nral des f vibratins f the spring. Thus the lngitudinal statinary waves will be set up n the spring. We will easure the frequencies f excitatin crrespnding t different nral des. Fr these, the fundaental frequency f scillatin with bth the ends fixed will be deterined. We will cpare this frequency with the frequency f scillatins with ne end fixed and the ther end free as deterined earlier in Part B. Thery: Part A: Let L be the length f the spring when the spring is kept hrizntal under n tensin, be the ass attached t the free end f the spring, L be the length f the spring when the ass is attached at its lwer end, S be the equilibriu extensin f the spring fr ass, s be the ass f the spring, K be the spring cnstant and g be the acceleratin due t gravity. Thus, S L L Nte that the tensin in the spring varies alng the spring fr ( + s )g at the tp t g at the btt. We can write, 2
T ( x) ( s ) g C g x where, C is a cnstant f prprtinality. C = s /L and x is the distance fr the tp f the given pint; x varies fr 0 t L. We can deterine the expressin fr S, by taking extensin f a sall eleent f length x and integrating ver the ttal length f the spring. The final expressin, which we get is, s g S 2 2 K where, ( s /2) is called the ass crrectin factr (static case) cs. Part B: The expressin fr the tie perid f scillatins T fr an ideal (ass-less) spring-ass syste is given by, T 2 In case f the sft assive springs, we cannt neglect the ass f the spring since these springs can scillate withut any attached ass. We thus need t dify the abve expressin fr T. This can be dne using the principle f cnservatin f energy, i.e. Ptential Energy + Kinetic Energy = cnstant. The dified expressin, which we get is, s 3 T 2 3 K where ( s /3) is again the ass crrectin factr (dynaic case) cd. Nte that the ass crrectin factrs in Part A and Part B are different. The crrespnding frequency f scillatins f ' is given by, f ' T Part C: The extended spring serves as a unifrly distributed ass syste. It has its wn natural frequencies like a hllw pipe clsed at bth the ends [Nte that, bth the ends f the spring ay be taken t be fixed. The upper end is fixed in any case and the aplitude f the lwer end is s sall, as cpared t the extended length f the spring that it can be taken t be zer]. The natural frequencies crrespnd t statinary waves; their wavelengths are given by 2 n L, n, 2,3,... n Nw, n f n = velcity V f the waves n the spring, where f n is the frequency f the lngitudinal statinary waves set up n the spring; n = is the fundaental, n = 2 the secnd harnic and s n: V V n f n 4 2 L n K, f 2,...., f V 2 L V L f n n f 3
This fundaental frequency f in this case shuld be twice that f the fundaental frequency f ' f the spring with zer ass attached t the spring. Experiental Setup: Fr Part A and B, yu will need a sft assive spring, a retrt stand with a clap, a set f ass, a easuring tape / scales and a digital stpwatch. Fr Part C, yu will need a sft assive spring, a lng and heavy retrt stand with a clap at the tp end and a echanical vibratr claped near the base f the stand. We will als need a functin generatr. In this case, the sft assive spring shuld be claped at the upper end n the lng retrt stand. The lwer end f the spring shuld be claped t the crcdile clip fixed at the centre f the echanical vibratr. The lwer end f the spring will be subjected t an up and dwn harnic tin using the echanical vibratr. It ust be ensured that the aplitude f this tin is sall enugh s that these ends culd be cnsidered t be fixed. Warning: ) D nt extend the spring beynd the elastic liit. Chse thughtfully the value f the axiu ass that ay be attached t the lwer end f the given spring. 2) Keep the aplitude f scillatins f the spring-ass syste just sufficient t get the required nuber f scillatins. 3) Reeber always t switch ON the pwer supply t any instruent befre applying the input t it. 4) Use the echanical vibratr very carefully. Yu shuld nt get hurt with the sharp edges/crners. Be extreely careful while claping the lwer end f the spring t the vibratr using the crcdile clip. 5) The aplitude f vibratins shuld be carefully adjusted t the required level using the vltage selectin and aplitude knb f the functin generatr. 6) Use the easuring tape carefully t avid any injury. The tape is etallic, and the edges are very sharp. Prcedural Instructins: Part A: (i) Measure the length L f the spring keeping it hrizntal n a table in an unstretched (all the cils tuching each ther) psitin. (ii) Hang the spring t the clap fixed t the tp end f the retrt stand. The spring gets extended under its wn weight. (iii) Take apprpriate asses and attach the t the lwer end f the spring. (Chse the range f the ass carefully, keeping in ind the elastic liit). (iv) Measure the length L f the spring in each case. (Fr better results yu ay repeat each easureent tw r three ties.) Thus deterine the equilibriu extensin S fr each value f ass attached. (v) Plt an apprpriate graph and deterine the spring cnstant K f the spring and als the ass crrectin factr cs. (Take g = 980 c/s 2 = 9.80 /s 2 ). Questin : State and justify the selectin f variables pltted n X and Y axes. Explain the bserved behavir and interpret the X and Y intercepts. Part B: 4
(i) Keep the spring claped t the retrt stand. (ii) Try t set the spring int scillatins withut any ass attached, yu will bserve that the spring scillates under the influence f its wn weight. (iii)attach different asses t the lwer end f the spring and easure the tie perid f scillatins f the spring ass syste fr each value f the ass attached. (Chse the asses carefully, keeping in ind the elastic liit). Yu ay use the ethd f easuring tie fr a nuber f (ay be 0, 20, 30..) scillatins and deterine the average tie perid. (iv) Perfr the necessary data analysis and deterine spring cnstant K and the ass crrectin factr cd using the abve data. (v) Als deterine frequency f ' fr zer ass attached t the spring fr the graph. Using the Eq. f ' 2 K c and the value f K fr Part A, deterine c (the ass crrectin factr). Check whether this is the sae ass crrectin factr cd btained earlier. Questin 2: Des the abve ethd f easuring the ttal tie fr a nuber f scillatins help us t increase the reliability f tie perid easureent? (5) Part C: (i) Keep the spring claped t the lng retrt stand. (ii)clap the lwer end f the spring t the crcdile clip attached t the vibratr. (iii) Cnnect the utput f the functin generatr t the input f the echanical vibratr using BNC cable (iv) Starting fr zer, slwly g n increasing the frequency f vibratins prduced by the vibratr by increasing the frequency f the sinusidal signal/wave generated by the funtin generatr. At a particular frequency yu will bserve that the idpint f the spring will scillate with large aplitude indicating an antinde there. (Yu ay use a sall piece f tissue paper t bserve the aplitude at the antinde.) This is the fundaental de (first harnic) f scillatin f the spring. Adjust the frequency t get the axiu pssible aplitude at the antinde. Measure and recrd this frequency using the display n the functin generatr. (v) Increase the frequency further and bserve higher harnics identifying the n the basis f the nuber f lps yu can see between the fixed ends. (vi) Plt a graph f frequency fr different nuber f lps (i.e. the harnics) versus the nuber f lps (harnics). Deterine this fundaental frequency f fr the slpe f this graph. (vii) Cpare this fundaental frequency f with the frequency f ' f the spring ass syste with ne end fixed and the zer ass attached (as deterined in Part B) and shw that f ' = (f /2). Questin 3: Explain why the tw frequencies shuld be related by a factr f tw? (Take the analgy between the spring and an air clun.) References: ) J. Christensen, A. J. Phys, 2004, 72(6), 88-828. 2) T. C. Heard, N. D. Newby Jr, Behavir f a Sft Spring, A. J. Phys, 45 (), 977, pp. 02-06. 5
3) H. C. Pradhan, B. N. Meera, Oscillatins f a Spring With Nn-negligible Mass, Physics Educatin (India), 3, 996, pp. 89-93. 4) B. N. Meera, H. C. Pradhan, Experiental Study f Oscillatins f a Spring with Mass Crrectin, Physics Educatin (India), 3, 996, pp. 248-255. 5) Rajesh B. Khaparde, B. N. Meera, H. C. Pradhan, Study f Statinary Lngitudinal Oscillatins n a Sft Spring, Physics Educatin (India), 4, 997, pp. 30-9. 6) H. J. Pain, The Physics f Vibratins and Waves, 2 nd Ed, Jhn Wiley & Sns, Ltd., 98. 7) D. Halliday, R. Resnick, J. Walker, Fundaentals f Physics, 5 th Ed, Jhn Wiley & Sns, Inc., 997. 8) K. Raa Reddy, S. B. Badai, V. Balasubraanian, Oscillatins and Waves, University Press, Hyderabad, 994. 6