Chapter 3: Q3-3.4-16E (page 116)

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

Short Answer

Expert verified

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

Step by step solution

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

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

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

Short Answer

Step by step solution

Find the equation for the angular velocity

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" ω

Short Answer

Step by step solution

Find the equation for the angular velocity

Find the value of A

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Applied Mathematics

Calculus

Logic and Functions

Geometry

Statistics

Mechanics Maths

Study anywhere. Anytime. Across all devices.

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