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17.3E: Exercises for Section 17.3

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    72451
    • Gilbert Strang & Edwin “Jed” Herman
    • OpenStax

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    1. A mass weighing 4 lb stretches a spring 8 in. Find the equation of motion if the spring is released from the equilibrium position with a downward velocity of 12 ft/sec. What is the period and frequency of the motion?

    2. A mass weighing 2 lb stretches a spring 2 ft. Find the equation of motion if the spring is released from 2 in. below the equilibrium position with an upward velocity of 8 ft/sec. What is the period and frequency of the motion?

    Answer
    differential equation: \(x″+16x=0,\)
    equation of motion: \(x(t)=\frac{1}{6} \cos (4t)−2 \sin (4t),\)
    period \(=\,\frac{π}{2} \text{ sec},\)
    frequency \(=\,\frac{2}{π} \text{ Hz}\)

    3. A 100-g mass stretches a spring 0.1 m. Find the equation of motion of the mass if it is released from rest from a position 20 cm below the equilibrium position. What is the frequency of this motion?

    4. A 400-g mass stretches a spring 5 cm. Find the equation of motion of the mass if it is released from rest from a position 15 cm below the equilibrium position. What is the frequency of this motion?

    Answer
    differential equation: \(x″+196x=0,\)
    equation of motion: \(x(t)=0.15 \cos (14t),\)
    period \(=\,\frac{π}{7} \text{ sec},\)
    frequency \(=\,\frac{7}{π} \text{ Hz}\)

    5. A block has a mass of 9 kg and is attached to a vertical spring with a spring constant of 0.25 N/m. The block is stretched 0.75 m below its equilibrium position and released.

    1. Find the position function \(x(t)\) of the block.
    2. Find the period and frequency of the vibration.
    3. Sketch a graph of \(x(t)\).
    4. At what time does the block first pass through the equilibrium position?

    6. A block has a mass of 5 kg and is attached to a vertical spring with a spring constant of 20 N/m. The block is released from the equilibrium position with a downward velocity of 10 m/sec.

    1. Find the position function \(x(t)\) of the block.
    2. Find the period and frequency of the vibration.
    3. Sketch a graph of \(x(t)\).
    4. At what time does the block first pass through the equilibrium position?
    Answer

    a. \(x(t)=5 \sin (2t)\)
    b. period \(=π \text{ sec},\)
    frequency \(=\frac{1}{π} \text{ Hz}\)
    c.
    This figure is the graph of a function. It is a periodic function with consistent amplitude. The horizontal axis is labeled in increments of 1. The vertical axis is labeled in increments of 1.5.
    d. \(t=\frac{π}{2} \text{ sec}\)

    7. A 1-kg mass is attached to a vertical spring with a spring constant of 21 N/m. The resistance in the spring-mass system is equal to 10 times the instantaneous velocity of the mass.

    1. Find the equation of motion if the mass is released from a position 2 m below its equilibrium position with a downward velocity of 2 m/sec.
    2. Graph the solution and determine whether the motion is overdamped, critically damped, or underdamped.

    8. An 800-lb weight (25 slugs) is attached to a vertical spring with a spring constant of 226 lb/ft. The system is immersed in a medium that imparts a damping force equal to 10 times the instantaneous velocity of the mass.

    1. Find the equation of motion if it is released from a position 20 ft below its equilibrium position with a downward velocity of 41 ft/sec.
    2. Graph the solution and determine whether the motion is overdamped, critically damped, or underdamped.
    Answer

    a. \(x(t)=e^{−t/5}(20 \cos (3t)+15 \sin(3t))\)

    b. underdamped

    9. A 9-kg mass is attached to a vertical spring with a spring constant of 16 N/m. The system is immersed in a medium that imparts a damping force equal to 24 times the instantaneous velocity of the mass.

    1. Find the equation of motion if it is released from its equilibrium position with an upward velocity of 4 m/sec.
    2. Graph the solution and determine whether the motion is overdamped, critically damped, or underdamped.

    10. A 1-kg mass stretches a spring 6.25 cm. The resistance in the spring-mass system is equal to eight times the instantaneous velocity of the mass.

    1. Find the equation of motion if the mass is released from a position 5 m below its equilibrium position with an upward velocity of 10 m/sec.
    2. Determine whether the motion is overdamped, critically damped, or underdamped.
    Answer

    a. \(x(t)=5e^{−4t}+10te^{−4t}\)

    b. critically damped

    11. A 32-lb weight (1 slug) stretches a vertical spring 4 in. The resistance in the spring-mass system is equal to four times the instantaneous velocity of the mass.

    1. Find the equation of motion if it is released from its equilibrium position with a downward velocity of 12 ft/sec.
    2. Determine whether the motion is overdamped, critically damped, or underdamped.

    12. A 64-lb weight is attached to a vertical spring with a spring constant of 4.625 lb/ft. The resistance in the spring-mass system is equal to the instantaneous velocity. The weight is set in motion from a position 1 ft below its equilibrium position with an upward velocity of 2 ft/sec. Is the mass above or below the equation position at the end of \(π\) sec? By what distance?

    Answer
    \(x(π)=\frac{7e^{−π/4}}{6}\) ft below

    13. A mass that weighs 8 lb stretches a spring 6 inches. The system is acted on by an external force of \(8 \sin 8t \)lb. If the mass is pulled down 3 inches and then released, determine the position of the mass at any time.

    14. A mass that weighs 6 lb stretches a spring 3 in. The system is acted on by an external force of \(8 \sin (4t) \) lb. If the mass is pulled down 1 inch and then released, determine the position of the mass at any time.

    Answer
    \(x(t)=\frac{32}{9} \sin (4t)+ \cos (\sqrt{128}t)−\frac{16}{9\sqrt{2}} \sin (\sqrt{128}t)\)

    15. Find the charge on the capacitor in an RLC series circuit where \(L=40\) H, \(R=30\,Ω\), \(C=1/200\) F, and \(E(t)=200\) V. Assume the initial charge on the capacitor is 7 C and the initial current is 0 A.

    16. Find the charge on the capacitor in an RLC series circuit where \(L=2\) H, \(R=24\,Ω,\) \(C=0.005\) F, and \(E(t)=12 \sin 10t\) V. Assume the initial charge on the capacitor is 0.001 C and the initial current is 0 A.

    Answer
    \(q(t)=e^{−6t}(0.051 \cos (8t)+0.03825 \sin (8t))−\frac{1}{20} \cos (10t)\)

    17. A series circuit consists of a device where\(L=1\) H, \(R=20\,Ω,\) \(C=0.002\) F, and \(E(t)=12\) V. If the initial charge and current are both zero, find the charge and current at time \(t.\)

    18. A series circuit consists of a device where \(L=12\) H, \(R=10\,Ω\), \(C=\frac{1}{50}\) F, and \(E(t)=250\) V. If the initial charge on the capacitor is 0 C and the initial current is 18 A, find the charge and current at time \(t.\)

    Answer
    \(q(t)=e^{−10t}(−32t−5)+5,I(t)=2e^{−10t}(160t+9)\)

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