The average net torque τ on the wheel during time interval t is
[tex]{\tau= l( \dfrac{\omega_f}{t} )}[/tex].
Given to us:
Initial angular speed of the wheel, [tex]\bold{\omega_i} =0[/tex]
Final angular speed of the wheel, [tex]\bold{\omega_f} =\bold{\omega_f}[/tex]
time interval needed to increase the angular speed of the wheel, t = t
Angular acceleration of the wheel,
[tex]\alpha=\dfrac{\bold{\omega_f} -\bold{\omega_i}}{t},\\\alpha=\dfrac{\bold{\omega_f} -0}{t},\\\alpha=\dfrac{\bold{\omega_f}}{t}[/tex]
According to Newton's Second Law for Rotation:
If more than one torque acts on a rigid body about a fixed axis, then the sum of the torques equals the moment of inertia times the angular acceleration.
[tex]\begin{aligned}\\\sum_{0}^{\bold i}\tau&= \bold{l\alpha}\\\end{aligned}[/tex]
where,
τ is the torque
I is the moment of inertia,
α is the angular acceleration.
Putting the values into the second law, to calculate the average net torque τ on the wheel during this time interval;
[tex]\begin{aligned}\\\sum_{0}^{\bold i}\tau&= \bold{l\alpha}\\\tau&= \bold{l\alpha}\\&= l( \frac{\omega_f}{t} )\end{aligned}[/tex]
Hence, the average net torque τ on the wheel during time interval t is
[tex]{\tau= l( \dfrac{\omega_f}{t} )}[/tex].
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