Angle turned by the wheel is given by the kinematics as
[tex]\theta = \omega t + \frac{1}{2}\alpha t^2[/tex]
now here we know that
[tex]\omega = 26 rad/s[/tex]
[tex]\alpha = 25 rad/s^2[/tex]
t = 2.50 seconds
[tex]\theta = 26(2.50) + \frac{1}{2}(25)(2.50)^2[/tex]
[tex]\theta = 143.1 rad[/tex]
So here total angle that the wheel turn is given as
[tex]\theta = 143.1 + 440 = 583.1 rad[/tex]
Speed of the wheel after 2.5 s
[tex]\omega = \omega_0 + \alpha t[/tex]
[tex]\omega = 26 + (25)(2.5) [/tex]
[tex]\omega = 88.5 rad/s[/tex]
now angular deceleration is given as
[tex]\alpha_1 = \frac{\omega^2 - \omega_0^2}{2\theta}[/tex]
[tex]\alpha_1 = \frac{0 - 88.5^2}{2(440)}[/tex]
[tex]\alpha_1 = -8.9 rad/s^2[/tex]
time taken to stop
[tex]t = \frac{\omega - \omega_0}{\alpha}[/tex]
[tex]t = \frac{0 - 88.5}{-8.9} = 9.94 s[/tex]
now total time is given as
[tex]T = 2.50 + 9.94 = 12.44 s[/tex]
