Short Answer

Find the equation for the angular velocity $ω$ in Problem15, assuming that the retarding torque is proportional to role="math" localid="1663966970646" $\sqrt{ω}$

The equation of angular velocity is $(K \sqrt{ω} - T \ln |T - K \sqrt{ω}|) = (K \sqrt{ω_{o}} - T \ln |T - K \sqrt{ω_{o}}|) - \frac{K^{2} t}{2 I}$

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step1: Find the equation for the angular velocity

Here the notations are T= torque for motor, $ω$ = angular velocity, I = moment of inertia and $ω_{0}$ = initial angular velocity.

According To the question retarding torque due to friction is proportional to the angular velocity so, $T_{1} = - K \sqrt{ω}$ (K is proportionality constant)

Now moment of inertia × angular velocity = sum of the torques

$I \frac{d ω}{dt} = T - K \sqrt{ω} \frac{I dω}{T - K \sqrt{ω}} = dt (Variable separating) \frac{- 2 I \sqrt{ω}}{K} - \frac{2 IT \ln |T - K \sqrt{ω}|}{K^{2}} = t + C (Integrating on both sides) \frac{2 I}{K} (\sqrt{ω} - \frac{IT \ln |T - K \sqrt{ω}|}{K}) = t + C (\sqrt{ω} - \frac{IT \ln |T - K \sqrt{ω}|}{K}) = - \frac{Kt}{2 I} + C_{1} (K \sqrt{ω} - T \ln |T - K \sqrt{ω}|) = \frac{- K^{2} t}{2 I} + A$

Step 2: Find the value of A

Put $ω (0) = ω_{0}$ then value of A.

$A = (K \sqrt{ω_{o}} - T \ln |T - K \sqrt{ω_{o}}|) (K \sqrt{ω} - T \ln |T - K \sqrt{ω}|) = (K \sqrt{ω_{o}} - Tln |T - K \sqrt{ω_{o}}|) - \frac{K^{2} t}{2 I}$

Hence, the equation of angular velocity is $(K \sqrt{ω} - T l n |T - K \sqrt{ω}|) = (K \sqrt{ω_{o}} - T l n |T - K \sqrt{ω_{o}}|) - \frac{K^{2} t}{2 I}$ .

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Fundamentals Of Differential Equations And Boundary Value Problems

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Short Answer

Find the equation for the angular velocity $ω$ in Problem15, assuming that the retarding torque is proportional to role="math" localid="1663966970646" $\sqrt{ω}$

Step by Step Solution

Step1: Find the equation for the angular velocity

Step 2: Find the value of A

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Fundamentals Of Differential Equations And Boundary Value Problems

Answers without the blur.

Short Answer

Find the equation for the angular velocity ω in Problem15, assuming that the retarding torque is proportional to role="math" localid="1663966970646" ω

Step by Step Solution

Step1: Find the equation for the angular velocity

Step 2: Find the value of A

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Find the equation for the angular velocity $ω$ in Problem15, assuming that the retarding torque is proportional to role="math" localid="1663966970646" $\sqrt{ω}$