The motion of an oscillating flywheel is defined by the relation0 = 0ge* cos 4nt, where 0 is expressed in radians and t in seconds. Knowingthat 0, = 0.5 rad, determine the angular coordinate, the angular velocity, and theangular acceleration of the flywheel when (a) t = 0, (b) t = 0.125 s.-371

Answered: The motion of an oscillating flywheel…

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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The motion of an oscillating flywheel is defined by the relation
0 = 0ge* cos 4nt, where 0 is expressed in radians and t in seconds. Knowing
that 0, = 0.5 rad, determine the angular coordinate, the angular velocity, and the
angular acceleration of the flywheel when (a) t = 0, (b) t = 0.125 s.
-371

Expert Solution

Step 1

Given data: Motion of flywheel is represented by the relation $θ = θ_{0} e^{- 3 πt} \cos 4 πt$

$θ_{0} = 0.5 rad$

To determine: Angular coordinate at t =0 & t = 0.125 s

Angular velocity at t =0 & t = 0.125 s

Angular acceleration at t =0 & t = 0.125 s

Step 2

As equation of flywheel is given as $θ = θ_{0} e^{- 3 πt} \cos 4 πt$

Case 1: First we will calculate angular coordinate at t = 0 s & t = 0.125 s

$Putting t = 0 in equation of flywheel we get θ = θ_{0} e^{_{- 3 πt}} \cos 4 πt = 0.5 \times e^{_{0}} \cos 4 π \times 0 = 0.5 radians For t = 0.125 s we get θ = θ_{0} e^{_{- 3 πt}} \cos 4 πt = 0.5 \times e^{_{- 3 π \times 0.125}} \cos 4 π \times 0.125 = 0 radians$

Case 2: Now we will calculate the angular velocity at t = 0 s & t = 0.125 s

$Differentiating the equation of flywheel once we get ω = \frac{dθ}{dt} = \frac{d}{dt} (θ_{0} e^{- 3 πt} \cos 4 πt) = 0.5 (- 3 {πe}^{- 3 πt} \cos 4 πt - 4 {πe}^{- 3 πt} \sin 4 πt) For t = 0 we have ω = - 0.5 (3 {πe}^{- 3 πt} \cos 4 πt + 4 {πe}^{- 3 πt} \sin 4 πt) = - 0.5 \times 3 π = - 4.71 rad / s For t = 0.125 s we get ω = - 0.5 (3 {πe}^{- 3 πt} \cos 4 πt + 4 {πe}^{- 3 πt} \sin 4 πt) = - 0.5 \times 4 π \times 0.30786 = - 1.934 rad / s$

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