Answer
$\alpha=60rad/s^2$
$\omega=90rad/s$
$\theta=90 rad$
Work Step by Step
We can determine the required angular acceleration, angular velocity and the angular displacement as follows:
As $a_t=\alpha r$
We plug in the known values to obtain:
$10t=\alpha \times 0.5$
This simplifies to:
$\alpha=20t$
At $t=3s$ $\alpha=20(3)=60rad/s^2$
The angular velocity can be calculated as
$d\omega=\alpha dt$
$\implies \int_0^\omega d\omega=\int_0^ t 20t dt$
This simplifies to:
$\omega=10t^2$
We plug in the known values to obtain:
$\omega=10(3)^2=90rad/s$
Now we calculate the angular displacement as
$d\theta=\omega dt$
$\int_0 ^\theta d\theta=\int_0^t 10t^2 dt$
$\theta=(\frac{10}{3})t^3$
At $t=3s$
$\implies \theta=(\frac{10}{3})(3)^3$
$\implies \theta=90 rad$
