6.2.1: 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)=4cos2t 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)=8cos3t 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)=3cos4t−5sin4t 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)=18cost−9sint 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)=8sin2t−6cos2t N.
21. A mass m is suspended from a spring with constant k and subjected to an external force F(t)=αcosω0t+βsinω0t, where ω0 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 c1 and c2 are not both zero then
y=er1t(c1+c2t)
can’t equal zero for more than one value of t.
23. Show that if c1 and c2 are not both zero then
y=c1er1t+c2er2t
can’t equal zero for more than one value of t.
24. Find the solution of the initial value problem
my″
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