A wheel rotates with an angular acceleration given by α = 4at^3 – 3bt^2 , where t is the time and a and b are constants…

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.

Sol: 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 $

$\large \theta = \omega_0 t + \frac{a t^5}{5} – \frac{b t^4}{4}$