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)=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″+cy′+ky=0,y(0)=y0,y′(0)=v0,
given that the motion is underdamped, so the general solution of the equation is
y=e−ct/2m(c1cosω1t+c2sinω1t).
25. Find the solution of the initial value problem
my″+cy′+ky=0,y(0)=y0,y′(0)=v0,
given that the motion is overdamped, so the general solution of the equation is
y=c1er1t+c2er2t(r1,r2<0).
26. Find the solution of the initial value problem
my″+cy′+ky=0,y(0)=y0,y′(0)=v0,
given that the motion is critically damped, so that the general solution of the equation is of the form
y=er1t(c1+c2t)(r1<0).