4.3 Constant Angular Acceleration

In the case where we have a constant angular acceleration, we can derive a set of equations analogous to the SUVAT equations for linear motion derived in section 1.3. From the equation for average acceleration above, we get

	$\displaystyle\Delta\omega=\omega(t)-\omega_{0}$	$\displaystyle=\alpha t$		(4.44)
	$\displaystyle\omega(t)$	$\displaystyle=\omega_{0}+\alpha t,$		(4.45)

which is analogous to equation 1.14. Then from the definition of angular displacement,

$\displaystyle\Delta\theta$	$\displaystyle=\int_{t_{1}}^{t_{2}}(\omega_{0}+\alpha t)\operatorname{\mathrm{d% }}\!t$	(4.46)
	$\displaystyle=\omega_{0}t+\frac{1}{2}\alpha t^{2}$	(4.47)
$\displaystyle\implies\theta(t)$	$\displaystyle=\theta_{0}+\omega_{0}t+\frac{1}{2}\alpha t^{2}.$	(4.48)

This is like equation 1.17 Finally, squaring equation 4.45 and substituting it into equation 4.48 gives the last equation:

\omega^{2}(t)=\omega_{0}^{2}+2\alpha\theta(t),

(4.49)

which is an angular version of equation 1.18. These equations are useful in solving problems where we don’t have to worry (or don’t care) about motion in the radial direction.