HYIC 534 XRCI-4 ANWR Hke s Law (prings) DAVION Clintn Davissn was awarded the Nbel prize fr physics in 1937 fr his wrk n the diffractin f electrns. A spring is a device that stres ptential energy. When a spring is stretched (r cmpressed), a frce is applied thrugh a distance. Thus, wrk is dne. The wrk dne ges t the spring in the frm f ptential energy ( ). The spring frce is a restring frce which, in bringing the spring back t its rest pint, des wrk. Nte that every spring has an elastic limit. If the spring is stretched within its elastic limit, it springs back t its rest pint. If a spring is stretched beynd its elastic limit, it becmes defrmed. As illustrated belw, the distance a spring is stretched is called the elngatin. Rbert Hke was the first t discver that the spring frce is directly prprtinal t the elngatin. Tday, we call this law Hke s Law. Rest int lastic limit F Nte that the spring frce (F ) is ppsite the applied frce (F A ). lngatin F A Defrmed The frmula fr the spring is: F = (Hke s Law) where F is the spring frce, in newtns (N) k is the spring cnstant, in N/m x is the elngatin, in metres The spring cnstant is different fr different springs and depends upn the type f material the spring is made f as well as the thickness f the spring cil. The greater the value f the spring cnstant, the stiffer the spring. The frmula fr the ptential energy stred in a spring is: where = ½ is the ptential energy stred in the spring, in jules (J) k is the spring cnstant, in N/m x is the elngatin, in metres Remember that in an ideal spring, there is n lss f energy ( ) due t frictin. When the spring frce (F ) is pltted versus the elngatin (x) f the spring, the resulting graph is a linear relatin. The slpe f the curve represents the spring cnstant while the area under the curve represents the ptential energy stred in the spring. Frce (N) pring cnstant lngatin (m) 1. What determines the spring cnstant (k) f a spring? The type f material the cil is made frm and the thickness f the cil. HYIC 534 NRGY Anshysics x-4.doc 1999. Lancine age 1 /5
. A ciled spring is stretched 0.05 m by a weight f 0.50 N hung frm ne end. a) Hw far will the spring stretch if a 1.0 N weight replaces the 0.50 N weight? [0.1 m] First find the spring cns tan t : F k F / x 0.5 N / 0.05 m 10 N / m Nw find hw far the spring will stretch: F 1.0 N F x 0.1 m k 10 N / m b) What weight will stretch the spring a distance f 0.03 m? [0.3 N] Nte: The weight equals the spring frce. F (10 N / m)(0.03 m) 0.3 N 3. When the spring frce is pltted versus the elngatin f the spring: a) Hw d yu determine the spring cnstant (k) frm the graph? By finding the slpe f the line pltted. b) Hw d yu determine the ptential energy ( ) stred in a spring frm the graph? By finding the area (frm the curve t the x-axis) f the line pltted. 4. An ideal spring, whse spring cnstant is 14.0 N/m, is stretched 0.40 m when a mass f 0.560 kg is hung frm it, hw much ptential energy ( ) is stred in this spring? [1.1 J] (14 N / s)(0.40) 1.1 J 5. The wrk dne t cmpress a spring frm 0 t 0.15 m is 8.0 J. Hw much wrk is required t cmpress this spring frm 0 t 0.30 m? [3 J] Find the value f k : k ( ) / x (8.0 J) /(0.15 m) 711 N / m Find the new : (711 N / m)(0.30 m) 3 J HYIC 534 NRGY Anshysics x-4.doc Hke s Law age /5
6. The graph n the right represents the frce-cmpressin graph f an ideal spring. a) Determine the spring cnstant. [100 N/m] 10 N lpe 100 N / m 0.1 m Therefre, the spring cns tan t (k) is 100 N / m. b) Calculate the lss in kinetic energy f an bject that cllides with this spring and cmpresses it a distance f 0.60 m. [18 J] 10 5 0 Frce (N) 0.05 0.1 Length (m) Nte: The lss in f the bject gain in f the spring. (100 N / m)(0.60 m) Thus : 18 J _ 7. A spring is cmpressed 0.10 m when a frce f.0 N is exerted upn it. What frce must be used t cmpress this spring 0.40 meters? [8.0 N] Find the spring cns tan t : F k F / x.0 N / 0.10 m 0 N / m Nw find the spring frce required : F (0 N / m)(0.40 m) 8 N 8. A mass f 10 kg, having a velcity f 5 m/s, slides int a spring as illustrated in the diagram n the right. Assuming the system is frictinless, which f the fllwing statements is crrect? 5 m/s 10 kg I- At maximum cmpressin, the f the mass is zer. II- At maximum cmpressin, the f the spring equals the f the mass. III- After the interactin, the mass mves with a velcity f 5 m/s. a) I b) II c) III d) I and III e) all are crrect Velcity = -5 m/s HYIC 534 NRGY Anshysics x-4.doc Hke s Law age 3 /5
9. A 5 kg blck is frced against a hrizntal spring a distance f 10 cm. When released, the blck mves a hrizntal distance f cm then stps. If the spring cnstant f this spring is 150 N/m, what is the cefficient f frictin between the blck and the hrizntal surface? [0.06] Nte: The spring frce is the frictinal frce (F f ). Find the spring frce : F (150 N / m)(0.0 m) 3 N pringcnstant CAUTION: D nt cnfuse the spring cnstant with the cefficient f frictin. f 3 N Find the cefficien t f frictin : f kfn k 0.06 FN 50 N Ceficientfrictin 5 kg 10. Blck-A is placed n a spring and cmpresses it a distance x. As illustrated belw, when Blck-B is placed n the same spring, it is cmpressed a distance f x. Rest int Blck-A Blck-B x x What is the value f the fllwing rati? tential energy f spring with Blck-B n tp tential energy f spring with Blck-A n tp f B f A B x B (x) 4x 4 A x A x x 1 11. A blck f unknwn mass mves at m/s twards a hrizntal spring whse spring cnstant is 1000 N/m. m/s If the spring is cmpressed a maximum distance f 0.4 m, what is the mass f the blck? [40 kg] Nte: At max imum cmpressin, Initial f mass. mv Thus, r (1000 N / m)(0.4 m) Therefre : m 40 kg v ( m / s) HYIC 534 NRGY Anshysics x-4.doc Hke s Law age 4 /5
Frce (N) 1. The frce-cmpressin graph f a spring is illustrated n the right. Hw Nte: much kinetic Frm the energy givenis graph, lst by the an slpe bject that cllides f with the this line spring is the spring and cmpresses cns tan t. it a distance f 0.8 m? [3 J] 100 N Thus, k slpe 100 N / m 1.0 m Nw find the f the spring which equals the lst by the bject. (100 N / m)(0.8 m) 3 J 13. When a 4 kg mass is attached t the end f a spring hanging vertically, the spring experiences an elngatin f 5.0 cm. Hw much ptential energy is stred in the spring? [6 J] Find the spring cns tan t : F k F / x mg / x (4 kg)(10 m / s ) / 0.05 m 4800 N / m Find the ptentialenergy stred in the spring: (4800 N / m)(0.05 m) 6 J 14. A spring is fixed alng an inclined plane whse angle f incline is 30 as illustrated in the diagram n the right. A 1 kg blck is attached t the spring thereby stretching it 15 cm. Find the spring cnstant f this spring. [400 N/m] 30 Nte: Re lative t the inclined plane, the spring frce (F ) equals the mg in 30 hrizntal cmpnent f the weight (mg in 30 ). F mg in 30 (1 kg)(10 m / s )in 30 60 N Nw find the spring cns tan t : F k F / x 60 N / 0.15 m 400 N / m 15. The graph n the left displays the frce-cmpressin curve f fur springs labeled pring-a, pring-b, pring-c and pring D. Which spring is the springiest? pring-d (Least slpe) Fs pring-a 1 kg pring-b pring-c pring-d Cmpressin (m) HYIC 534 NRGY Anshysics x-4.doc Hke s Law age 5 /5