Page 147 - Schaum's Outline of Differential Equations

130 SECOND-ORDER LINEAR DIFFERENTIAL EQUATIONS [CHAP. 14

14.62. An RCL circuit connected in series with a resistance of 2 ohms, a capacitor of 1/260 farad, and an inductance of
0.1 henry has an applied voltage E(t) = 100 sin (X)t. Assuming no initial current and no initial charge on the capacitor,
find an expression for the charge on the capacitor at any time t.
14.63. Determine the steady-state charge on the capacitor in the circuit described in Problem 14.62 and write it in the form
of Eq. (14.13).
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14.64. An RCL circuit connected in series has R = 5 ohms, C = 10~ farad, L = | henry, and no applied voltage. Find the
subsequent steady-state current in the circuit. Hint. Initial conditions are not needed.
14.65. An RCL circuit connected in series with R = 5 ohms, C = 10~ farad, and L = | henry has applied voltage E(t) = sin t.
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Find the steady-state current in the circuit. Hint Initial conditions are not needed.
14.66. Determine the equilibrium position of a cylinder of radius 3 in, height 20 in, and weight 57rlb that is floating with
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its axis vertical in a deep pool of water of weight density 62.5 lb/ft .
14.67. Find an expression for the motion of the cylinder described in Problem 14.66 if it is disturbed from its equilibrium
position by submerging an additional 2 in of height below the water line and with a velocity of 1 ft/sec in the
downward direction.
14.68. Write the harmonic motion of the cylinder described in Problem 14.67 in the form of Eq. (14.13).

14.69. Determine the equilibrium position of a cylinder of radius 2 ft, height 4 ft, and weight 600 Ib that is floating with
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its axis vertical in a deep pool of water of weight density 62.5 lb/ft .
14.70. Find an expression for the motion of the cylinder described in Problem 14.69 if it is released from rest with 1 ft of
its height submerged in water.

14.71. Determine (a) the circular frequency, (b) the natural frequency, and (c) the period for the vibrations described in
Problem 14.70.

14.72. Determine (a) the circular frequency, (b) the natural frequency, and (c) the period for the vibrations described in
Problem 14.67.

14.73. Determine the equilibrium position of a cylinder of radius 3 cm, height 10 cm, and mass 700 g that is floating with
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its axis vertical in a deep pool of water of mass density 1 g/cm .
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14.74. Solve Problem 14.73 if the liquid is not water but another substance with mass density 2 g/cm .
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14.75. Determine the equilibrium position of a cylinder of radius 30 cm, height 500 cm, and weight 2.5 X 10 dynes that
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is floating with its axis vertical in a deep pool of water of weight density 980 dynes/cm .
14.76. Find an expression for the motion of the cylinder described in Problem 14.75 if it is set in motion from its equilib-
rium position by striking it to produce an initial velocity of 50 cm/sec in the downward direction.
14.77. Find the general solution to Eq. (14.10) and determine its period.

14.78. Determine the radius of a cylinder weighing 5 Ib with its axis vertical that oscillates in a pool of deep water
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(p = 62.5 lb/ft ) with a period of 0.75 sec. Hint: Use the results of Problem 14.77.
14.79. Determine the weight of a cylinder having a diameter of 1 ft with its axis vertical that oscillates in a pool of deep
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water (p = 62.5 lb/ft ) with a period of 2 sec. Hint: Use the results of Problem 14.77.
14.80. A rectangular box of width w, length /, and height h floats in a pool of liquid of weight density p with its height
parallel to the vertical axis. The box is set into motion by displacing it x 0 units from its equilibrium position and
giving it an initial velocity of v 0. Determine the differential equation governing the subsequent motion of the box.

14.81. Determine (a) the period of oscillations for the motion described in Problem 14.80 and (b) the change in that period
if the length of the box is doubled.