6.2.1: Spring Problems II (Exercises)

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$trench-1995.jpg$
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

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