Problem 1: Recall the two-mass system from a previous problem set, in which you deter- mined the (stable) equilibrium point of the system. (a) Let y be the distance of the bead from the ceiling. From the earlier problem, rewrite U(0) so that the potential energy is a function of this variable y, instead of 0. Also, given the expression for the equilibrium angle, find the equilibrium position yo for this system. (b) Expand U(y) in a Taylor series around Yo to show that for small oscillations, U(y) can be written in the Hooke's Law form ky², and determine the "effective spring constant" k. (c) Write the total kinetic energy T of the system as a function of y. Compare your kinetic and potential energy for this system to the energy expressions in the SHO, and determine this system's angular frequency for small oscillations. M b Ꮎ m

Problem 1: Recall the two-mass system from a previous problem set, in which you deter- mined the (stable) equilibrium point of the system. (a) Let y be the distance of the bead from the ceiling. From the earlier problem, rewrite U(0) so that the potential energy is a function of this variable y, instead of 0. Also, given the expression for the equilibrium angle, find the equilibrium position yo for this system. (b) Expand U(y) in a Taylor series around Yo to show that for small oscillations, U(y) can be written in the Hooke's Law form ky², and determine the "effective spring constant" k. (c) Write the total kinetic energy T of the system as a function of y. Compare your kinetic and potential energy for this system to the energy expressions in the SHO, and determine this system's angular frequency for small oscillations. M b Ꮎ m

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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Concept explainers

Forced undamped vibrations

In mechanical engineering, vibration is expressed as the term that occurs due to an object's backward and forward movement about a point of equilibrium. Generally, there are three types of vibrations which are,

Question

Problem 1: Recall the two-mass system from a previous problem set, in which you deter-
mined the (stable) equilibrium point of the system. (a) Let y be the distance of the bead
from the ceiling. From the earlier problem, rewrite U(0) so that the potential energy is a
function of this variable y, instead of 0. Also, given the expression for the equilibrium angle,
find the equilibrium position yo for this system. (b) Expand U(y) in a Taylor series around
yo to show that for small oscillations, U(y) can be written in the Hooke's Law form ky²,
and determine the "effective spring constant" k. (c) Write the total kinetic energy T of the
system as a function of y. Compare your kinetic and potential energy for this system to
the energy expressions in the SHO, and determine this system's angular frequency for small
oscillations.
M
b
Ꮎ
m

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