A particle moves in a circle of radius \( 0.5\text{ m} \) with an angular displacement varying as \( \theta(t) = 2t^3 - 3t^2 + 4 \) (where \( \theta \) is in radians and \( t \) is in seconds). At \( t = 2\text{ s} \), what is the magnitude of the total linear acceleration of the particle?

\( \omega = \frac{d\theta}{dt} = 6t^2 - 6t \). At \( t = 2\text{ s} \), \( \omega = 24 - 12 = 12\text{ rad/s} \). \( \alpha = \frac{d\omega}{dt} = 12t - 6 \). At \( t = 2\text{ s} \), \( \alpha = 18\text{ rad/s}^2 \). Tangential acceleration \( a_t = \alpha r = 18(0.5) = 9\text{ m/s}^2 \). Radial acceleration \( a_c = \omega^2 r = (144)(0.5) = 72\text{ m/s}^2 \). Total acceleration \( a = \sqrt{a_t^2 + a_c^2} = \sqrt{9^2 + 72^2} = 9\sqrt{1^2 + 8^2} = 9\sqrt{65}\text{ m/s}^2 \).