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# 6.2.1: Spring Problems II (Exercises)

• • William F. Trench
• Andrew G. Cowles Distinguished Professor Emeritus (Mathematics) at Trinity University
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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

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

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

$y=c_1e^{r_1t}+c_2e^{r_2t}$

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

24. Find the solution of the initial value problem

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

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

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

25. Find the solution of the initial value problem

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

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

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

26. Find the solution of the initial value problem

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

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

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