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Mathematics LibreTexts

6.2E: Spring Problems II (Exercises)

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    18283
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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

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