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6.2E: Spring Problems II (Exercises)

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Q6.2.1

1. A 64 lb object stretches a spring 4 ft in equilibrium. It is attached to a dashpot with damping constant c=8 lb-sec/ft. The object is initially displaced 18 inches above equilibrium and given a downward velocity of 4 ft/sec. Find its displacement and time–varying amplitude for t>0.

2. A 16 lb weight is attached to a spring with natural length 5 ft. With the weight attached, the spring measures 8.2 ft. The weight is initially displaced 3 ft below equilibrium and given an upward velocity of 2 ft/sec. Find and graph its displacement for t>0 if the medium resists the motion with a force of one lb for each ft/sec of velocity. Also, find its time–varying amplitude.

3. An 8 lb weight stretches a spring 1.5 inches. It is attached to a dashpot with damping constant c=8 lb-sec/ft. The weight is initially displaced 3 inches above equilibrium and given an upward velocity of 6 ft/sec. Find and graph its displacement for t>0.

4. A 96 lb weight stretches a spring 3.2 ft in equilibrium. It is attached to a dashpot with damping constant c=18 lb-sec/ft. The weight is initially displaced 15 inches below equilibrium and given a downward velocity of 12 ft/sec. Find its displacement for t>0.

5. A 16 lb weight stretches a spring 6 inches in equilibrium. It is attached to a damping mechanism with constant c. Find all values of c such that the free vibration of the weight has infinitely many oscillations.

6. An 8 lb weight stretches a spring .32 ft. The weight is initially displaced 6 inches above equilibrium and given an upward velocity of 4 ft/sec. Find its displacement for t>0 if the medium exerts a damping force of 1.5 lb for each ft/sec of velocity.

7. A 32 lb weight stretches a spring 2 ft in equilibrium. It is attached to a dashpot with constant c=8 lb-sec/ft. The weight is initially displaced 8 inches below equilibrium and released from rest. Find its displacement for t>0.

8. A mass of 20 gm stretches a spring 5 cm. The spring is attached to a dashpot with damping constant 400 dyne sec/cm. Determine the displacement for t>0 if the mass is initially displaced 9 cm above equilibrium and released from rest.

9. A 64 lb weight is suspended from a spring with constant k=25 lb/ft. It is initially displaced 18 inches above equilibrium and released from rest. Find its displacement for t>0 if the medium resists the motion with 6 lb of force for each ft/sec of velocity.

10. A 32 lb weight stretches a spring 1 ft in equilibrium. The weight is initially displaced 6 inches above equilibrium and given a downward velocity of 3 ft/sec. Find its displacement for t>0 if the medium resists the motion with a force equal to 3 times the speed in ft/sec.

11. An 8 lb weight stretches a spring 2 inches. It is attached to a dashpot with damping constant c=4 lb-sec/ft. The weight is initially displaced 3 inches above equilibrium and given a downward velocity of 4 ft/sec. Find its displacement for t>0.

12. A 2 lb weight stretches a spring .32 ft. The weight is initially displaced 4 inches below equilibrium and given an upward velocity of 5 ft/sec. The medium provides damping with constant c=1/8 lb-sec/ft. Find and graph the displacement for t>0.

13. An 8 lb weight stretches a spring 8 inches in equilibrium. It is attached to a dashpot with damping constant c=.5 lb-sec/ft and subjected to an external force F(t)=4\cos2t lb. Determine the steady state component of the displacement for t>0.

14. A 32 lb weight stretches a spring 1 ft in equilibrium. It is attached to a dashpot with constant c=12 lb-sec/ft. The weight is initially displaced 8 inches above equilibrium and released from rest. Find its displacement for t>0.

15. A mass of one kg stretches a spring 49 cm in equilibrium. A dashpot attached to the spring supplies a damping force of 4 N for each m/sec of speed. The mass is initially displaced 10 cm above equilibrium and given a downward velocity of 1 m/sec. Find its displacement for t>0.

16. A mass of 100 grams stretches a spring 98 cm in equilibrium. A dashpot attached to the spring supplies a damping force of 600 dynes for each cm/sec of speed. The mass is initially displaced 10 cm above equilibrium and given a downward velocity of 1 m/sec. Find its displacement for t>0.

17. A 192 lb weight is suspended from a spring with constant k=6 lb/ft and subjected to an external force F(t)=8\cos3t lb. Find the steady state component of the displacement for t>0 if the medium resists the motion with a force equal to 8 times the speed in ft/sec.

18. A 2 gm mass is attached to a spring with constant 20 dyne/cm. Find the steady state component of the displacement if the mass is subjected to an external force F(t)=3\cos4t-5\sin4t dynes and a dashpot supplies 4 dynes of damping for each cm/sec of velocity.

19. A 96 lb weight is attached to a spring with constant 12 lb/ft. Find and graph the steady state component of the displacement if the mass is subjected to an external force F(t)=18\cos t-9\sin t lb and a dashpot supplies 24 lb of damping for each ft/sec of velocity.

20. A mass of one kg stretches a spring 49 cm in equilibrium. It is attached to a dashpot that supplies a damping force of 4 N for each m/sec of speed. Find the steady state component of its displacement if it is subjected to an external force F(t)=8\sin2t-6\cos2t N.

21. A mass m is suspended from a spring with constant k and subjected to an external force F(t)=\alpha\cos\omega_0t+\beta\sin\omega_0t, where \omega_0 is the natural frequency of the spring–mass system without damping. Find the steady state component of the displacement if a dashpot with constant c supplies damping.

22. Show that if c_1 and c_2 are not both zero then

y=e^{r_1t}(c_1+c_2t) \nonumber

can’t equal zero for more than one value of t.

23. Show that if c_1 and c_2 are not both zero then

y=c_1e^{r_1t}+c_2e^{r_2t} \nonumber

can’t equal zero for more than one value of t.

24. Find the solution of the initial value problem

my''+cy'+ky=0,\quad y(0)=y_0,\;y'(0)=v_0, \nonumber

given that the motion is underdamped, so the general solution of the equation is

y=e^{-ct/2m}(c_1\cos\omega_1t+c_2\sin\omega_1t). \nonumber

25. Find the solution of the initial value problem

my''+cy'+ky=0,\quad y(0)=y_0,\;y'(0)=v_0, \nonumber

given that the motion is overdamped, so the general solution of the equation is

y=c_1e^{r_1t}+c_2e^{r_2t}\;(r_1,r_2<0). \nonumber

26. Find the solution of the initial value problem

my''+cy'+ky=0,\quad y(0)=y_0,\;y'(0)=v_0, \nonumber

given that the motion is critically damped, so that the general solution of the equation is of the form

y=e^{r_1t}(c_1+c_2t)\,(r_1<0). \nonumber


This page titled 6.2E: Spring Problems II (Exercises) is shared under a CC BY-NC-SA 3.0 license and was authored, remixed, and/or curated by William F. Trench via source content that was edited to the style and standards of the LibreTexts platform.

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