Question 16 Spring-Mass SystemsHarmonic Motion Basics -Spring Constant (k in N/m) – The spring constant tells youhow strong (stiff) a spring is. A stiffer spring has a bigger k.Amplitude (A) - maximum displacement from theequilibrium position. The amount of energyin a spring-mass system is determined ONLYy the amplitude.Period (T)- time for one complete cycle.Frequency (f) - number of cycles in one second.Amplitude - 14 cmPeriod (T) = 4.2 seck-k=0:00.0start20 N/m20 N/mk = 20 N/m..faster Tk-faster TM MM M10 N/m20 N/m0:04.2stopWith the same spring constantmore mass causes a slowervibration (larger T).With the same mass thestronger spring (bigger k) willvibrate faster (maller T).28 cmA spring-mass system is called Simple Harmonic Motion (SHM)because the force is directly proportional to the displacement ofthe spring. Anything that follows this rule is known as SimpleHarmonic Motion. (A pendulum is only close to simple harmonicmotion.)Hooke's Law:Spring Constant (in N/m):bigger k= stiffer spring.Force (in N)of thespring»F=-kxdisplacement (in m) fromthe equilibrium position.Equilibrium position (where it will cometo rest; where it was before disturbed)The restoring force(F) tries to return itto its equilibriumposition.Force and Position: As seen above in Hooke's LawF and x always oppose each other. If F is positive, x isnegative, etc. [k, though, is always positive]).StretchedwwwwX-+A; F=-maxStretched Spring–The spring gets bigger,so x is positive and F is negative.Compressed Spring-The spring gets smaller,so x is negative and F is positive.At equilibrium: Regardless if it is moving or not, atthe equilibrium position the spring does not applya force. x=0 and F=0.Ma--max; v -0PE = max; KE =0Relaxed[wwwx = 0; F =0a = 0; v=-maxPE = 0; KE = maxMAcceleration (a), Velocity (v), and Energy (E)It should be obvious that where the force is zero(equilibrium position), the acceleration is zero.At the ends the mass stops for an instant, so v=0and KE - 0, but PE is a maximum (PE- Vlox).The mass speeds up as it moves toward the center,where it has maximum v and KE and minimum PE,Compressedx=-A; F= +maxa =+max; v= 0PE = max; KE = 0+FPeriod of a Spring-Mass System:Ex. A 350 g mass is attached to a spring that hasa spring constant of 12 N/m. What is the period ofvibration?Notice that amplitudeis not in the equation.Just as in all harmonicmotion-amplitudedoes not affect theperiod or frequency ofa spring-mass system.PeriodMass(in sec)(in kg)mT = 2n,Variables:m = 0.35 kg(1000 g =1 kg)k= 12 N/mT = 2x.Springconstant(in N/m)T= 6.28.0292kT= 6.28(.1709)35T =1.07 seçT = 6.2812T =cstephenmurray.comCopyright © 2013, C. Stephen MurraywwwwwwwwwwwwwwmD 12.Spring-Mass Systems p.24. Two different masses are suspendedfrom springs with the same springconstant, which will have the fasterperiod?1.Which of the springs has thebigger spring constant?1Right2. How do you know for certain?200 8Not as streached5. Why?200g3. Which will vibrate faster?400 gtether thO200 gk = 20 N/mk= 20 N/mk= 10 N/mk = 10 N/mB. 2M MM2MM WMC.M W MD.2M W 2MA.L20 cmL20 emL20 cm -20 cm-0:00.00:01.610What is the amplitude of spring A?10. What is the period of spring D26.1.6xZ =13.257. What is the amplitude of spring B?11. Find the mass on spring D.8.A or B will have the fastest period?9.A or C will have the slowest period?18. Find the period of a spring mass system that has a 5.4 kgmass and a 60 N/m spring constant.12. Which position or positions?A. 13 Has maximum velocity?B. SMaximum + force?Acceleration = 0?Maximum kinetic energy?No potential elastic energy?X is a negative maximum?Zero amplitude.wwwA[wwwBC.D.Е.F.G.13. Positive or negative displacement (x).A. -B. ŁA stretched spring (the spring is longer)C. 1 Pulling on a spring so it stretches.D. Pushing on a spring so that it compresses.E.Position vs. TimeA compressed spring (the spring gets shorter)t Hanging a mass on a spring.14. In F=-kx, is F the spring or what's pulling on the spring?15. If I pull on a spring with 20 N, then the spring pulls backwith how much force?Time (sec)16. A 5 kg object stretches a spring 20 cm.A. How much force is pulling on the spring ?19. What is the amplitude of the above graph?B. Find the spring constant.20. What is the period of the above graph?21. If it has a 3 kg mass on it, what is its spring constant?17. A 300 g mass stretches a spring 50 cm, find the springconstant.Copyright © 2013, C. Stephen Murraycstephenmurray.com

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5. A 5 kg object stretches a spring 20 cm. A. How much force is pulling on the spring ? B. Find the spring constant.