17.3E: Exercises for Section 17.3
- Page ID
- 72451
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.
- Find the position function \(x(t)\) of the block.
- Find the period and frequency of the vibration.
- Sketch a graph of \(x(t)\).
- 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.
- Find the position function \(x(t)\) of the block.
- Find the period and frequency of the vibration.
- Sketch a graph of \(x(t)\).
- At what time does the block first pass through the equilibrium position?
- Answer
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a. \(x(t)=5 \sin (2t)\)
b. period \(=π \text{ sec},\)
frequency \(=\frac{1}{π} \text{ Hz}\)
c.
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.
- 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.
- 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.
- 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.
- 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.
- Find the equation of motion if it is released from its equilibrium position with an upward velocity of 4 m/sec.
- 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.
- 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.
- 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.
- Find the equation of motion if it is released from its equilibrium position with a downward velocity of 12 ft/sec.
- 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
- \(q(t)=e^{−10t}(−32t−5)+5,I(t)=2e^{−10t}(160t+9)\)