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9Gustav Kirchhoff (1824-1887), professor at Breslau, Heidelberg, and Berlin, was one of the leading physicists of the nineteenth century. He discovered the basic laws of electric circuits about 1845 while still a student at K?nigsberg. He is also famous for fundamental work in electromagnetic absorption and emission and was one of the founders of spectroscopy.
ldR + RI + 1Q = E (t )
LQ' + RQ + C Q = E(t)
LI" + RI' + CI = E'(t),
with the initial conditions
I(t0) = I0 I (t0) = I0
From Eq. (32) it follows that
E(t0) RI0 (I/ C) Q0
3.8 Mechanical and Electrical Vibrations
? 5. A mass weighing 2 lb stretches a spring 6 in. If the mass is pulled down an additional 3 in.
and then released, and if there is no damping, determine the position u of the mass at any time t. Plot u versus t. Find the frequency, period, and amplitude of the motion.
6. A mass of 100 g stretches a spring 5 cm. If the mass is set in motion from its equilibrium position with a downward velocity of 10 cm/sec, and if there is no damping, determine the position u of the mass at any time t. When does the mass first return to its equilibrium position?
7. A mass weighing 3 lb stretches a spring 3 in. If the mass is pushed upward, contracting the spring a distance of 1 in., and then set in motion with a downward velocity of 2 ft/sec, and if there is no damping, find the position u of the mass at any time t. Determine the frequency, period, amplitude, and phase of the motion.
8. A series circuit has a capacitor of 0.25 x 10-6 farad and an inductor of 1 henry. If the initial charge on the capacitor is 10-6 coulomb and there is no initial current, find the charge Q on the capacitor at any time t.
? 9. A mass of 20 g stretches a spring 5 cm. Suppose that the mass is also attached to a
viscous damper with a damping constant of 400 dyne-sec/cm. If the mass is pulled down an additional 2 cm and then released, find its position u at any time t. Plot u versus t. Determine the quasi frequency and the quasi period. Determine the ratio of the quasi period to the period of the corresponding undamped motion. Also find the time t such that | u(t) | < 0.05 cm for all t > t .
? 10. A mass weighing 16 lb stretches a spring 3 in. The mass is attached to a viscous damper
with a damping constant of 2 lb-sec/ft. If the mass is set in motion from its equilibrium position with a downward velocity of 3 in./sec, find its position u at any time t. Plot u versus t. Determine when the mass first returns to its equilibrium position. Also find the time t such that | u(t) | < 0.01 in. for all t > t .
11. A spring is stretched 10 cm by a force of 3 newtons. A mass of 2 kg is hung from the spring and is also attached to a viscous damper that exerts a force of 3 newtons when the velocity of the mass is 5 m/sec. If the mass is pulled down 5 cm below its equilibrium position and
given an initial downward velocity of 10 cm/sec, determine its position u at any time t.
Find the quasi frequency p and the ratio of p to the natural frequency of the corresponding undamped motion.
12. A series circuit has a capacitor of 10-5 farad, a resistor of 3 x 102 ohms, and an inductor of 0.2 henry. The initial charge on the capacitor is 10-6 coulomb and there is no initial current. Find the charge Q on the capacitor at any time t.
13. A certain vibrating system satisfies the equation u" + y u' + u = 0. Find the value of the damping coefficient y for which the quasi period of the damped motion is 50% greater than the period of the corresponding undamped motion.
14. Show that the period of motion of an undamped vibration of a mass hanging from a vertical spring is 2n L/g, where L is the elongation of the spring due to the mass and g is the acceleration due to gravity.
15. Show that the solution of the initial value problem
mu" + y u + ku = 0, u(t0) = u0, u'(t0) = u'0
can be expressed as the sum u = v + w, where v satisfies the initial conditions v(t0) = u0, v'(t0) = 0, w satisfies the initial conditions w(t0) = 0, w'(t0) = u'0, and both v and w satisfy the same differential equation as u. This is another instance of superposing solutions of simpler problems to obtain the solution of a more general problem.
In each of Problems 1 through 4 determine rnQ, R, and S so as to write the given expression in the form u = Rcos(o)Qt S).
1. u = 3cos2t + 4sin2t 2. u =cos t + \/3sin t
3. u = 4cos3t 2sin3t 4. u= 2cosnt 3 sinnt
Chapter 3. Second Order Linear Equations
16. Show that A cos rn01 + B sinrn0t can be written in the form r sin(ซ0t 9). Determine r and 9 in terms of A and B. If Rcos(ซ0t S) = r sin(ซ0t 9), determine the relationship among R, r, S, and 9.
17. A mass weighing 8 lb stretches a spring 1.5 in. The mass is also attached to a damper with coefficient y . Determine the value of y for which the system is critically damped; be sure to give the units for y.
18. If a series circuit has a capacitor of C = 0.8 x 106 farad and an inductor of L = 0.2 henry, find the resistance R so that the circuit is critically damped.