Chapter 3: Q3-3.4-15E (page 116)

A rotating flywheel is being turned by a motor that exerts a constant torque T (see Figure 3.10). A retarding torque due to friction is proportional to the angular velocity v. If the moment of inertia of the flywheel, is Iand its initial angular velocity is, find the equation for the angular velocity v as a function of time. [Hint:Use Newton’s second law for rotational motion, that is, moment of inertia * angular acceleration = sum of the torques.]

Short Answer

Expert verified

The equation of angular velocity is $ω (t) = \frac{T}{K} + (ω_{o} - \frac{T}{K}) e^{\frac{- KT}{I}}$ .

Step by step solution

Find the equation for the angular velocity

Here the notations are T= torque for motor, = angular velocity, I = moment of inertia and

$ω_{o}$ = initial angular velocity.

Acc. 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

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

Find the value of C

Put $ω (0) = ω_{0}$ then $C = T - K ω_{o}$

$ω (t) = \frac{T}{K} + (ω_{o} - \frac{T}{K}) e^{\frac{- KT}{I}}$

Hence, the equation of angular velocity is role="math" localid="1663966683149" $ω (t) = \frac{T}{K} + (ω_{o} - \frac{T}{K}) e^{\frac{- K T}{I}}$ .

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Find the equation for the angular velocity

Find the value of C

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