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Spring, Mass, and Dashpot Exercises


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


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