# 6.2E: Spring Problems II (Exercises)

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- 18283

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[exer:6.2.1] 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 \(t>0\).

[exer:6.2.2] 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.

[exer:6.2.3] 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 \(t>0\).

[exer:6.2.4] 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 \(t>0\).

[exer:6.2.5] 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.

[exer:6.2.6] 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.

[exer:6.2.7] 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 \(t>0\).

[exer:6.2.8] 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.

[exer:6.2.9] 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.

[exer:6.2.10] 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.

[exer:6.2.11] 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 \(t>0\).

[exer:6.2.12] 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 \(t>0\).

[exer:6.2.13] 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 \(t>0\).

[exer:6.2.14] 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 \(t>0\).

[exer:6.2.15] 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 \(t>0\).

[exer:6.2.16] 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 \(t>0\).

[exer:6.2.17] 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.

[exer:6.2.18] 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.

[exer:6.2.19] 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.

[exer:6.2.20] 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’s subjected to an external force \(F(t)=8\sin2t-6\cos2t\) N.

[exer:6.2.21] 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.

[exer:6.2.22] 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 \(t\).

[exer:6.2.23] 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 \(t\).

[exer:6.2.24] 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

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

[exer:6.2.25] 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

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

[exer:6.2.26] 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

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