An object of mass 3 grams hanging at the bottom of a spring with spring constant 2 grams per second square is moving in a liquid with damping constant 3 grams per second. Denote by y vertical coordinate, positive downwards, and y = 0 is the spring-mass resting position. (a) Write the differential equation y" = f(y, y) satisfied by this system. f(x, y) = Note: Write t for 1, write y for y(t), and yp for y' (t). (b) Find the mechanical energy E of this system. Σ Σ E(1) = Note: Write t for 1, write y for y(1), and yp for y' (t). (c) Find the differential equation E' = F(y) satisfied by the mechanical energy. F(y) = Σ Note: Write t for 1, write y for y(1), write yp for y' (t), write E for E(t).

An object of mass 3 grams hanging at the bottom of a spring with spring constant 2 grams per second square is moving in a liquid with damping constant 3 grams per second. Denote by y vertical coordinate, positive downwards, and y = 0 is the spring-mass resting position. (a) Write the differential equation y" = f(y, y) satisfied by this system. f(x, y) = Note: Write t for 1, write y for y(t), and yp for y' (t). (b) Find the mechanical energy E of this system. Σ Σ E(1) = Note: Write t for 1, write y for y(1), and yp for y' (t). (c) Find the differential equation E' = F(y) satisfied by the mechanical energy. F(y) = Σ Note: Write t for 1, write y for y(1), write yp for y' (t), write E for E(t).

Elements Of Electromagnetics

7th Edition

ISBN:9780190698614

Author:Sadiku, Matthew N. O.

Publisher:Sadiku, Matthew N. O.

ChapterMA: Math Assessment

Section: Chapter Questions

Problem 1.1MA

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Free Damped Vibrations

Free vibration is a type of vibration in which the external force is removed after giving the initial displacement to a body/system. In other words, in this type of vibration, force is applied at the initial displacement of the system, and after the initial displacement, force is removed, and the system vibrates at the natural frequency.

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An object of mass 3 grams hanging at the bottom of a spring with a spring constant 2 grams per second square moving in a liquid with damping constant 3 grams per second. Denote by y vertical coordinate, positive
downwards, and y = 0 is the spring-mass resting position.
(a) Write the differential equation y" = f(y, y) satisfied by this system.
f(x, y) =
Note: Write t for t, write y for y(t), and yp for y' (t).
(b) Find the mechanical energy E of this system.
E(t) =
Note: Write t for 1, write y for y(t), and yp for y' (t).
(c) Find the differential equation E' = F(y) satisfied by the mechanical energy.
F(y) =
Σ
Note: Write t for 1, write y for y(t), write yp for y' (t), write E for E(t).
Σ

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