Q: A wheel rotates with an angular acceleration given by α = 4at3 – 3bt2 , where t is the time and a and b are constants. If the wheel Has initial angular speed ω0, write the equations for the
(a) angular speed
(b) angular displacement
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Solution: Since $\large \alpha = \frac{d\omega}{dt}$
$\large \int_{\omega_0}^{\omega}d\omega = \int_{0}^{t} \alpha dt $
$\large \int_{\omega_0}^{\omega}d\omega = \int_{0}^{t} (4a t^3 – 3b t^2) dt $
$\large \omega – \omega_0 = [\frac{4at^4}{4}-\frac{3bt^3}{3}]_{0}^{t}$
$\large \omega = \omega_0 + a t^4 – b t^3 $
(b) Since $\large \omega = \frac{d\theta}{dt}$
$\large \int_{0}^{\theta}d\theta = \int_{0}^{t} \omega dt $
$\large \int_{0}^{\theta}d\theta = \int_{0}^{t} (\omega_0 + a t^4 – b t^3) dt $
$\large \theta = \omega_0 t + \frac{a t^5}{5} – \frac{b t^4}{4}$