A wheel initially at rest is subjected to a uniform angular acceleration about its axis. In the first \(2\text{ s}\) it rotates through an angle \(\theta_1\) and in the next \(2\text{ s}\) it rotates through an angle \(\theta_2\). The ratio \(\dfrac{\theta_2}{\theta_1}\) is _______.

Using the equation of rotational kinematics \(\theta = \omega_0 t + \dfrac{1}{2} \alpha t^2\).

Since the wheel is initially at rest, \(\omega_0 = 0\).

Angle rotated in the first \(2\text{ s}\):

\(\theta_1 = \dfrac{1}{2} \alpha (2)^2 = 2\alpha\)

Angle rotated in the first \(4\text{ s}\):

\(\theta_{total} = \dfrac{1}{2} \alpha (4)^2 = 8\alpha\)

Angle rotated in the next \(2\text{ s}\):

\(\theta_2 = \theta_{total} - \theta_1 = 8\alpha - 2\alpha = 6\alpha\)

The ratio is:

\(\dfrac{\theta_2}{\theta_1} = \dfrac{6\alpha}{2\alpha} = 3\)

Answer: \(3\)